Representations, Wavelets and Frames: A Celebration of the Mathematical Work of Lawrence W. Baggett

Applied and Numerical Harmonic Analysis Series Editor John J. Benedetto University of Maryland Editorial Advisory Board...

Author: Palle E. T. Jorgensen | Kathy D. Merrill | Judith A. Packer

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Applied and Numerical Harmonic Analysis Series Editor John J. Benedetto University of Maryland Editorial Advisory Board Akram Aldroubi Vanderbilt University

Douglas Cochran Arizona State University

Ingrid Daubechies Princeton University

Hans G. Feichtinger University of Vienna

Christopher Heil Georgia Institute of Technology

Murat Kunt Swiss Federal Institute of Technology, Lausanne

James McClellan Georgia Institute of Technology

Wim Sweldens Lucent Technologies, Bell Laboratories

Michael Unser Swiss Federal Institute of Technology, Lausanne

Martin Vetterli Swiss Federal Institute of Technology, Lausanne

M. Victor Wickerhauser Washington University

Representations, Wavelets, and Frames A Celebration of the Mathematical Work of Lawrence W. Baggett

Palle E.T. Jorgensen Kathy D. Merrill Judith A. Packer Editors

Birkhäuser Boston • Basel • Berlin

Palle E.T. Jorgensen Department of Mathematics The University of Iowa Iowa City, IA 52242-1419 USA [email protected]

Kathy D. Merrill Department of Mathematics Colorado College Colorado Springs, CO 80903-3294 USA [email protected]

Judith A. Packer Department of Mathematics University of Colorado at Boulder Boulder, CO 80309-0395 USA [email protected]

ISBN: 978-0-8176-4682-0 DOI: 10.1007/978-0-8176-4683-7

e-ISBN: 978-0-8176-4683-7

Library of Congress Control Number: 2008927259 Mathematics Subject Classiﬁcation (2000): 22D10, 22E45, 28A80, 32A70, 34A45, 35M99, 37B05, 41A58, 42B35, 42C10, 42C40, 42C99, 45L05, 44A05, 43A65, 54H10, 46E22, 46C07, 46A20, 46A35, 46L60, 46E35, 47L10, 47L40, 47A25, 47A20, 58D25, 60B15, 60G35, 65T50, 65T60, 68U10, 68W25, 81Q10, 92C55, 93A13, 94A08, 94A12, 94A24 ©2008 Birkhäuser Boston All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Birkhäuser Boston, c/o Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identiﬁed as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Printed on acid-free paper. 9 8 7 6 5 4 3 2 1 www.birkhauser.com

Larry Baggett (Photograph by James Jesudason)

To Larry

Mathematical relatives, colleagues, and descendants of Larry Baggett at the May 18–20, 2006, conference “Current Trends in Harmonic Analysis and Its Applications: Wavelets and Frames” (University of Colorado, Boulder) held in Larry’s honor. Top row, left to right: Alexander Powell, Jonas D’Andrea, Gail Ratcliﬀ, Dorin Dutkay, Eberhard Kaniuth (partially blocked), Kuzman Adzievski, Kevin Manley, Christian Roldan-Santos, John Benedetto, Darrin Speegle, Chris Brislawn, Brody Johnson, Keri Kornelson, Ernesto Acosta, Eric Weber, Shannon Bishop, Jens Christensen, Kenneth Joy, Chal Benson, Casey Leonetti, Jeﬀ Hogan, Kasso Okoudjou, Chris Heil, Menaissie Ephrem. Middle row, left to right: Keith Taylor, S. Zubin Gautam, Vera Furst (partially blocked), Ying Wang, Constantin Pirvulescu, Herbert Medina, Myung-Sin Song, Larry Baggett, En-Bing Lin, Dave Larson, Bin Han, Xingde Dai, Palle Jorgensen. Kneeling, left to right: Marcin Bownik, Demetrio Labate, Judy Packer, Kathy Merrill, Karla Oty, Wannapa Ruangthanakorn, Fumiko Futamura, April Hong Yu, Le Gui. (Photo taken by Marysia Mycielski using the camera of Ying Wang.)

ANHA Series Preface

The Applied and Numerical Harmonic Analysis (ANHA) book series aims to provide the engineering, mathematical, and scientiﬁc communities with signiﬁcant developments in harmonic analysis, ranging from abstract harmonic analysis to basic applications. The title of the series reﬂects the importance of applications and numerical implementation, but richness and relevance of applications and implementation depend fundamentally on the structure and depth of theoretical underpinnings. Thus, from our point of view, the interleaving of theory and applications and their creative symbiotic evolution is axiomatic. Harmonic analysis is a wellspring of ideas and applicability that has ﬂourished, developed, and deepened over time within many disciplines and by means of creative cross-fertilization with diverse areas. The intricate and fundamental relationship between harmonic analysis and ﬁelds such as signal processing, partial diﬀerential equations (PDEs), and image processing is reﬂected in our state-of-the-art ANHA series. Our vision of modern harmonic analysis includes mathematical areas such as wavelet theory, Banach algebras, classical Fourier analysis, time-frequency analysis, and fractal geometry, as well as the diverse topics that impinge on them. For example, wavelet theory can be considered an appropriate tool to deal with some basic problems in digital signal processing, speech and image processing, geophysics, pattern recognition, biomedical engineering, and turbulence. These areas implement the latest technology from sampling methods on surfaces to fast algorithms and computer vision methods. The underlying mathematics of wavelet theory depends not only on classical Fourier analysis, but also on ideas from abstract harmonic analysis, including von Neumann algebras and the aﬃne group. This leads to a study of the Heisenberg group and its relationship to Gabor systems, and of the metaplectic group for a meaningful interaction of signal decomposition methods. The unifying inﬂuence of wavelet theory in the aforementioned topics illustrates the justiﬁcation for providing a means for centralizing and disseminating information from the

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ANHA Series Preface

broader, but still focused, area of harmonic analysis. This will be a key role of ANHA. We intend to publish with the scope and interaction that such a host of issues demands. Along with our commitment to publish mathematically signiﬁcant works at the frontiers of harmonic analysis, we have a comparably strong commitment to publish major advances in the following applicable topics in which harmonic analysis plays a substantial role: Antenna theory Biomedical signal processing Digital signal processing Fast algorithms Gabor theory and applications Image processing Numerical partial diﬀerential equations

Prediction theory Radar applications Sampling theory Spectral estimation Speech processing Time-frequency and time-scale analysis Wavelet theory

The above point of view for the ANHA book series is inspired by the history of Fourier analysis itself, whose tentacles reach into so many ﬁelds. In the last two centuries Fourier analysis has had a major impact on the development of mathematics, on the understanding of many engineering and scientiﬁc phenomena, and on the solution of some of the most important problems in mathematics and the sciences. Historically, Fourier series were developed in the analysis of some of the classical PDEs of mathematical physics; these series were used to solve such equations. In order to understand Fourier series and the kinds of solutions they could represent, some of the most basic notions of analysis were deﬁned, e.g., the concept of “function.” Since the coeﬃcients of Fourier series are integrals, it is no surprise that Riemann integrals were conceived to deal with uniqueness properties of trigonometric series. Cantor’s set theory was also developed because of such uniqueness questions. A basic problem in Fourier analysis is to show how complicated phenomena, such as sound waves, can be described in terms of elementary harmonics. There are two aspects of this problem: ﬁrst, to ﬁnd, or even deﬁne properly, the harmonics or spectrum of a given phenomenon, e.g., the spectroscopy problem in optics; second, to determine which phenomena can be constructed from given classes of harmonics, as done, for example, by the mechanical synthesizers in tidal analysis. Fourier analysis is also the natural setting for many other problems in engineering, mathematics, and the sciences. For example, Wiener’s Tauberian theorem in Fourier analysis not only characterizes the behavior of the prime numbers, but also provides the proper notion of spectrum for phenomena such as white light; this latter process leads to the Fourier analysis associated with correlation functions in ﬁltering and prediction problems, and these problems, in turn, deal naturally with Hardy spaces in the theory of complex variables. Nowadays, some of the theory of PDEs has given way to the study of Fourier integral operators. Problems in antenna theory are studied in terms

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of unimodular trigonometric polynomials. Applications of Fourier analysis abound in signal processing, whether with the fast Fourier transform (FFT), or ﬁlter design, or the adaptive modeling inherent in time-frequency-scale methods such as wavelet theory. The coherent states of mathematical physics are translated and modulated Fourier transforms, and these are used, in conjunction with the uncertainty principle, for dealing with signal reconstruction in communications theory. We are back to the raison d’ˆetre of the ANHA series!

John J. Benedetto Series Editor University of Maryland College Park

Foreword

A group of friends and admirers of Lawrence W. Baggett (Larry) gathered in Boulder, May 18–20, 2006, for a C’estLarrybration, celebrating a special time in the life of Larry, namely his (semi-)retirement, and reﬂecting on his contributions to mathematics. A large part of the celebration consisted of talks on a number of subjects, in keeping with Larry’s broad interests, and illustrating the fact that he is a valued colleague and friend of many mathematicians. The speakers were distinguished mathematicians, and the talks were thought provoking. Among those speakers were four of Larry’s students: Kathy Merrill, Eric Weber, Keri Kornelson, and Veronica Furst. The organizers of the conference realized that the talks could be turned into papers that would be appreciated by a large-enough audience to warrant the publication of a volume in honor of Larry. In this way, the C’estLarrybration can be enjoyed by many others. Those of us fortunate enough to have attended the conference will also ﬁnd interesting reading here. This volume is in recognition of the many and varied contributions made by Larry to the advancement of mathematics and the health of the mathematics community. His calm and friendly manner enhance the atmosphere and functioning of the mathematics department in Boulder, as they do his interactions outside the department. His own creativity and his encouragement of that of others are widely known and greatly appreciated. The high quality of the speakers and their talks at the conference bear witness to the deservedly high regard in which Larry is held. Publications on which Larry was sole or co-author give some indication of the vigor and inventiveness of his thinking. His early papers were about topology in duals of locally compact groups. His expertise of course included the subject of induced representations, and it is natural that one of his students, Kenneth Joy, would write a thesis on the topology of duals of nilpotent groups. Larry’s interests took a tour through the subject of cocycles of ergodic transformations, perhaps, in part, because of his student Kathy Merrill, whose thesis was about cocycles for irrational rotations on the circle. In recent years, Larry has explored various aspects of wavelet theory, and his papers

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and those of his students reﬂect his interests, which extend the frontiers and probe hidden corners of the subject. It is great fun to see how he induces his students to master the topic of spectral multiplicity and put it to use. What is known only to those who have had the privilege of collaboration or discussion with him is the extent to which his ideas and attitude pervade these papers and contribute to the work of others. I mention only the insight he gave me into the subtle topic of ﬁnding the Mackey extensions that occur in using the Mackey Machine, and our joint paper on selection lemmas with proofs based on his clever idea to use extensions of linear functionals as in the Hahn–Banach Theorem. I am happy to be one of Larry’s co-authors and can testify to the fact that he makes a great collaborator. His lively sense of humor does a lot to lighten the load when the going gets hard, and adds to the pleasure all the time. One of our joint papers was primarily about a nonseparable locally compact group that we regarded as proof that direct integral decompositions of unitary representations should be avoided completely in the nonseparable case. As submitted, we claimed to have found “the smoking gun.” This was Larry’s suggestion, and I considered it a positive contribution to the value of the paper. Sadly enough, the referee did not approve of such levity. Certainly his other collaborators will agree that working with Larry is both fun and proﬁtable. One of the reasons for this is that he is generous with his ideas all the time, and he thoroughly enjoys the kind of interaction with fellow mathematicians that is necessary for collaboration or for helping another mathematician with a project. This is also the way he deals with his Ph.D. students. He is exceptionally generous with his time and attention, in addition to having a wealth of mathematical ideas. It is only natural that his students think of him as the ideal advisor. He gave them a great example of how to be a mathematician. On a personal note, one of my main reasons for joining the faculty at CU Boulder was my awareness that a mathematical cousin was already in Boulder. That choice paid oﬀ to a degree that would have been diﬃcult to predict. Certainly, few would have anticipated the way Larry has kept harmonic analysis and its interactions alive and well in Boulder. Because of Larry, we have had numerous visitors who brought their own contributions to progress. Because of Larry, himself, the department has maintained a healthy level of activity in the area and a lively pursuit of new ideas and connections for harmonic analysis. Find here for your enjoyment and stimulation some mathematics that hints at the value added to our discipline by Larry. And do stay tuned for future developments as Larry continues to explore. Boulder, Colorado, September 2007

Arlan B. Ramsay

Preface

Lawrence Wasson Baggett was born in Moorehead, Mississippi, in 1939 and lived with his family in the South during his early years. After a childhood accident left him blind at the age of ﬁve, his mother, Katherine Wasson Baggett, and father, Lawrence Witherspoon Baggett, refused to let the loss of his sight discourage him or the rest of his family about his future. Instead, in 1945 they moved with Larry up to Boston, Massachusetts, where Larry’s father worked as an electrical scientist for Raytheon and where Larry was enrolled in the celebrated Perkins School, located in Watertown, Massachusetts. Perkins School, established in 1829, is well-known as the alma mater of Helen Keller. Larry thrived as a student there, both in his studies and at the piano. Larry had a younger sister, Linda, and after she was born, the Baggett family moved in 1948 to the Orlando, Florida, area, where Larry’s father worked for the U.S. Navy. Larry credits his mother with persevering to persuade the school district in Gotha, Florida, to allow him to attend the regular public schools there; at the time it was highly unusual for students with physical challenges to be “mainstreamed.” Larry also credits his grade school teacher in Gotha, Helen Watson, for her contribution to his education; “I often think that everything I know, I learned in fourth grade from Mrs. Watson,” he remarked recently. Larry attended Davidson College in North Carolina, where he did so well in his mathematical studies that he decided to enroll in mathematics graduate school after receiving his B.A. He subsequently began his graduate studies under the guidance of J.M.G. Fell at the University of Washington in Seattle, receiving his Ph.D. in 1966. Immediately upon graduating, he joined the Department of Mathematics of the University of Colorado at Boulder. In 1979, Larry married Christy Sweet, a scientiﬁc editor at NOAA. They have a lovely family consisting of four daughters and three grandchildren. Larry notes that Christy has played a crucial role in making his professional years productive and happy. Anyone who has come to a conference or workshop in Boulder, and has been to one of the wonderful parties at the Baggetts’ home on Cedar Avenue, organized as if by magic, knows that she has made all

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of our professional years more productive and happy as well. She has become a friend to all of us as we have worked with Larry. The rest, as they say, is history. This volume continues a celebration of Larry Baggett’s mathematical work that was begun with our C’estLarrybration, a three-day workshop held in Larry’s honor at the CU campus in Boulder in May 2006. The editors and authors of the chapters in this volume were inspired by this occasion to write up work of their own that was stimulated by Larry’s growing mathematical legacy. Thus, this volume took shape, and like the mathematics of Larry through the years, it is broad and innovative, and also contains deep and technical results from our many generous contributors. As a fortunate sideeﬀect of soliciting works related to Larry’s career, we have been able to carry out two separate but related aims in this volume. The chapters here present trends and new results in a wide range of areas. Yet, at that same time, they stress the many interactions that have been such an important theme in Larry’s mathematical work. Larry’s early work done under the inﬂuence of Fell was on harmonic analysis and group representations, so it is appropriate that the ﬁrst section of this volume contains papers from this area. The ﬁrst article of this section is a paper by Victor Guillemin and Daniel Stroock entitled “Some Riemann sums are better than others,” containing a very intriguing result on the approximation of integrals by means of Fourier series. The theory of Fourier series and its applications to physics, probability and number theory can be viewed as part of the origins of harmonic analysis as we know it today. The second paper in this initial section, “Gelfand pairs associated with ﬁnite Heisenberg groups,” by Chal Benson and Gail Ratcliﬀ, deals with the representation theory of ﬁnite Heisenberg groups using the theory of Gelfand pairs. The theory of Gelfand pairs and the Kirillov theory in general have played a large role in some of the most recent developments in the theory of representations of Lie groups (cf. [29, 42]). As mentioned above, Larry’s keen eye for connections has led to many of his most notable contributions; most recently he has brought tools of abstract harmonic analysis to bear on the study of wavelets, yielding startling new insights even in the classical setting of Rn [15, 20, 21, 22]. The remaining papers in Section I reﬂect this achievement in tying the theory of wavelets together with that of representations of non-Abelian groups. Keith Taylor, who worked as a postdoc with Larry in the mid to late 1970s, contributed “Groups with atomic regular representation,” which includes a summary of some of Larry’s early work on group representations [1, 4] as well as some of Keith’s joint work with Larry [2, 3]. After giving an overview of the theory of group representations, including, in particular, regular representations and square-integrable representations, Taylor focuses on those locally compact groups that are “[AR]” groups, that is, those groups whose left regular representations can be decomposed as direct sums of irreducible representations. All compact groups and the two-dimensional aﬃne group are [AR],

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and Larry proved in [1] that any connected unimodular [AR] group must be compact. Baggett and Taylor gave certain conditions under which semidirect products were [AR]. They also showed that the Fourier algebra A(G) of a locally compact group G is of interest. Taylor discusses the problem of constructing projections in L1 (G) for G [AR], and, using the wavelet transform constructed from the aﬃne group as a motivator, constructs a general family of wavelet transforms of L2 (Rn ) using certain semidirect products of [AR] groups. Eric Weber, Larry’s eighth Ph.D. student, wrote the next article, entitled “Wavelet transforms and admissible group representations.” In Weber’s paper, the discrete wavelet transform is discussed in terms of sampling the continuous wavelet transform, which comes from an admissible representation of a locally compact group. This notion of Weber generalizes the concept of square-integrable group representation. Weber, mirroring Taylor’s work, characterizes those groups with an aﬃne structure admitting a general discrete wavelet transform and then relates this concept to various discussions of multiresolution analysis. The classical multiresolution analysis (MRA) construction due to S. Mallat and Y. Meyer [47, 48, 49] gave a standard way of building orthonormal wavelets, although it was known at the time that not every orthonormal wavelet is associated with an MRA. One of the key concepts used to generalize the classical MRA work is the essential notion of frames, which is also an important theme in this volume. Frames are a generalization of orthonormal basis and thus closely related to the problem of ﬁnding a good basis for a particular signal or function, usually in a Hilbert space. The subject of frames dates back to the work of R. Duﬃn and A. Schaeﬀer [38] in the 1950s; frames more closely related to wavelets, including “Weyl–Heisenberg” or “Gabor” frames as well as dilation/translation frames, were studied by I. Daubechies, A. Grossman, and Y. Meyer [32]. Whereas the construction of Mallat and Meyer was too specialized to allow for frames, in 1989 W. Lawton [46] used ﬁnite-impulseresponse (FIR) ﬁlters to give a method for building tight normalized frames (or “Parseval frames,” as they are now known, thanks to Larry) that did not use MRAs. Univariate wavelet frames for dilation by 2 had implicitly been studied by M. Frazier, G. Garrig´ os, K. Wang, and G. Weiss [40]. J. Benedetto and S. Li [28] explicitly used frames to generalize MRAs, in work that appeared in 1998. In this paper, they developed the notion of “frame multiresolution analyses,” known as “FMRAs.” A basic result of their theory was a characterization of frames of integer translates of a function φ in terms of the study of a computable periodization of the Fourier transform of φ. Closely related to this work on frames and FMRAs is the notion of “generalized multiresolution analysis” (abbreviated “GMRA”), a concept initially due to Larry Baggett, Kathy Merrill, and Herbert Medina, which ﬁrst appeared in [20]. Given dilation and translation operators acting on a Hilbert space such as L2 (Rn ), they deﬁne a generalized multiresolution analysis or GMRA to be a nested sequence of closed subspaces {Vj }j∈Z with the standard properties of multiresolution analyses except that the so-called core subspace

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V0 need not have a scaling function, but need only be invariant under the translation operators. It was here that Larry’s use of techniques from abstract harmonic analysis yielded important innovations. The representation of the group of translation operators (Zn in the standard case) on the core subspace V0 determined what Baggett, Medina, and Merrill called the multin = Tn → N ∪ {0} of the GMRA, which Weber later plicity function m : Z showed to be equivalent to the “wavelet dimension” function of Weiss et al. Jennifer Courter (Larry’s seventh Ph.D. student), Baggett, and Merrill were able to use the GMRA concept in [22] to develop the notion of generalized ﬁlters for GMRAs that allowed for the construction of a wide class of frames in L2 (Rn ). The articles in Section II all tap into this rich vein of work on frames and GMRAs in one way or another. Christopher Heil, a collaborator on an NSF FRG grant with Larry, is the author of “The density theorem and the homogeneous approximation property for Gabor frames.” This paper begins with Larry’s proof that a rectangular lattice Gabor system {e2πiβnt g(t − αk)}n,k∈Z must be incomplete in L2 (R) whenever αβ > 1 (found in [15]) and leads into a discussion of the work of Ramanathan and Steger [51] and the general homogeneous approximation property concerning density for irregular Gabor frames in higher dimensions. Bin Han’s paper “Recent developments on dual wavelet frames,” applies the oblique extension principle (OEP) to the construction of pairs of dual wavelet frames in Sobelev spaces, after giving an overview of this principle as ﬁrst developed by I. Daubechies, A. Ron, Z. Shen, and himself in [33]. Wavelet frames in such spaces are very useful for signal and image processing. The OEP is a generalization of the unitary extension principle (UEP) for wavelet frames ﬁrst developed by A. Ron and Z. Shen [52, 53]. The article by Veronika Furst, Larry’s twelfth Ph.D. student, “Characteristic wavelet equations and generalizations of the spectral function,” further develops work done in her Ph.D. thesis completed under the direction of Larry Baggett in 2006. In her paper in this volume, Furst uses the spectral function of M. Bownik and Z. Rzeszotnik [30] to generalize two equations due to Gripenberg and Wang [41, 54], which characterize orthonormal wavelets, to the case of GMRAs in abstract Hilbert spaces. This section concludes with an article by Marcin Bownik of the University of Oregon entitled “Baggett’s problem for frame wavelets.” In this paper, Bownik discusses the problem, ﬁrst posed by Larry Baggett in [26], of whether or not every Parseval wavelet is associated with a GMRA in the sense of [20]. Bownik, improving a result of Rzeszotnik and himself [31], shows that it is possible to construct a Parseval wavelet ψ in L2 (R) for dilation by 2, with arbitrary degrees of (smoothness, etc.) such that the intersection of the respective family of negative dilates of translates of ψ is equal to all of L2 (R), thus answering in the negative the question posed by Baggett. Section III contains papers related to the concept of a wavelet set, introduced in 1994 by David Larson and Xingde Dai in their study of wavelets from the point of view of operator algebras [35]. These sets have the property

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that (a scalar multiple of) the inverse Fourier transform of their characteristic functions are wavelets in L2 (Rn ) with respect to a given expansive dilation matrix. Wavelets arising from wavelet sets are sometimes called MSF wavelets (for “minimally supported frequency wavelets”), as up to multiples by certain unitary functions, these are precisely the wavelets whose Fourier transforms have support with minimal measure [39]. Wavelets corresponding to a wavelet set do not need to come from a classical multiresolution analysis in the sense of S. Mallat and Y. Meyer; the Journ´e wavelet is a prime example of a wavelet associated with a wavelet set yet not coming from a multiresolution analysis. Moreover, if the dilation matrix has a determinant whose absolute value is greater than 2, the wavelets so generated cannot come from multiresolution analyses. In spite of this, Dai, Larson, and Darrin Speegle [36] showed in 1997 that wavelet sets exist for any integer dilation matrix, and they gave several examples in R2 , including the famous “wedding cake set” [37]. After their pioneering work, many others took up the study of wavelet sets, including Larry Baggett, Herbert Medina, and Kathy Merrill in [20], who came up with a procedure that in theory allowed one to construct all wavelet sets. About the same time, another construction technique was discovered independently by J. Benedetto and M. Leon [27]. All of the two-dimensional examples for dilation by 2I constructed by these authors have a self-similar or fractal-like appearance, as their construction techniques are iterative in nature. It was widely conjectured that a wavelet set for dilation by 2I in R2 without this fractal-like appearance was impossible to construct. In the article “Simple wavelet sets for scalar dilations in R2 ,” Kathy Merrill, Larry’s third Ph.D. student and long-time collaborator, disproves this conjecture with a construction of wavelet sets in R2 for dilation by dI, where d is any scalar greater than 1, that are ﬁnite unions of convex polygons. One of the original motivations behind Dai and Larson’s invention of the concepts of wavelet sets was the study of connectedness properties of wavelets, and to this end, Dai and Larson in [35] introduced a method of interpolating between two diﬀerent wavelets by using a unitary operator called the “interpolation unitary.” In the case of MSF wavelets, the interpolation unitary gives rise to a measure-preserving map called the interpolation map. If, in addition, the unitary operator interpolating between the two wavelets is involutive, the two wavelets are called an “interpolation pair.” David Larson, who also collaborated with Larry on his NSF FRG grant, and his co-author Xiaofei Zhang have contributed an article to this section entitled “Interpolation maps and congruence domains for wavelet sets” that continues along this vein of study. They show that if an interpolation map between two wavelet sets preserves the union of the sets, the corresponding wavelets must be an interpolation pair, thus solving a ten-year-old conjecture of Larson. They also answer in the negative a question of D. Han by showing by means of a counterexample that the equality of the congruence domains of an interpolation map and its inverse does not imply that the associated pair of wavelet sets is an interpolation pair.

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One of Larry’s greatest strengths as a mathematician is his ability to bring together techniques from a wide variety of seemingly disparate ﬁelds. Two areas that have explicitly informed his approach are dynamical systems, through his work on cocycles of an irrational rotation [5, 6, 7, 8, 9, 11, 13, 16, 18], and C ∗ -algebras, as evidenced by his collaborations with his colleague Judith Packer [12, 14]. The connections between these areas and wavelets and frames are emphasized in the papers of Section IV. Keri Kornelson, Karen Shuman, and Palle Jorgensen’s paper, “Orthogonal exponentials for Bernoulli iterated function systems,” intertwines dynamical systems and frame theory, while the paper by Judy Packer, “A survey of projective multiresolution analyses and a projective multiresolution analysis corresponding to the quincunx lattice,” relates C ∗ -module theory to frame theory. The paper by Kornelson (Larry’s eleventh Ph.D. student), Shuman, and Jorgensen (a frequent collaborator of Larry who also worked with Larry on the NSF FRG grant) considers measures on fractal spaces constructed from iterated function systems and to what extent exponential functions can be used to form orthonormal bases on these spaces. A prime example of such a space is the Cantor set. Constructing orthonormal bases by considering the geometric underpinnings of the spaces and their respective measures has been a key theme in harmonic analysis through the years, dating back to the work of Fourier, but most recently Jorgensen et al. [34, 43, 44, 45] have developed an extensive literature on the topic as it relates to iterated function systems and have intertwined the results of their work with engineering and theoretical physics. In this paper, the authors consider Bernoulli iterated aﬃne systems and the associated fractal sets Xλ with rational parameter λ. If λ is rational, say λ = ab , a, b ∈ Z in lowest terms, the authors determine completely to what extent Xλ supports a family of mutually orthogonal exponential functions. The article by Packer (a colleague and frequent collaborator of Larry) gives an overview of projective multiresolution analyses, as ﬁrst deﬁned by M. Rieﬀel and as formalized by Packer and Rieﬀel in [50]. After presenting a survey of Rieﬀel’s original approach, she concentrates on two 2 × 2 dilation matrices that are not similar to diagonal matrices with integer entries. The columns of one of these matrices generate the quincunx lattice, and Packer constructs a projective multiresolution analysis corresponding to this matrix having non-free initial module, with free wavelet module. The matrix corresponding to the quincunx lattice has positive determinant, and Packer also considers another nondiagonalizable 2 × 2 dilation matrix with negative determinant whose initial module is not free and whose wavelet module is not free; these two examples mirror the constructions in Packer and Rieﬀel’s work on 2×2 diagonal dilation matrices with integer entries. Projective multiresolution analyses can be completed to form ordinary multiresolution analyses in Hilbert spaces, and several observations are made about the multiresolution analyses that arise in this fashion. Section V includes some of the more applied articles in the volume. The article by Jeﬀrey Hogan and Joseph Lakey, “Sampling and time-frequency localization of band-limited and multiband signals,” begins with a discussion of

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the classical Shannon sampling theorem for band-limited signals from Fourier analysis, at ﬁrst specializing to functions in Paley–Wiener space, and then extending their analysis to a wider class of spaces by using time-frequency localization. The chapter by Myung-Sin Song, “Entropy encoding in wavelet image compression,” analyzes computational schemes. Song gives an overview of the wavelet image compression process, including its relationship to multiresolution analysis, and gives a detailed analysis of various algorithms and thresholding processes used. She then studies the role of entropy encoding, which is a lossless method of compression, in data compression and reconstruction algorithms. The interrelationship of wavelet and frame theory to signal and image processing is stressed throughout this article. A discerning reader will note that the crucial point Professor Arlan Ramsay has mentioned in his foreword to this volume comes through in every article: Larry has been an essential contributor to a truly diverse group of fundamental research areas in harmonic analysis, both pure and applied, both through direct collaboration with many of the authors in this volume, and through his enthusiasm and welcome listening ear to anyone who wants to discuss mathematics. He has been both a generous collaborator of and wonderful friend to all of us. The three of us look forward to working with him for many years to come. Iowa City, Colorado Springs, Boulder September 2007

Palle E. T. Jorgensen Kathy D. Merrill Judith A. Packer

References [1] L. W. Baggett, A separable group having a discrete dual space is compact, J. Funct. Analysis 10 (1972), 131–148. [2] L. W. Baggett and K. F. Taylor, Riemann–Lebesgue subsets of Rn and representations which vanish at inﬁnity, J. Funct. Analysis 28 (1978), 168–181. [3] L. W. Baggett and K. F. Taylor, A suﬃcient condition for the complete reducibility of the regular representation, J. Funct. Anal. 34 (1979), 250–265. [4] L. W. Baggett, Unimodularity and atomic Plancherel measure, Math. Ann. 266 (1984), 513–518. [5] L. W. Baggett, W. E. Mitchell, and A. B. Ramsay, Representations of the discrete Heisenberg group and cocycles of an irrational rotation, Michigan Math. J. 31 (1984), 263–273. [6] L. W. Baggett and K. D. Merrill, Representations of the Mautner group and cocycles of an irrational rotation, Michigan Math. J. 33 (1986), 221–229. [7] L. W. Baggett, On circle-valued cocycles of an ergodic measure-preserving transformation, Israel J. Math. 61 (1988), 29–38. [8] L. W. Baggett, On functions that are trivial cocycles for a set of irrationals, Proc. Amer. Math. Soc. 104 (1988), 1212–1215.

xxii

Preface

[9] L. W. Baggett and K. D. Merrill, Equivalence of cocycles under an irrational rotation, Proc. Amer. Math. Soc. 104 (1988), 1050–1053. [10] L. W. Baggett, Processing a radar signal and representations of the discrete Heisenberg group, Colloq. Math. 60/61 (1990), 195–203. [11] L. W. Baggett and K. D. Merrill, On the cohomological equivalence of a class of functions under an irrational rotation of bounded type, Proc. Amer. Math. Soc. 111 (1991), 787–793. [12] L. W. Baggett and J. A. Packer, C ∗ -algebras associated to two-step nilpotent groups, in “Selfadjoint and nonselfadjoint operator algebras and operator theory (Fort Worth, TX, 1990),” pp. 1–6, Contemp. Math. 120, Amer. Math. Soc., Providence, RI, 1991. [13] L. W. Baggett and K. D. Merrill, Smooth cocycles for an irrational rotation, Israel J. Math. 79 (1992), 281–288. [14] L. W. Baggett and J. A. Packer, The primitive ideal space of two-step nilpotent group C ∗ -algebras, J. Funct. Anal. 124 (1994), 389–426. [15] L. W. Baggett, A. L. Carey, W. Moran and P. Ohring, General existence theorems for orthonormal wavelets, an abstract approach, Publ. Res. Inst. Math. Sci. 31 (1995), 95–111. [16] L. W. Baggett, H. A. Medina and K. D. Merrill, On functions that are trivial cocycles for a set of irrationals, II, Proc. Amer. Math. Soc. 124 (1996), 89–93. [17] L. W. Baggett, E. Kaniuth, and W. Moran, Primitive ideal spaces, characters, and Kirillov theory for discrete nilpotent groups, J. Funct. Anal. 150 (1997), 175–203. [18] L. W. Baggett, H. A. Medina and K. D. Merrill, Cohomology of polynomials under an irrational rotation, Proc. Amer. Math. Soc. 126 (1998), 2909–2918. [19] L. W. Baggett and K. D. Merrill, Abstract harmonic analysis and wavelets in Rn , in “The functional and harmonic analysis of wavelets and frames (San Antonio, TX, 1999),” pp. 17–27, Contemp. Math. 247, Amer. Math. Soc., Providence, RI, 1999. [20] L. W. Baggett, H. A. Medina and K. D. Merrill, Generalized multi-resolution analyses and a construction procedure for all wavelet sets in Rn , J. Fourier Anal. Appl. 5 (1999), 563–573. [21] L. W. Baggett, An abstract interpretation of the wavelet dimension function using group representations, J. Funct. Anal. 173 (2000), 1–20. [22] L. W. Baggett, J. E. Courter and K. D. Merrill, The construction of wavelets from generalized conjugate mirror ﬁlters in L2 (Rn ), Appl. Comput. Harmon. Anal. 13 (2002), 201–223. [23] L. W. Baggett, P. E. T. Jorgensen, K. D. Merrill, and J. A. Packer, An analogue of Bratteli–Jorgensen loop group actions for GMRA’s, “Wavelets, frames and operator theory,” pp. 11–25, Contemp. Math. 345, Amer. Math. Soc., Providence, RI, 2004. [24] L. W. Baggett, P. E. T. Jorgensen, K. D. Merrill, and J. A. Packer, Construction of Parseval wavelets from redundant ﬁlter systems, J. Math. Phys. 46 (2005), no. 8, 083502, 28 pp. [25] L. W. Baggett, P. E. T. Jorgensen, K. D. Merrill, and J. A. Packer, A non-MRA C r frame wavelet with rapid decay, Acta Appl. Math. 89 (2005), 251–270. [26] L. W. Baggett, Redundancy in the frequency domain, in “Harmonic analysis and applications,” pp. 335–357, Appl. Numer. Harmon. Anal., Birkh¨ auser Boston, Boston, MA, 2006.

Preface

xxiii

[27] J. J. Benedetto, M. T. Leon, The construction of multiple dyadic minimally supported frequency wavelets on Rd , Contemp. Math. 247 (1999), 43–74. [28] J. J. Benedetto and S. Li, The theory of multiresolution analysis frames and applications to ﬁlter banks, Appl. Comput. Harmon. Anal. 5 (1998), 389–427. [29] Chal Benson, Joe Jenkins, and Gail Ratcliﬀ, On Gelfand pairs associated with solvable Lie groups, Trans. Amer. Math. Soc. 321 (1990), 85–116. [30] M. Bownik, Z. Rzeszotnik, The spectral function of shift-invariant spaces, Michigan Math. J. 51 (2003), 387–414. [31] M. Bownik, Z. Rzeszotnik, On the existence of multiresolution analysis for framelets, Math. Ann. 332 (2005), 705–720. [32] I. Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986),1271–1283. [33] I. Daubechies, B. Han, A. Ron, and Z. W. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (2003), 1–46. [34] D. E. Dutkay and P. E. T. Jorgensen, Wavelets on fractals, Rev. Mat. Iberoamericana 22 (2006), no. 1, 131–180. [35] X. Dai and D. R. Larson, Wandering vectors for unitary systems and orthogonal wavelets, Mem. Amer. Math. Soc. 134, no. 640 (1998). [36] X. Dai, D. R. Larson, and D. M. Speegle, Wavelet sets in Rn , J. Fourier Anal. Appl. 3 (1997), 451–456. [37] X. Dai, D. R. Larson, and D. M. Speegle, Wavelet sets in Rn II, Contemp. Math. 216 (1998), 15–40. [38] R. J. Duﬃn and A. C. Schaeﬀer , A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341–366. [39] X. Fang, X. Wang, Construction of minimally supported frequency wavelets, J. Fourier Anal. Appl. 2 (1996), 315–327. [40] M. Frazier, G. Garrig´ os, K. Wang, and G. Weiss, A characterization of functions that generate wavelet and related expansion, J. Fourier Anal. Appl. 3 (1997), 883–906. [41] G. Gripenberg, A necessary and suﬃcient condition for the existence of a father wavelet, Studia Math. 114 (1995), 207–226. [42] R. Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, pp. 1–82, in “The Schur lectures (1992),” Bar-Ilan Univ., Ramat Gan, 1995. [43] P. E. T. Jorgensen, “Analysis and Probability: Wavelets, Signals, Fractals,” Graduate Texts in Mathematics, 234, Springer, New York, 2006. [44] P. E. T. Jorgensen and S. Pedersen, Harmonic analysis of fractal measures, Constr. Approx. 12 (1996), no. 1, 1–30. [45] P. E. T. Jorgensen and S. Pedersen, Dense analytic subspaces in fractal L2 spaces, Journal d’Analyse Math´ematique 75 (1998), 185–228. [46] W. M. Lawton, Tight frames of compactly supported aﬃne wavelets, J. Math. Phys. 31 (1990), 1898–1901. [47] S. Mallat, A theory for multiresolution signal decomposition: the wavelet representation, IEEE Trans. Patt. Recog. and Mach. Intell. 11 (1989), 674–693. [48] S. Mallat, Multiresolution approximations and wavelet orthonormal bases of L2 (R), Trans. Amer. Math. Soc. 315 (1989), 69–87. [49] Y. Meyer, “Wavelets and Operators,” Cambridge University Press, Cambridge, 1992. [50] J. Packer and M. A. Rieﬀel, Projective multi-resolution analyses for L2 (R2 ). J. Fourier Anal. Appl. 10 (2004), 439–464.

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[51] J. Ramanathan and T. Steger, Incompleteness of sparse coherent states, Appl. Comput. Harmon. Anal. 2 (1995), 148–153. [52] A. Ron and Z. Shen, Aﬃne systems in L2 (Rd ): the analysis of the analysis operator, J. Funct. Anal. 148 (1997), 408–447. [53] A. Ron and Z. Shen, Aﬃne systems in L2 (Rd ) II: dual systems, J. Fourier Anal. Appl. 3 (1997), 617–637. [54] X. Wang, The study of wavelets from the properties of their Fourier transforms, Ph.D. thesis, Washington University in St. Louis (1995).

Acknowledgments

The three named editors thank Brian Treadway for expert help: TEX, fonts, graphics, organization, style ﬁles, index, and more. Chris Heil kindly sent us helpful additional suggestions for organization. Our anonymous referees were generous with timely and detailed suggestions for improvements. We also thank John Benedetto, who was the ﬁrst to suggest that a volume should be published in honor of Larry Baggett, for his help to us in arranging for the volume to appear in the Applied and Numerical Harmonic Analysis series. Many individuals helped with workshop organization. Christy Baggett helped organize several social events, including a wonderful cocktail party, and both she and Marysia Mycielski were the primary organizers of the conference banquet. The staﬀ at the University of Colorado Math Department, including the assistant to the Chair, Donna Maes, the undergraduate secretary, Marysia Mycielski, and the department accounting technician, Denise Rodriguez, were very generous with their time and concern for visitors. We thank Dr. Rick Clelland for arranging for computer access for visitors, and we thank the Administration at the University, in particular the Chair of the Department of Mathematics, Dr. Lynne Walling, and the Acting Associate Vice Chancellor for Research and Dean of the Graduate School, Stein Sture, for their friendly introduction to our visitors. We also thank the Boulder Fire Department for their aid. The workshop was funded in part by grants from the University of Colorado Council for Research and Creative Work, the University of Colorado Arts and Sciences Dean’s Fund for Excellence, and the U.S. National Science Foundation by grant number 0600718.

xxv

Mathematical Family Tree of Lawrence W. Baggett

J. M. G. Fell, University of California, Berkeley, 1951 Lawrence W. Baggett, University of Washington, 1966 Kenneth Joy, University of Colorado at Boulder, 1977 Wesley Mitchell, University of Colorado at Boulder, 1979 Kathy Merrill, University of Colorado at Boulder, 1983 Peter Ohring, University of Colorado at Boulder, 1987 Mark Willis, University of Colorado at Boulder, 1993 Melissa Richey, University of Colorado at Boulder, 1999 Jennifer Courter, University of Colorado at Boulder, 1999 Eric Weber, University of Colorado at Boulder, 1999 Sharon Schaﬀer, University of Colorado at Boulder, 2000 Curtis Caravone, University of Colorado at Boulder, 2001 Keri Kornelson, University of Colorado at Boulder, 2001 Veronika Furst, University of Colorado at Boulder, 2006

xx vii

Publications of Lawrence W. Baggett

A. Books 1. “Fourier analysis” (with W. Fulks), Anjou Press, Inc., Boulder, Colo., 1979. viii+183 pp. ISBN: 0-88446-001-0. 2. “Functional analysis: A primer,” Monographs and Textbooks in Pure and Applied Mathematics 153, Marcel Dekker, Inc., New York, 1992. xii+267 pp. ISBN: 0-8247-8598-3. 3. “The functional and harmonic analysis of wavelets and frames: Proceedings of the AMS Special Session on the Functional and Harmonic Analysis of Wavelets held in San Antonio, TX, January 13–14, 1999” (edited with D. R. Larson), Contemporary Mathematics 247, American Mathematical Society, Providence, RI, 1999, x+306 pp., ISBN: 0-8218-1957-7. B. Papers 1. A weak containment theorem for groups with a quotient R-group, Trans. Amer. Math. Soc. 128 (1967), 277–290. 2. A description of the topology on the dual spaces of certain locally compact groups, Trans. Amer. Math. Soc. 132 (1968), 175–215. 3. Hilbert–Schmidt representations of groups, Proc. Amer. Math. Soc. 21 (1969), 502–506. 4. A note on groups with ﬁnite dual spaces, Paciﬁc J. Math. 31 1969, 569– 572. 5. A separable group having a discrete dual space is compact, J. Funct. Analysis 10 (1972), 131–148. 6. Multiplier representations of abelian groups (with A. Kleppner), J. Funct. Analysis 14 (1973), 299–324. 7. An ergodic theorem for Poisson processes on a compact group with applications to random evolutions (with D. W. Stroock), J. Funct. Analysis 16 (1974), 404–414. 8. Multiplier extensions other than the Mackey extension, Proc. Amer. Math. Soc. 56 (1976), 351–356. xxix

xxx

Publications of Lawrence W. Baggett

9. Operators arising from representations of nilpotent Lie groups, J. Funct. Analysis 24 (1977), 379–396. 10. Riemann–Lebesgue subsets of Rn and representations which vanish at inﬁnity (with K. F. Taylor), J. Funct. Analysis 28 (1978), 168–181. 11. A characterization of “Heisenberg groups”; when is a particle free? Rocky Mountain J. Math. 8 (1978), 561–582. 12. Groups with completely reducible regular representation (with K. F. Taylor), Proc. Amer. Math. Soc. 72 (1978), 593–600. 13. Representations of the Mautner group, I, Paciﬁc J. Math. 77 (1978), 7–22. 14. A suﬃcient condition for the complete reducibility of the regular representation (with K. F. Taylor), J. Funct. Anal. 34 (1979), 250–265. 15. Some pathologies in the Mackey analysis for a certain nonseparable group (with A. B. Ramsay), J. Funct. Anal. 39 (1980), 375–380. 16. A functional analytic proof of a selection lemma (with A. B. Ramsay), Canad. J. Math. 32 (1980), 441–448. 17. The Hausdorﬀ dual problem for connected groups (with T. Sund), J. Funct. Anal. 43 (1981), no. 1, 60–68. 18. On asymptotic behavior of induced representations (with K. F. Taylor), Canad. J. Math. 34 (1982), 220–232. 19. Unimodularity and atomic Plancherel measure, Math. Ann. 266 (1984), 513–518. 20. On the continuity of Mackey’s extension process, J. Funct. Anal. 56 (1984), 233–250. 21. Representations of the discrete Heisenberg group and cocycles of an irrational rotation (with W. E. Mitchell and A. B. Ramsay), Michigan Math. J. 31 (1984), 263–273. 22. Measures invariant under a linear group, Proc. Amer. Math. Soc. 94 (1985), 179–186. 23. Representations of the Mautner group and cocycles of an irrational rotation (with K. D. Merrill), Michigan Math. J. 33 (1986), 221–229. 24. Nonmonomial representations of abelian groups with multipliers (announcement) (with A. L. Carey, W. Moran and A. B. Ramsay), in “Miniconference on harmonic analysis and operator algebras (Canberra, 1987),” pp. 1–5, Proc. Centre Math. Anal. Austral. Nat. Univ. 15, Austral. Nat. Univ., Canberra, 1987. 25. On circle-valued cocycles of an ergodic measure-preserving transformation, Israel J. Math. 61 (1988), 29–38. 26. On functions that are trivial cocycles for a set of irrationals, Proc. Amer. Math. Soc. 104 (1988), 1212–1215. 27. Equivalence of cocycles under an irrational rotation (with K. D. Merrill), Proc. Amer. Math. Soc. 104 (1988), 1050–1053. 28. On the notion of virtual amenability for groups, Colloq. Math. 56 (1988), 129–136. 29. A functional analytic proof of a Borel selection theorem, J. Funct. Anal. 94 (1990), 437–450.

Publications of Lawrence W. Baggett

xxxi

30. Processing a radar signal and representations of the discrete Heisenberg group, Colloq. Math. 60/61 (1990), 195–203. 31. On the cohomological equivalence of a class of functions under an irrational rotation of bounded type (with K. D. Merrill), Proc. Amer. Math. Soc. 111 (1991), 787–793. 32. Nonmonomial multiplier representations of abelian groups (with A. L. Carey, W. Moran and A. B. Ramsay), J. Funct. Anal. 97 (1991), 361– 372. 33. C ∗ -algebras associated to two-step nilpotent groups (with J. A. Packer), in “Selfadjoint and nonselfadjoint operator algebras and operator theory (Fort Worth, TX, 1990),” pp. 1–6, Contemp. Math. 120, Amer. Math. Soc., Providence, RI, 1991. 34. Smooth cocycles for an irrational rotation (with K. D. Merrill), Israel J. Math. 79 (1992), 281–288. 35. The primitive ideal space of two-step nilpotent group C ∗ -algebras (with J. A. Packer), J. Funct. Anal. 124 (1994), 389–426. 36. General existence theorems for orthonormal wavelets, an abstract approach (with A. L. Carey, W. Moran and P. Ohring), Publ. Res. Inst. Math. Sci. 31 (1995), 95–111. 37. On functions that are trivial cocycles for a set of irrationals, II (with H. A. Medina and K. D. Merrill), Proc. Amer. Math. Soc. 124 (1996), 89–93. 38. Simultaneously symmetric functions (with H. A. Medina and K. D. Merrill), Amer. Math. Monthly 104 (1997), 520–528. 39. Primitive ideal spaces, characters, and Kirillov theory for discrete nilpotent groups (with E. Kaniuth and W. Moran), J. Funct. Anal. 150 (1997), 175–203. 40. Cohomology of polynomials under an irrational rotation (with H. A. Medina and K. D. Merrill), Proc. Amer. Math. Soc. 126 (1998), 2909–2918. 41. Abstract harmonic analysis and wavelets in Rn (with K. D. Merrill), in “The functional and harmonic analysis of wavelets and frames (San Antonio, TX, 1999),” pp. 17–27, Contemp. Math. 247, Amer. Math. Soc., Providence, RI, 1999. 42. Generalized multi-resolution analyses and a construction procedure for all wavelet sets in Rn (with H. A. Medina and K. D. Merrill), J. Fourier Anal. Appl. 5 (1999), 563–573. 43. An abstract interpretation of the wavelet dimension function using group representations, J. Funct. Anal. 173 (2000), 1–20. 44. The construction of wavelets from generalized conjugate mirror ﬁlters in L2 (Rn ) (with J. E. Courter and K. D. Merrill), Appl. Comput. Harmon. Anal. 13 (2002), 201–223. 45. An analogue of Bratteli–Jorgensen loop group actions for GMRA’s (with P. E. T. Jorgensen, K. D. Merrill, and J. A. Packer), “Wavelets, frames and operator theory”, pp. 11–25, Contemp. Math. 345, Amer. Math. Soc., Providence, RI, 2004.

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Publications of Lawrence W. Baggett

46. Construction of Parseval wavelets from redundant ﬁlter systems (with P. E. T. Jorgensen, K. D. Merrill, and J. A. Packer), J. Math. Phys. 46 (2005), no. 8, 083502, 28 pp. 47. A non-MRA C r frame wavelet with rapid decay (with P. E. T. Jorgensen, K. D. Merrill, and J. A. Packer), Acta Appl. Math. 89 (2005), 251–270. 48. Redundancy in the frequency domain, in “Harmonic analysis and applications,” pp. 335–357, Appl. Numer. Harmon. Anal., Birkh¨ auser Boston, Boston, MA, 2006.

Co-workers of Lawrence W. Baggett

CO-AUTHORS OF LAWRENCE W. BAGGETT 1. Alan Carey, University of Adelaide/Flinders University/Australian National University, Australia 2. Jennifer E. Courter, University of Colorado/Colorado College 3. Watson Fulks, University of Colorado 4. Palle E. T. Jorgensen, University of Iowa 5. Eberhard Kaniuth, University of Paderborn, Germany 6. Adam Kleppner, University of Maryland 7. David R. Larson, Texas A & M University 8. Herbert A. Medina, Loyola Marymount University 9. Kathy D. Merrill, Colorado College 10. Wesley E. Mitchell 11. William Moran, Flinders University, Australia 12. Peter Ohring, SUNY College at Purchase 13. Judith A. Packer, University of Colorado 14. Arlan Ramsay, University of Colorado 15. Daniel W. Stroock, University of Colorado/Massachusetts Institute of Technology 16. Terje Sund, University of Oslo, Norway 17. Keith F. Taylor, University of Saskatchewan/Dalhousie University, Canada CO-INVESTIGATORS WITH LAWRENCE W. BAGGETT ON RESEARCH GRANTS 1. 2. 3. 4. 5. 6. 7.

Akram Aldroubi, Vanderbilt University John J. Benedetto, University of Maryland Christopher Heil, Georgia Tech University Palle E. T. Jorgensen, University of Iowa David R. Larson, Texas A & M University Gestur Olafsson, Louisiana State University Yang Wang, Georgia Tech University xxxiii

Titles of All Talks

Current Trends in Harmonic Analysis and Its Applications: Wavelets and Frames May 18–20, 2006, University of Colorado, Boulder Kathy Merrill, Colorado College Smooth single Parseval wavelets from ﬁlters in L2 (R2 ), an almost smooth wavelet set, and a tribute to Larry Baggett Keith F. Taylor, Dalhousie University Groups with atomic regular representations Yang Wang, Georgia Institute of Technology Denoising natural color images Joe Lakey, New Mexico State University On periodic non-uniform sampling in shift-invariant spaces Jeﬀ Hogan, University of Arkansas Wavelets and sampling in the Zak domain Casey Leonetti, Vanderbilt University Non-uniform sampling and reconstruction from sampling sets with unknown jitter Keri Kornelson, Grinnell College Non-integer translation invariant spaces Chal Benson, East Carolina University A geometric model for the space of bounded spherical functions on a 2-step nilpotent Lie group (joint work with Gail Ratcliﬀ)

xxxv

xxxvi

Titles of All Talks

Wannapa Ruangthanakorn, Suranaree University of Technology Cross-sections and continuous wavelets associated with matrix groups Christopher Brislawn, Los Alamos National Laboratory Recent research on multirate ﬁlter banks at Los Alamos National Lab Dorin Dutkay, Rutgers University Oversampling generates superwavelets Palle E. T. Jorgensen, University of Iowa From signals to operator algebra to wavelets Eberhard Kaniuth, University of Paderborn Induced characters, Mackey analysis and primitive ideal spaces of nilpotent discrete groups Christopher Heil, Georgia Institute of Technology The density theorem for Gabor systems and localized frames David Larson, Texas A&M University Wavelets, frames and unitary operators Bin Han, University of Alberta The projection method in wavelet analysis Kasso Okoudjou, Cornell University Unimodular Fourier multipliers for modulation spaces Jens Christensen, Louisiana State University Modulation spaces and co-orbit spaces Demetrio Labate, North Carolina State University Optimally sparse multidimensional representations using shearlets Marcin Bownik, University of Oregon Recent progress on Baggett’s problem Xingde Dai, University of North Carolina, Charlotte The path connectivity of s-elementary tight frame wavelets Brody Johnson, St. Louis University Orthogonal wavelet frames and vector-valued wavelet transforms Eric Weber, Iowa State University My failed attempts at the HRT conjecture

Titles of All Talks xxxvii

Daniel W. Stroock, MIT Some queer diﬀusions Gerald Folland, University of Washington The discrete Heisenberg group: The abstruse meets the applicable Dennis Sullivan, CUNY/SUNY Stony Brook Algebraic topology of functions on the circle with values in a manifold Herbert Medina, Loyola Marymount University Challenges and opportunities in developing some heretofore untapped American mathematical talent Jonas D’Andrea, University of Colorado Fractal wavelets of Dutkay–Jorgensen type for the Sierpinski gasket space Myung-Sin Song, Southern Illinois University Wavelet image compression Amy Chambers, University of Colorado Conditional expectations from the tensor product of graph C ∗ -algebras to certain subalgebras Alex Powell, Vanderbilt University Alternate dual frames for Sigma-Delta quantization Fumiko Futamura, Vanderbilt University Symmetrically localized frames Kuzman Adzievski, South Carolina State University Non-isotropic Hausdorﬀ capacity of exceptional sets for Pluri–Green potentials in the unit ball of Cn

Contents

ANHA Series Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv Mathematical Family Tree of Lawrence W. Baggett . . . . . . . . . . . xxvii . Publications of Lawrence W. Baggett . . . . . . . . . . . . . . . . . . . . . . . . . .xxix Co-workers of Lawrence W. Baggett . . . . . . . . . . . . . . . . . . . . . . . . . .xxxiii . Titles of All Talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxv . I Classical and Abstract Harmonic Analysis 1

Some Riemann Sums Are Better Than Others Victor W. Guillemin and Daniel W. Stroock . . . . . . . . . . . . . . . . . . . . .

3

2

Gelfand Pairs Associated with Finite Heisenberg Groups Chal Benson and Gail Ratcliﬀ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3

Groups with Atomic Regular Representation Keith F. Taylor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4

Wavelet Transforms and Admissible Group Representations Eric Weber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

II Frames and Multiresolution Structures 5

The Density Theorem and the Homogeneous Approximation Property for Gabor Frames Christopher Heil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

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Contents

6

Recent Developments on Dual Wavelet Frames Bin Han . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7

Characteristic Wavelet Equations and Generalizations of the Spectral Function Veronika Furst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

8

Baggett’s Problem for Frame Wavelets Marcin Bownik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

III Wavelet Sets 9

Simple Wavelet Sets for Scalar Dilations in R2 Kathy D. Merrill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

10 Interpolation Maps and Congruence Domains for Wavelet Sets Xiaofei Zhang and David R. Larson . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 IV Applications to Dynamical Systems and C ∗ -Algebras 11 Orthogonal Exponentials for Bernoulli Iterated Function Systems Palle E. T. Jorgensen, Keri Kornelson, and Karen Shuman . . . . . . . 217 12 A Survey of Projective Multiresolution Analyses and a Projective Multiresolution Analysis Corresponding to the Quincunx Lattice Judith A. Packer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 V Signal and Image Processing 13 Sampling and Time-Frequency Localization of Band-Limited and Multiband Signals Jeﬀrey A. Hogan and Joseph D. Lakey . . . . . . . . . . . . . . . . . . . . . . . . . . 275 14 Entropy Encoding in Wavelet Image Compression Myung-Sin Song . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

Chapter 1

Some Riemann Sums Are Better Than Others Victor W. Guillemin and Daniel W. Stroock

Abstract This note contains some results obtained while ruminating about Riemann sums. We know that nothing here is truly new, but we know of no other place in which these ideas are presented in the way that they occurred to us. In particular, some of them are so elementary that we hope they will find their way into calculus texts.

1.1 Introduction Our aim in this little note is to point out an easy, but seemingly ignored, fact about Riemann integration. Namely, given a smooth, periodic function i : R $ C of period 1, the dierence between the Riemann sum UQ (i )

Q 1 X ¡p¢ i Q p=1 Q

(1.1)

R1 and the integral 0 i ({) g{ tends to 0 as Q $ 4 faster than any power Rieof Q 1 . Of course, this result is patently false if one takes arbitrary s mann sums to approximate the integral. To wit, take i ({) = h 1 2{ . Then R1 i ({) g{ = 0. On the other hand, if Q = 1 Q1 , then the Riemann sum 0

Victor W. Guillemin Department of Mathematics, M.I.T., 2-170, Cambridge, MA 02139 e-mail: [email protected] Daniel W. Stroock Department of Mathematics, M.I.T., 2-272, Cambridge, MA 02139 e-mail: [email protected]

3

4

Victor W. Guillemin and Daniel W. Stroock s

Q 1 X ¡ pQ ¢ h i Q = Q p=1

1 2

Q

Q Q

s 1 2Q s 1 2 QQ h

1h 1

is asymptotic to Q1 as Q $ 4. Thus, if one uses general Riemann sums, the order of approximation will be no better than Q1 . As we will see, there are many ways in which the rapid convergence can be derived. What surprises us is that, even though some of these derivations are completely elementary, none of them appears in any of the standard calculus texts with which we are familiar. In the hope that our doing so may persuade someone to include it in the next generation of calculus books, we will present two entirely dierent approaches to its derivation. The first of these entails nothing more sophisticated than integration by parts, and the second is based on elementary properties of Fourier series.

1.2 Via Integration by Parts Let i : R $ C be a smooth function that is periodic with period 1. Given n 0 and Q 1, define (n) Q (i )

¶n Q Z p µ £ ¡ ¢¤ p1 1 X Q i ({) i p g{= { = Q n! p=1 p1 Q Q

R1 (0) Obviously, Q (i ) = 0 i ({) g{ UQ (i ). Integration by parts of each summand gives (n) Q (i )

Next, because is equal to

¶n+1 Q Z p µ X Q p1 1 { = i 0 ({) g{= (n + 1)! p=1 p1 Q Q

R1 0

(1.2)

i 0 ({) g{ = 0, the expression on the right-hand side of (1.2)

"µ # ¶n+1 Q Z p X Q p1 1 1 { i 0 ({) g{ n+1 (n + 1)! p=1 p1 Q (n + 2)Q Q "µ # ¶n+1 Z p Q £ 0 ¡ ¢¤ p1 1 1 { i ({) i 0 p g{ = Q n+1 (n + 1)! p1 Q (n + 2)Q Q =

1 (0) (n+1) (i 0 ) Q (i 0 )= (n + 2)!Q n+1 Q

That is, (n)

Q (i ) =

1 (0) (n+1) (i 0 ) Q (i 0 )= (n + 2)!Q n+1 Q

(1.3)

1 Some Riemann Sums Are Better Than Others

5

Starting from (1.3), one can use induction on c 0 to prove that (0)

Q (i ) = d0>0 = 1>

d0>c+1 =

c X

n=0

1 Q c+1

c X

(n)

dn>c Q n+1 Q (i (c) ) where

n=0

dn>c > (n + 2)!

and dn>c+1 = dn1>c for 1 n c + 1=

Notice that a simpler expression can be obtained after observing that dn>c = (1)n d0>cn , which allows us to rewrite the preceding as (0)

Q (i ) =

1 Q c+1

c X (n) (1)n ecn Q n+1 Q (i (c) ) where n=0

e0 = 1 and ec+1 =

c X (1)n ecn = (1.4) (n + 2)!

n=0

At the same time, from (1.2) one has that1 ¯ (n) ¯ ki (c+1) ku > Q n+1 ¯Q (i (c) )¯ (n + 2)!

and so we have now shown that ¯ ¯ Z 1 ¯ ¯ Nc+1 (c+1) ¯UQ (i ) ¯ i ({) g{ ku ¯ ¯ Q c+1 ki 0

where Nc+1

(1.5)

c X |ecn | = (1.6) (n + 2)!

n=0

We now want to examine the behavior of ec and Nc as c $ 4. To this end, s (0) first note that if i ({) = h 1 2{ , then 1 (i ) = 1, ki (c+1) ku = (2)c+1 , and so (1.6) shows that 1 = (1.7) Nc+1 (2)c+1 To get a preliminary upper bound, use induction on c to see that |ec | c , 1 where is the unique element of (0> 1) that satisfies h = 1+ 2 . In particular, the generating function 4 X en n1 E() n=1

is well defined for 5 C in the open unit disk. Furthermore, by taking advantage of the convolution structure in the definition of the ec ’s, we see that

1

We use k*ku to denote the uniform (i.e., supremum) norm of * on [0> 1].

6

Victor W. Guillemin and Daniel W. Stroock

E() =

4 X

ec+1 c =

c=0

4 4 X X (1)n ecn

n=0 c=n

(n + 2)!

c

¡ ¢ h 1 + = 1 + E() > 2

and therefore that

E() =

g 1 h + h = log (h 1) g

µ

h 1

¶

>

(1.8)

where log is the primary branch of the logarithm function. By inspection, it is obvious that 2 is the radius of convergence of the Taylor’s expansion of the right-hand side at the origin. Hence, 2 is also the radius of convergence for E(). Equivalently, we now know that 1

lim |ec | c =

c$4

1 = 2

(1.9)

Finally, after plugging this into the expression for Nc+1 , one concludes that 1 1 limc$4 (Nc ) c 2 , which, in conjunction with (1.8), proves that 1

lim (Nc ) c =

c$4

1 = 2

(1.10)

As we will see in Section 1.3, one can get much more precise information about the numbers ec , but (1.10) together with (1.6) are sucient to show that Z 1 ¡ (c) ¢ 1c lim ki ku ? 2Q =, i ({) g{ = UQ (i )= (1.11) c$4

0

It is of some interest to see what can be said when i is not periodic. In this case, (1.2) continues to hold, but (1.3) must be replaced by (n)

Q (i ) =

£ (0) 0 ¡ ¢¤ 1 (n+1) Q (i ) i (1) i (0) Q (i 0 )= n+1 (n + 2)!Q

(1.12)

Starting from (1.12), one can use induction on c 1 to show that (0) Q (i )

=

1 Q c+1

c X ¢ (n) ¡ (1)n ecn Q n+1 Q i (c) n=0

c X ¢ en ¡ (n1) i (1) i (n1) (0) > (1.13) n Q

n=1

where the en ’s are those in (1.4). In particular, this leads to the estimate

1 Some Riemann Sums Are Better Than Others

7

¯Z ¯ c ¯ 1 X ¢¯¯ en ¡ (n1) ¯ (n1) i i ({) g{ UQ (i ) + (1) i (0) ¯ ¯ ¯ 0 ¯ Qn n=1

Nc+1 (c+1) ki ku = (1.14) Q c+1

Finally, just as before, one can derive from (1.14) the conclusion that (cf. (1.9) and (1.10)) ¡ ¢1 lim ki (c) kF([0>1)] c ? 2Q c$4 Z 1 4 X ¢ en ¡ (n1) =, i i ({) g{ = UQ (i ) (1) i (n1) (0) > n Q 0 n=1

where the series on the right-hand side is absolutely convergent.

1.3 Via Fourier Series As most, if not all, our readers will have guessed, Fourier series provides a much more elegant approach to the results in Section 1.2. Indeed, the basic observation out of which this note grew was made by the first author while he was thinking about the Poisson summation formula and realized that the rapid convergence of these Riemann sums is an easy consequence. Our reason for not putting a Fourier-grounded proof first is purely pedagogic: as distinguished from one involving Fourier series, the proof in Section 1.2 is one appropriate for a calculus class. Be that as it may, here is a simple proof using Fourier series. s Set hq ({) = h 1 2q{ , and note that ( 1 if Q divides q UQ (hq ) = 0 otherwise= Thus, if i is a smooth, periodic function with period 1, 4 X ¡ ¢ UQ (i ) = i> hpQ > p=4

R1 where (*> #) 0 *({)#({) g{ is the standard, Hermitian inner product for O2 ([0> 1]; C). Equivalently, for any smooth i with period 1, UQ (i )

Z

0

1

i ({) g{ =

X

(i> hpQ )=

p6=0

8

Victor W. Guillemin and Daniel W. Stroock

s Because (i> hq ) = ( 1 2q)c (i (c) > hq ) for q 6= 0, this gives UQ (i )

Z

1

i ({) g{ = 0

¡s ¢c X 1 1 2Q (i> hpQ ) for c 0= (1.15) pc p6=0

Hence, by Schwarz’s inequality and Parseval’s identity, ¯ p ¯Z 1 ¯ ¯ 2(2c) (c) ¯ i ({) g{ UQ (i )¯¯ ki k2 > ¯ (2Q )c

(1.16)

0

where k*k2 is the O2 ([0> 1]; C)-norm of *. Clearly, (1.16) represents a sharpening of (1.6) and confirms the conclusion drawn in (1.11). We close this section by using Fourier series to show that, for c 2, ( c 2(c) (1) 2 +1 (2) if c is even c (1.17) ec = 0 if c is odd= To this end, take Q = 1 in (1.4) to see that ! Z 1 ÃX Z 1 c ¤ (1)n ecn n £ (c) i ({) i (c) (0) g{= { i ({) g{ i (1) = n! 0 0 n=0

At the same time, we know that 4 3 Z 1 Z 1 X £ ¤ 1 C s i (1) = i ({) g{ + hp ({)D i (c) ({) i (c) (0) g{= c ( 1 2p) 0 0 p6=0 Thus, for any smooth i with compact support in (0> 1), ! Z 1 ÃX c (1)n ecn n { i ({) g{ n! 0 n=0 3 4 Z 1 X 1 C s h ({)D i ({) g{> = c p ( 1 2p) 0 p6=0 and so Sc ({) =

X

1 s hp ({) for c 2 and { 5 [0> 1] ( 1 2p)c p6=0 where Sc ({) (1)c

c X (1)n ecn

n=0

In particular, when { = 0, this says that

n!

{n

for c 0= (1.18)

1 Some Riemann Sums Are Better Than Others

9

X 1 (1)c ec = s > ( 1 2)c p6=0 pc from which (1.17) is an immediate consequence. Knowing that ec = 0 for odd c 3, the algorithm in (1.4) can be replaced by ; c 2 if c 5 {0> 1} A A A c2 A ? 2 X 1 e2n (1.19) ec = if c 2 is even= A 2c! (c 2n + 1)! A A n=0 A = 0 if c 3 is odd=

1.4 Euler Was Here Before As we admitted at the outset, we have been treading on well-trod ground. To make the connection with familiar results, we begin by noting that S0 ({) 1>

Sc0 = Sc for c 1>

and Sc (0) = Sc (1) for c 2=

(1.20)

The first two of these follow easily from the definition of the Sc ’s given in the second line of (1.18), and the third follows from the first line in (1.18). In addition, it is obvious from the first line of (1.18) that |S2c | achieves its maximum at 0, and therefore that kS2c ku = (1)c+1 e2c =

2(2c) = (2)2c

(1.21)

Although it is easy to see that kS1 ku = 12 , it is not clear how to compute kS2c+1 ku when c 1. Nonetheless, an estimate can be obtained from the fact that Z { |S2c ()| g (1)c+1 e2c {> |S2c+1 ({)| 0

combined with the observation, which follows for c 1 from the first line in (1.18), S2c+1 ({) = S2c+1 (1 {). Thus, kS2c+1 ku

(1)c+1 e2c = 2

(1.22)

It is not clear whether, when c 1, this is better than the estimate 2(2c+1) (2)2c+1 that one gets directly from (1.18), but, of course, neither one is precise. We next observe that the properties in (1.20) determine the Sc ’s uniquely. Indeed, suppose that {Tc : c 0} satisfy the properties in (1.20). Clearly Tc P is an cth order polynomial. Moreover, if Tc ({) = cn=0 tn>c {n , then t0>0 = 1,

10

Victor W. Guillemin and Daniel W. Stroock

and, for c 0, c X tn>c n+1 { + t0>c+1 n+1

Tc+1 ({) =

n=0

and c X

n=0

Z

tn>c + t0>c+1 = (n + 1)(n + 2)

1

Tc+1 ({) g{ = Tc+2 (1) Tc+2 (0) = 0=

0

Thus, all the coecients {tn>c : c 0 & 0 n c} are uniquely determined. Using this characterization of the Sc ’s, we can show that Ec ({) = c!Sc ({) where Ec ({) is the cth Bernoulli polynomial=

(1.23)

Indeed (cf. Chapter XIII in [1] for basic facts about the Bernoulli polynomials), E©0 ({) 1, Ec0 ({)ª= cEc1 ({) for c 1, and Ec (0) = Ec (1) for c 2. Thus, c!1 Ec ({) : c 0 satisfies (1.20). Of course, (1.23) says that Ec = c!ec

where Ec Ec (0) is the cth Bernoulli number=

(1.24)

Our final goal is the show that (1.13) for even c 2 is the Euler—MacLaurin formula given in (13.5.4) of [1]. To this end, define S˜c to be the periodic extension of (1)c Sc ¹ [0> 1). Then, for c 1, c Q Z X X (n) (1)n ecn Q n Q (i ) = p=1

n=0

p Q p1 Q

£ ¡ ¢¤ g{= S˜c (Q {) i ({) i p q

But, for each 1 p Q , Z

p Q p1 Q

1 S˜c (Q {) g{ = Q

Z

1

0

S˜c+1 (0) S˜c+1 (1) = 0= S˜c ({) g{ = Q

Thus, (1.13) can be rewritten as Z

0

1

c ¢ X ¢ 1 ¡ en ¡ (n1) i i (1) i (0) (1) i (n1) (0) n 2Q Q n=2 Z 1 1 + c S˜c (Q {)i (c) ({) g{> Q 0

i ({) g{ UQ (i ) =

where the sum on the right is understood to be 0 when c = 1. Equivalently,

1 Some Riemann Sums Are Better Than Others

Z

11

1

i ({) g{

0 Q 1 X i = Q p=1

¡p¢ Q

+i 2

¡ p1 ¢ Q

X

¢ e2n ¡ (2n1) i (1) i (2n1) (0) 2n Q

1n 2c

+

1 Qc

Z

1

S˜c (Q {)i (c) ({) g{= (1.25)

0

Finally, replacing c by 2c in (1.25) and putting the final summand together with the integral on the right, one gets the Euler—MacLaurin formula Z

1

i ({) g{

0 Q 1 X i = Q p=1

¡p¢ Q

+i 2

¡ p1 ¢ Q

+

X

1n?c

1 Q 2c

Z

0

¢ e2n ¡ (2n1) i (1) i (2n1) (0) 2n Q

1¡

¢ S˜2c (Q {) e2c i (2c) ({) g{= (1.26)

The final term on the right of (1.26) deserves further comment. Namely, by (1.21), the integral on the right of (1.26) is dominated by 2|e2c | times ki 2c k1 , R1 where k*k1 0 |*({)| g{. Hence, (1.26) yields the estimate ¡ p1 ¢ ¡ ¢ ¯Z 1 Q X ¯ i p Q +i Q ¯ i ({) g{ 1 ¯ Q p=1 2 0 ¯ X e2n ¡ ¢¯ (2n1) (2n1) i + (1) i (0) ¯¯ Q 2n 1n?c

4(2c) ki (2c) k1 = (2Q )2c

(1.27)

¡ Second, again from (1.21), we know that (1)c S˜2c e2c ) 0. In addition, ¢ R 1¡ S˜2c (Q {)e2c g{ = e2c . Thus, when i (2c) is real-valued, the mean value 0 theorem for integrals allows one to replace (1.26) by Z

0

1

¡ p1 ¢ ¡ ¢ Q 1 X i p Q +i Q i ({) g{ Q p=1 2 X e2n ¡ ¢ e2c i (2n1) (1) i (2n1) (0) 2c i (2c) ()> (1.28) = 2n Q Q 1n?c

for some 5 [0> 1].

12

Victor W. Guillemin and Daniel W. Stroock

Acknowledgments The authors are grateful for support provided to them, respectively, by NSF grants DMS 0104116 and DMS 0244991.

References 1. G.H. Hardy, Divergent Series. AMS Chelsea Series. American Math. Soc., Providence, 1991.

Chapter 2

Gelfand Pairs Associated with Finite Heisenberg Groups Chal Benson and Gail Ratcli

Abstract We examine a family of finite Gelfand pairs which arise in connection with Heisenberg groups K = Kq (F) over finite fields of odd characteristic. The symplectic group Vs(q> F) acts on K by automorphisms. A subgroup N of Vs(q> F) yields a Gelfand pair (N> K) when the N-invariant functions on K commute under convolution. This is equivalent to the restriction of the oscillator representation to N being multiplicity free. An interesting example of this type occurs with N a finite analog of the unitary group X (q).

A topological group J together with a compact subgroup N are said to form a Gelfand pair if the set O1 (N\J@N) of N-bi-invariant integrable functions on J is a commutative algebra under convolution. The situation where J and N are Lie groups has been the focus of extensive and ongoing investigation. Riemannian symmetric spaces J@N furnish the most widely studied and best understood examples. ([Hel84] is a standard reference.) Apart from these, key examples arise as semidirect products J = N n Q , of compact Lie groups N with two-step nilpotent Lie groups Q . Such pairs are the focus of [BJR90], [Vin03] and [Yak06], among other works. There are many examples where Q = Kq (R), a (real) Heisenberg group, and N is a subgroup of the unitary group X (q). Gelfand pairs also arise in connection with analysis on finite groups but, to our knowledge, have been studied less extensively. Known examples include the symmetric group modulo the hyperoctahedral group [Mac] and finite analogues of the hyperbolic plane [SA87, Ter99]. In this paper, we introduce a family of Gelfand pairs associated with finite Heisenberg groups. They provide finite analogues for the Gelfand pairs associated with Kq (R). Our Chal Benson, Gail Ratclig Department of Mathematics, East Carolina University, Greenville, NC 27858 e-mail: [email protected], [email protected]

13

14

Chal Benson and Gail Ratclig

examples appear elsewhere, in the literature on the oscillator representation, but their relevance to the study of Gelfand pairs has not, however, been previously emphasized.

2.1 Preliminaries To begin, we must establish notation and recall some ideas concerning the representation theory for Heisenberg groups over finite fields. For a finite set V, the symbol C[V] will denote the set of all C-valued functions on V. This is a complex vector space of dimension |V| that carries a positive-definite Hermitian inner product hi> jiV =

1 X i ({)j({)= |V| {5V

2.1.1 Heisenberg Groups Let F be a field of odd characteristic. The polarized Heisenberg group Kq (F) is the set Kq (F) = Fq × Fq × F with product µ ¶ 1 0 0 0 0 0 (x> y> w)(x > y > w ) = x + x > y + y > w + w + (x · y y · x ) = 2 0

0

0

(2.1)

Inclusion of the factor 1/2 is motivated by the use of exponential coordinates in connection with the real Heisenberg group Kq (R). (See [Fol89].) Some authors omit this factor, but the resulting group is isomorphic with that defined here via the mapping (x> y> w) 7$ (x> y> 2w). An alternate notation is useful in connection with certain examples and constructions. We write W = Wq = Fq × Fq > so that Kq (F) = W × F, and Equation (2.1) becomes (z> w)(z0 > w0 ) = (z + z0 > w + w0 + 21 [z> z0 ])

(2.2)

where [z> z0 ] denotes the usual symplectic form on W, namely [z> z0 ] = [(x> y)> (x0 > y0 )] = x · y0 y · x0 =

(2.3)

2 Gelfand Pairs Associated with Finite Heisenberg Groups

15

More generally, for any finite dimensional symplectic vector space (W> [·> ·]) over F, we let KW = W × F with product given by (2.2).

(2.4)

2.1.2 Unitary Dual of Hn (Fq ) Throughout this paper, we take F = Ft > the finite field with t elements where t = sp for some odd prime s. The field F is an extension of its prime field Zs = Z@(sZ). The characters on F are b = {#d : d 5 F} F

where

#d (w) = exp

µ

¶ 2l TrF@Zs (dw) s

(2.5)

and TrF@Zs : F $ Zs is the trace map for the field extension F@Zs . Explicitly, one can write p1

2

TrF@Zs (w) = w + ws + ws + · · · + ws

=

(See Chapter 2 in [LN].) The basic identity ½ ¾ X t if d = 0 #d (w) = = td>0 = 0 if d 6= 0

(2.6)

w5F

shows that the characters are pairwise orthogonal unit vectors in C[F]. The mappings a>b : Kq (F) $ T>

a>b (x> y> w) =

q Y

l=1

#dl ({l )

q Y

#em (|m )

(2.7)

m=1

for a> b 5 Fq give t 2q distinct one-dimensional representations of Kq (F). One can verify, moreover, that for 5 F× , the formula ¡ ¢ (2.8) (x> y> w)i (u) = # w + y · u + 21 x · y i (u + x)

defines a unitary representation (analogous to the Schr¨odinger model in the real case) of Kq (F) in the the inner product space C[Fq ]. The trace character

16

Chal Benson and Gail Ratclig

" (x> y> w) = wu( (x> y> w)) for is " (x> y> w) = t q x>0 y>0 # (w)> which yields h" > "0 iKq (F) = h# > #0 iF = >0 in view of orthogonality for the characters of F. It follows that the representations { : 5 F× } are inequivalent and irreducible. Summing the squares of the dimensions for the representations (2.7) and (2.8) gives t 2q × 12 + (t 1) × (t q )2 = t 2q+1 = |Kq (F)|= Thus, (2.7) and (2.8) exhaust the unitary dual of Kq (F): q × \ K q (F) = { a>b : a> b 5 F } ] { : 5 F }=

(2.9)

On the center of Kq (F), we have (0> 0> w) = # (w)LC[Fq ] . So the t q dimensional irreducible representations are determined by their central characters. This proves: Theorem 2.1 (Stone—von Neumann Theorem). Let 5 F× and : Kq (I ) $ X (Y ) be an irreducible unitary representation with central character # . (That is, (0> 0> w) = # (w)LY .) Then is unitarily equivalent to the Schr¨ odinger representation defined by Equation (2.8).

2.1.3 Oscillator Representation ¡ ¢ The symplectic group for W = Fq × Fq > [·> ·] ,

Vs(q> F) = {j 5 JO(2q> F) : [jz> jz0 ] = [z> z0 ]}>

acts by automorphisms on Kq (F) via j · (z> w) = (jz> w)= Fix 5 F× . For given j 5 Vs(q> F), (z> w) 7$ j(z> w) = (jz> w) is an irreducible representation with central character # . The Stone—von Neumann Theorem ensures that j is unitarily equivalent to . Thus, there is a unitary operator $ (j) on C[Fq ] satisfying (jz> w) = $ (j) (z> w)$ (j)1 =

(2.10)

2 Gelfand Pairs Associated with Finite Heisenberg Groups

17

Schur’s Lemma shows that (2.10) defines $ (j) up to a multiplicative scalar of modulus one. In the context of finite fields, there is a systematic choice of scalars for which $ : Vs(q> F) $ X (C[Fq ]) is a representation of the group Vs(q> F). In the literature, $ is variously called the oscillator, metaplectic, or Weil—Segal—Shale representation. It is known that Vs(q> F) coincides with its commutator subgroup provided q A 1 or t A 3. Thus, (2.10) completely determines the representation $ , except when q = 1 and t = 3. (See [How].) The contragredient representation for has central character # . Thus, is unitarily equivalent to , by the Stone—von Neumann Theorem. Moreover, the contragredient $ of the oscillator representation satisfies (jz> w) = $ (j) (z> w)$ (j)1 . It follows that $ is unitarily equivalent to $ .

(2.11)

There are, in fact, just two distinct oscillator representations $ , up to unitary equivalence. Indeed Proposition 2.2 (See [How], [Neu02]). For > 0 5 F× one has $0 ' $ if and only if 0 @ is a square in F× . The oscillator representation can be rendered explicitly, at least on a set of generators for Vs(q> F). The formulas are given below in Theorem 2.3. Writing (2q) × (2q)-matrices in block form, ¸ DE (D> E> F> G of size q × q)> j= FG one has j 5 Vs(q> F) +,

ª © w D F = F w D> E w G = Gw E> Dw G F w E = L =

The group Vs(q> F) is generated by the subset © ª © ª © ª Dgldj : D 5 JO(q> F) ^ Forzhu : F w = F ^ M

where

Dgldj =

¸ D 0 > 0 (Dw )1

Forzhu =

¸ L 0 > FL

M=

¸ 0 L = L 0

(2.12)

p Theorem 2.3 (See [Neu02]). For 5 F× t (t = s ), the oscillator representation $ : Vs(q> F) $ X (C[Fq ])

is given on the generators (2.12) for Vs(q> F) as follows. • $ (Dgldj )i (u) = vjq(det D)i (D1 u) where

18

Chal Benson and Gail Ratclig

½

+1 if t is a square in F, 1 otherwise. ¡ 1 w ¢ • $ (Forzhu )i (u) = # 2 u Fu i (u). • $ (M)i (u) = (1)q(p+1) (l)qp(s1)@2 vjq()F i (u) where vjq(w) =

1 X F i (u) = s q i (x)# (x · u)= t q x5F

Note that F i is a -weighted variant of the (q-dimensional) inverse discrete Fourier transform (DFT).

2.2 The Group Algebra C[Hn (Fq )] and Gelfand Pairs Let K denote the Heisenberg group K = Kq (F) = W × F where, as before, W = Fq × Fq > t = sp >

F = Ft >

s an odd prime=

The convolution product on C[K] is X ¢ ¡ i (z> w)(z0 > w0 )1 j(z0 > w0 ) (i B j)(z> w) = (z0 >w0 )5K

=

X

(z0 >w0 )5K

¡ ¢ i z z0 > w w0 21 [z> z0 ] j(z0 > w0 )=

2.2.1 Twisted Convolution on C[W] Twisted convolution is well-known in connection with analysis on the real Heisenberg group Kq (R). (See [Fol89].) Here we require its discrete analog. Definition 2.4. For i 5 C[K] and d 5 F, define id 5 C[W] via 1 X i (z> w)#d (w)= id (z) = s t w5F

For fixed z 5 W, id (z) is the (one-dimensional) inverse discrete Fourier transform of w 7$ i (z> w) evaluated at d. The Fourier inversion formula yields 1 X id (z)#d (w)= i (z> w) = s t d5F

(2.13)

2 Gelfand Pairs Associated with Finite Heisenberg Groups

19

In particular, a function i 5 C[K] is completely determined by {id : d 5 F}. So for given i> i 0 5 C[K], i = i 0 +, id = id0 for all d 5 F.

(2.14)

Definition 2.5. For functions i> j 5 C[W] and given d 5 F, we define the twisted convolution i _d j 5 C[W] via ¶ µ X 1 [z> w] = (i _d j)(z) = i (z w)j(w)#d 2 w5W

A straightforward calculation using (2.13) and (2.6) yields the following. Lemma 2.6. For i> j 5 C[K] and d 5 F, one has (i B j)d =

s tid _d jd =

2.2.2 K-invariant Functions on H and W The symplectic group Vs(q> F) acts on C[W] and C[K] via n · i (z) = i (n1 z) and n · i (z> w) = i (n 1 · (z> w)) = i (n 1 }> w)= For subgroups N of Vs(q> F), we let C[W]N and C[K]N denote the sets of N-fixed elements in C[W] and C[K], respectively. These are easily seen to be subalgebras of C[W] and C[K] with respect to the convolutions _d and B. Definition 2.7. Given a subgroup N of Vs(q> F), we say that (N> K) is a Gelfand pair when C[K]N is a commutative algebra under convolution. Remark 2.8. One can identify C[K]N with the algebra C[N\J@N] of N-biinvariant functions on the semidirect product J = N n K. So (J> N) is a Gelfand pair in the traditional sense when Definition 2.7 applies. Proposition 2.9. Let N be a subgroup of Vs(q> F). Then (N> K) is a Gelfand pair if and only if kk0 5 (Nk0 )(Nk) for all k> k0 5 K. Proof. This result is the discrete analog of Theorem 1.12 in [BJR90].

t u

Proposition 2.10. Let N be a subgroup of Vs(q> F). Then (N> K) is a Gelfand pair if and only if (C[W]N > _ ) is commutative for all 5 F× . Proof. First note that (C[W]N > _0 ) is, in any case, commutative since _0 is the standard (untwisted) convolution on C[W]. To complete the proof, use (2.14) together with Lemma 2.6 and the the obvious identity (n · i )d = n · id > (n 5 Vs(q> F)> i 5 C[K]> d 5 F).

t u

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Chal Benson and Gail Ratclig

Two immediate but useful properties of Gelfand pairs are noted in the following lemma. Lemma 2.11. Let N1 and N2 be a pair of subgroups of Vs(q> F) and suppose that (N1 > K) a Gelfand pair. (a) If N1 N2 , then (N2 > K) is a Gelfand pair. (b) If N1 , N2 are conjugate in Vs(q> F), then (N2 > K) is a Gelfand pair.

2.3 Gelfand Pairs and the Oscillator Representation 2.3.1 Operator-Valued Fourier Transform on C[W] For i 5 C[W] and 5 F× , let (i ) denote the operator X (i ) = i (z) (z)

(2.15)

z5W

odinger representation on C[W]. Here (z) = (z> 0), and is the Schr¨ (2.8). The following standard result is easily verified. Lemma 2.12. (i _ j) = (i ) (j) for i> j 5 C[W] and 5 F× . Lemma 2.13. The map : C[W] $ Hqg(C[Fq ]) is a vector space isomorphism for each 5 F× . In fact t 3q@2 is a unitary isomorphism of C[W] onto Hqg(C[Fq ]) equipped with the Hilbert—Schmidt inner product hW> ViKV = wu(W V )= Proof. The set {t q@2 u : u 5 Fq } is an orthonormal basis for C[Fq ] with (x> y)u = # (u · y 21 x · y)ux = So for i> i 0 5 C[W] we compute X h (i )> (i 0 )iKV = t q h (i )u > (i 0 )u iFq u5Fq

=

X

u5Fq ;z>z0 5W

i (z)> i 0 (z0 )t q h (z)u > (z0 )u iFq =

A calculation using (2.8) shows X h (z)u > (z0 )u iFq = z>z0 = u5Fq

Thus

2 Gelfand Pairs Associated with Finite Heisenberg Groups

h (i )> (i 0 )iKV = t q

X

z5W

21

i (z)i 0 (z) = t 3q hi> i 0 iW =

This shows t 3q@2 : C[W] $ Hqg(C[Fq ]) is unitary, hence injective. As the spaces C[W] and Hqg(C[Fq ]) have equal dimension, it follows that is t u an isomorphism of C[W] onto Hqg(C[Fq ]).

2.3.2 Oscillator Representation Recall that $ : Vs(q> F) $ X (C[Fq ]) denotes the oscillator representation, characterized by Equation (2.10). Now for n 5 Vs(q> F), let $ e (n) be the operator on Hqg(C[Fq ]) defined as $ e (n)W = $ (n)W $ (n)1

(2.16)

One checks that $ e defines a unitary representation of Vs(q> F) on the Hermitian vector space (Hqg(C[Fq ]> h·> ·iKV ). Moreover for n 5 Vs(q> F) and i 5 C[W], one has X X i (n1 z) (z) = i (w) (nw) = $ e (n) (i )> (2.17) (n · i ) = z5W

w5W

since (nw) = $ (n) (w)$ (n)1 . So the isomorphism : C[W] $ Hqg(C[Fq ])

intertwines the natural representation of Vs(q> F) on C[W] with $ e .

Definition 2.14. Let N be a subgroup of Vs(q> F). For 5 F× , we define e (N) in Hqg(C[Fq ]) as the commutant C>N of $

C>N = Hqg(C[Fq ])$e (N) = {W 5 Hqg(C[Fq ]) : $ (n)W = W $ (n) ;n 5 N}= Note that C>N is a subalgebra of Hqg(C[Fq ]).

Proposition 2.15. yields an algebra isomorphism of (C[W]N > _ ) onto C>N . Proof. Taken together, Lemmas 2.12 and 2.13 show that : C[W] $ Hqg(C[Fq ]) is an algebra isomorphism of (C[W]> _ ) onto Hqg(C[Fq ]). Equat u tion (2.17) shows that maps C[W]N onto C>N . Proposition 2.16. Let N be a subgroup of Vs(q> F) and 5 F× . Then $ |N is multiplicity free if and only if C>N is commutative. Proof. Suppose that $ |N is multiplicity free. So C[Fq ] has a canonical decomposition into pairwise inequivalent $ (N)-irreducible subspaces:

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Chal Benson and Gail Ratclig

C[Fq ] = S1 · · · Sp say. Schur’s Lemma shows that each operator W 5 C>N must preserve the Sm ’s and act by a scalar on each. Any two such operators commute with one another. Next suppose that $ |N is not multiplicity free. Hence C[Fq ] has a decomposition of the sort C[Fq ] = Z1 Z2 Y> where Z1 , Z2 , Y are $ (N)-invariant and Z1 , Z2 are $ (N)-irreducible and equivalent. Thus C>N contains a copy of JO(2> F), and it fails to be commutative. t u Definition 2.17. We say that a subgroup N of Vs(q> F) is $-multiplicity free if the restriction $ |N of the oscillator representation to N is multiplicity free for all 5 F× . Together, Propositions 2.10, 2.15, and 2.16 imply the following. Theorem 2.18. Let N be a subgroup of Vs(q> F). Then (N> K) is a Gelfand pair if and only if N is $-multiplicity free. When applying Definition 2.17, it suces to check that $ |N is multiplicity free for at most two values of . Proposition 2.19. A subgroup N of Vs(q> F) is $-multiplicity free if and only if $1 |N and $% |N are multiplicity free for any fixed choice of % 5 F× that is not a square. Moreover, when t 3 mod 4, it suces that $1 |N be multiplicity free. Proof. Proposition 2.2 implies that each oscillator representation $ is unitarily equivalent to one of $1 or $% . If 1 is not a square in F, equivalently when t 3 mod 4, we can take % = 1. But $1 is contragredient to $1 by t u (2.11). So when $1 |N is multiplicity free, so is $1 |N . Remark 2.20. We do not know of an example where $1 |N is multiplicity free but N fails to be $-multiplicity free.

2.4 Counting and Convolving K-orbits in W Let denote the natural (unitary) representation of Vs(q> F) on C[W]: (n)i (z) = n · i (z) = i (n1 z)> and $ e : Vs(q> F) $ X (Hqg(C[Fq ]) be as in (2.16). We have seen that and $ e are unitarily equivalent via t 3q@2 , a multiple of the operator-valued

2 Gelfand Pairs Associated with Finite Heisenberg Groups

23

Fourier transform. There is another viewpoint on this equivalence. Consider the standard isomorphism : C[Fq ] C[Fq ] $ Hqg(C[I q ])>

(i *)(j) = *(j)i=

One checks easily that • is unitary. (As before, Hqg(C[I q ]) carries the Hilbert—Schmidt inner product, and we give C[Fq ] C[Fq ] the tensor product of h·> ·iFq with its dual ¡ inner product.)¢ e (n)(i *). • $ (n)i $ (n)* = $ So establishes a unitary equivalence

$ $ ' $ e >

and the composite t 3q@2 1 yields

$ $ ' =

(2.18)

This basic fact plays a central role in [How].

2.4.1 Counting Orbits Now let N be a subgroup of Vs(q> F) and for 5 F× decompose $ |N : X $ |N ' p> > p> = pxow(> $ |N )= b 5N

Writing g = dim(), one has X p> g = dim(C[Fq ]) = t q = b 5N

(2.19)

Also, applying (2.18):

|N ' ($ $ )|N '

X

p> p0 > ( 0 ) =

b > 0 5N

We know that () has a one-dimensional space of N-fixed vectors and 6 . So that ( ( 0 ) )N = 0 when 0 ' X dim(C[W]N ) = p2> = b 5N

But dim(C[W]N ) = |W@N|, the number of N-orbits in W. (Indeed, if Nz1 > = = = > Nzu are the distinct N-orbits in W, then the characteristic functions {Nz1 > = = = > Nzu } form a basis for C[W]N .) So now

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Chal Benson and Gail Ratclig

X

b 5N

p2> = |W@N|=

(2.20)

Proposition 2.21. If N is $-multiplicity free (equivalently (N> K) is a Gelfand pair), then we must have • |W@N| t q and • |N| t q + 1. Proof. Suppose that |W@N| A t q . Using Equations (2.19) and (2.20), one obtains X X X p2> A p> g p> = b b b 5N

5N

5N

b Hence N fails to be $-multiplicity So we must have p> 2 for some 5 N. q free unless |W@N| t . Next observe that t 2q 1 > |W@N| 1 + |N|

since {0} is a N-orbit in W and W\{0} contains at least in view of the inequality |W@N| t q , we must have 1+

t 2q 1 |N|

N-orbits. So

t 2q 1 t 2q 1 t q =, q |N| =, t q + 1 |N|> |N| t 1

as claimed.

t u

Proposition 2.22. Let N be an Abelian subgroup of Vs(q> F). Then |W@N| t q and N is $-multiplicity free if and only if |W@N| = t q . Moreover, this is possible only when |N| t q + 1. b So Equations Proof. When N is Abelian, we have g = 1 for all 5 N. (2.19) and (2.20) become X X q p = t > p2> = |W@N|= > b b 5N

P

5N

P

q As 5Nb p2> b p> , we conclude that |W@N| t must hold. 5N P P 2 Also, N is $-multiplicity free if and only if 5Nb p> = 5Nb p> . Equivalently, |W@N| = t q must hold. Proposition 2.21 shows, moreover, that this t u implies |N| t q + 1.

2.4.2 Convolving Orbits The characteristic functions {Nz : Nz 5 W@N}

2 Gelfand Pairs Associated with Finite Heisenberg Groups

25

for the N-orbits in W yield an (orthogonal) basis for C[W]N . We compute X Nz _ Nz0 (w) = Nz (w v)Nz0 (v)# (21 [w> v]) v5W

=

X

# (21 [w> v])=

(2.21)

v5(wNz)_Nz0

In view of Proposition 2.10, (N> K) will be a Gelfand pair if and only if Nz _ Nz0 (w) = Nz0 _ Nz (w) for all z> z0 > w 5 W and all 5 F× . It is enough to consider z 6= 0 6= z0 since N0 = 0 is a two-sided identity in (C[W]> _ ). Moreover, we can take w 6= 0 because, in any case, i _ j(0) = i _ j(0) for functions i> j 5 C[W]. This discussion yields the following. Lemma 2.23. (N> K) is a Gelfand pair if and only if X X # ([w> v]) = # ([w> v]) v5(wNz)_Nz0

v5(wNz0 )_Nz

for all z> z0 > w 5 W\{0} and 5 F× .

Remark 2.24. By Proposition 2.19, it suces to check the condition in Lemma 2.23 for at most two values of the parameter . Lemma 2.25. Suppose that there are N-invariant subspaces X and Y in W with X _ Y = 0> [X > Y] 6= 0= Then (N> K) is not a Gelfand pair. Proof. Choose points z 5 X and z0 5 Y with [z> z0 ] = 1 and let w = z + z0 . We have (w Nz) _ Nz0 = {z0 } =

Indeed, suppose that v 5 (w Nz) _ Nz0 . Thus for some n> n0 5 N, w nz = v = n0 z0

and hence z nz = n0 z0 z0 =

But z nz 5 X and n 0 z0 z0 5 Y. So v = n0 z0 = z0 since X _ Y = 0. Thus for these choices of z, z0 , and w, we have X #1 ([w> v]) = #1 ([z + z0 > z0 ]) = #1 (1) = h2l@s = v5(wNz)_Nz0

Likewise X

#1 ([w> v]) = #1 ([z + z0 > z]) = #1 (1) = h2l@s =

v5(wNz0 )_Nz

As s is odd, these values are necessarily dierent. So (N> K) fails to be a Gelfand pair in view of Lemma 2.23. u t

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Corollary 2.26. Let (N> KW ) be a Gelfand pair where N acts reductively but nonirreducibly on W. So W decomposes as a direct sum of N-invariant subspaces, W = W1 · · · Wc say. Let Nm JO(Wm ) denote the restriction of N to Wm . Then:

(a) [Wl > Wm ] = 0 for all l 6= m. (b) Each Wm is a symplectic subspace of W and Nm Vs(Wm ). (c) Each (Nm > KWm ) is a Gelfand pair.

Corollary 2.27. Suppose we are given symplectic vector spaces (Wm > [·> ·]m ) and subgroups Nm Vs(Wm ) for m = 1> = = = c. We form the symplectic direct sum Z = W1 · · · Wc >

[(w1 > = = = > wc )> (w10 > = = = > wc0 )] =

c X [wm > wm0 ]m m=1

and let N Vs(W) denote the product N = N1 × · · · × Nc = Then, (N> KW ) is a Gelfand pair if and only if (Nm > KWm ) is a Gelfand pair for m = 1> = = = > c. Recall that a subspace X of the symplectic vector space W is said to be isotropic if [X > X ] = 0. Corollary 2.28. Suppose that N acts reductively on W and that the action preserves a nonzero isotropic subspace X . Then, (N> K) is not a Gelfand pair. Proof. As N acts reductively on W, we have W = X Y for some N-invariant subspace Y. As X is isotropic, we necessarily have [X > Y] 6= 0. t u

2.5 Examples 2.5.1 Symplectic Groups A trivial application of Lemma 2.23 shows that (Vs(q> F)> Kq (F)) is a Gelfand pair. Indeed, for N = Vs(q> F), we have only one nonzero N-orbit. That is, Nz = W\{0} for all z 6= 0 in W. Equation (2.20) now shows that the oscillator representation $ must decompose into exactly two inequivalent irreducible constituents. This fact is well-known. (See [How], [Neu02].) The $ (Vs(q> F))-irreducible subspaces of C[Fq ] can be identified as follows. The matrix L belongs to the center of Vs(q> F), and the first formula from Theorem 2.3 shows that $ (L) = vjq((1)q )W where

2 Gelfand Pairs Associated with Finite Heisenberg Groups

27

W i (u) = i (u)= So the eigenspaces for W must be $ (Vs(q> F))-invariant. These are the spaces of even and odd functions. As $ has exactly two irreducible components, these spaces are necessarily irreducible. To obtain more interesting examples, we must consider smaller subgroups of Vs(q> F).

2.5.2 General Linear Groups The group JO(q> F) embeds diagonally in Vs(q> F) via {Dgldj : D 5 JO(q> F)}. (See (2.12).) The action of JO(q> F) on W = Fq × Fq preserves the isotropic subspaces X = Fq × {0}>

Y = {0} × Fq =

So (JO(q> F)> Kq (F)) is not a Gelfand pair, in view of Corollary 2.28. More generally, if N Vs(q> F) is conjugate in Vs(q> F) to a subgroup of JO(q> F), then (N> K) is not a Gelfand pair. For q 2, there are t + 3 distinct JO(q> F)-orbits in W, namely {(0> 0)}>

(Fq \{0}) × {0}>

{0} × (Fq \{0})>

{(x> y) : x 6= 0 6= y> x · y = 0} and {(x> y) : x · y = d}

for each d 5 F× . So here |W@JO(q> F)| ? t q , but (JO(q> F)> Kq (F)) is not a Gelfand pair. This shows that, for N non-Abelian, there can be no converse for Proposition 2.21.

2.5.3 Borel Subgroups A subgroup of Vs(q> F) conjugate to ½ ¸ ¾ D 0 w E= : D 5 JO(q> F) lower triangular, D F symmetric F (Dw )1 is called a Borel subgroup. Proposition 2.29. If N is a subgroup of Vs(q> F) that contains a Borel subgroup, then (N> K) is a Gelfand pair. Proof. It suces to show that (E> K) is a Gelfand pair. This can be done using Proposition 2.9. There are exactly 2q nonzero E-orbits in W = Fq ×Fq :

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W\{0} =

q ]

m=1

q ]

Eum ]

Evm

where um = (em > 0)> vm = (0> em )

m=1

and {em } is the standard basis for Fq . Suppose that z 5 Eul and w 5 Eum . One has z ± w 5 Eumin(l>m) , and hence z + w = n(z w) for some n 5 E. Let n0 = n and note that n 0 5 E, since L 5 E. Now (z> v)(w> w) = (z + w> v + w + 21 [z> w]) = (nz nw> v + w + 21 [nz> nw]) = (n 0 w + nz> w + v + 21 [n0 w> nz]) = (n 0 w> w)(nz> v)= So (z> v)(w> w) 5 (E(w> w))(E(z> v)). Similar calculations apply for other combinations of orbits. t u

One can also give an explicit description of the algebra (C[W]E > _ ) by determining the twisted convolution of pairs of characteristic functions for E-orbits in W. We adopt the notation X # (21 [z> w]) (z 5 W> 5 F× )= H (z) = w5Ez

Brute force calculation yields the following. ½ H (umax(l>m) P )Eumin(l>m) Eul _ Eum = H (ul ) cl Euc Eul ½ H (vmin(l>m) P )Evmax(l>m) Evl _ Evm = H (vl ) cl Evc Evl

for l 6= m for l = m for l 6= m for l = m

Eum _ Evl = Evl _ Eum = H (vl )Eum

These formulas show, in particular, that (C[W]E > _ ) is commutative, as guaranteed by Proposition 2.29.

2.5.4 Unitary Groups e denote a quadratic extension of the field F = Ft . Up to isomorphism, Let F e F is a copy of Ft2 . More concretely, we choose any nonsquare % 5 F× \(F× )2

in F and take

s e = F( %)= F

2 Gelfand Pairs Associated with Finite Heisenberg Groups

29

e$F e will be written as } 7$ }. One has } = } t and The Galois involution F s s d + e % = d e % (d> e 5 F)= e f h·> ·i) be a (finite dimensional) Hermitian vector space over F. Let (W> e f f That is, h·> ·i : W × W $ F is

• F-bilinear and nondegenerate with • hz> z0 i = hz> z0 i and hz> z0 i = hz0 > zi

e f 5 F). (z> z0 5 W,

e f is the set of F-linear The unitary group X (W) operators preserving h·> ·i. f One obtains a symLet W denote the underlying F-vector space for W. plectic form on W via ´ ´ 1 ³ 1 ³ [z> z0 ] = s hz> z0 i hz> z0 i = s hz0 > zi hz> z0 i = 2 % 2 % s s (Writing hz> z0 i 5 F( %) as hz> z0 i = hz> z0 iu + hz> z0 il %, one has [z> z0 ] = f is a subgroup of Vs(W), the symplectic group for hz> z0 il .) Clearly, X (W) (W> [·> ·]). We will prove that: f KW ) is a Gelfand pair. Proposition 2.30. (X (W)>

e It is well-known that a given finite dimensional F-vector space admits exactly one Hermitian inner product, up to equivalence.1 In fact, one can find an orthonormal basis B = {h1 > = = = > hq }

eq , we have the f h·> ·i) with hhl > hm i = l>m . Using B to identify W f with F for (W> usual formula hz> z0 i = }1 }10 + · · · + }q }q0 = Let e m> fm = Fh W

Now

fm viewed as an F-vector space> Wm denote Z

im =

s %hm =

• {hm > im } is a basis for Wm , • {h1 > = = = > hq > i1 > = = = > iq } is a symplectic basis for W (i.e., [hl > hm ] = 0 = [il > im ], [hl > im ] = l>m ), and • W = W1 · · · Wq is a symplectic direct sum. fm . We consider fm is a Hermitian inner product on W The restriction of h·> ·i to W f the subgroups X (Wl ) Vs(Wm ) and their direct product

1 In contrast, the complex vector space Cq admits b(q+2)@2c inequivalent Hermitian inner products. These yield distinct unitary groups X (u> v) with u + v = q. The analogs for these Hermitian inner products in the finite fields context are, however, mutually equivalent.

30

Chal Benson and Gail Ratclig

f1 ) × · · · × X (W fq ) X (W) f Vs(W)> X (W

f preserving the decomposition W = W1 · · · Wq . the subgroup of X (W) The following result evidently implies Proposition 2.30. ¢ ¡ fq )> KW is a Gelfand pair. f1 ) × · · · × X (W Proposition 2.31. X (W

¡ ¢ fm )> KW Proof. In view of Corollary 2.27, it suces to show that each X (W m e K1 (F)) is a Gelfand is a Gelfand pair. This amounts to showing that (X (F)> e carries the Hermitian inner product pair, where F h}> } 0 i = }} 0 =

Now

e = { 5 F e× : = 1}> X (F)

is the kernel of the norm mapping

e $ F> Q :F

Q () =

e So X (F) e is, in pare× for the field F. restricted to the multiplicative group F e belong to a common ticular, Abelian. Moreover, a pair of points }> } 0 5 F e e if and only if Q (}) = Q (} 0 ). So X (F)-orbit in F e (F)| e = |Q (F)| e = |F| = t> |F@X

as it is well-known that Q is surjective. Proposition 2.22 now implies that e K1 (F)) is a Gelfand pair as desired. t u (X (F)> e is the kernel of the epimorphism Q : F e× $ F× , it is Remark 2.32. As X (F) cyclic of order t + 1.

Our final result asserts that the Gelfand pair in Proposition 2.31 is minimal. The analogous theorem for real Heisenberg groups Kq (R) is due to Leptin [Lep85].

Proposition 2.33. (N> KW ) fails to be a Gelfand pair for all proper subf1 ) × · · · × X (W fq ). groups N of the torus X (W

e and W eq with the usual Hermitian inner f= F fm = F Proof. We can take W product. Now the torus e × · · · × X (F) e W = X (F)

f Namely, one has coincides with a W -orbit in W.

W · z = W for z = (1> 1> = = = > 1).

2 Gelfand Pairs Associated with Finite Heisenberg Groups

31

Let N denote a proper subgroup of W . The W -orbit W ·z is a disjoint union of |W @N| orbits for the subgroup N. These correspond to the cosets of N in W . As (W> KW ) is a Gelfand pair, we have |W@W | = t q and |W@N| t q 1 + |W @N| A t q = Thus (N> KW ) fails to be a Gelfand pair by Proposition 2.22.

t u

References [BJR90] Chal Benson, Joe Jenkins, and Gail Ratclig. On Gel0 fand pairs associated with solvable Lie groups. Trans. Amer. Math. Soc., 321(1):85—116, 1990. [Fol89] Gerald B. Folland. Harmonic analysis in phase space, volume 122 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1989. [G´ er77] Paul G´ erardin. Weil representations associated to finite fields. J. Algebra, 46(1):54—101, 1977. [Hel84] Sigurdur Helgason. Groups and geometric analysis, volume 113 of Pure and Applied Mathematics. Academic Press Inc., Orlando, FL, 1984. (Integral geometry, invariant digerential operators, and spherical functions.) [How] Roger E. Howe. Invariant theory and duality for classical groups over finite fields, with applications to their singular representation theory. Unpublished manuscript. [How73] Roger E. Howe. On the character of Weil’s representation. Trans. Amer. Math. Soc., 177:287—298, 1973. [Lep85] Horst Leptin. A new kind of eigenfunction expansions on groups. Pacific J. Math., 116(1):45—67, 1985. [LN] Rudolf Lidl and Harald Niederreiter. Finite fields, volume 20 of Encyclopedia of Mathematics and its Applications. Second edition. Cambridge University Press, Cambridge, 1997. [Mac] I. G. Macdonald. Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. Second edition. Oxford University Press, New York, 1995. [Neu02] Markus Neuhauser. An explicit construction of the metaplectic representation over a finite field. J. Lie Theory, 12(1):15—30, 2002. [SA87] Jorge Soto-Andrade. Geometrical Gel0 fand models, tensor quotients, and Weil representations. In The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986), volume 47 of Proc. Sympos. Pure Math., pages 305—316. Amer. Math. Soc., Providence, RI, 1987. [Shi80] Ken-ichi Shinoda. The characters of Weil representations associated to finite fields. J. Algebra, 66(1):251—280, 1980. [Ter99] Audrey Terras. Fourier analysis on finite groups and applications, volume 43 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1999. ` B. Vinberg. Commutative homogeneous spaces of Heisenberg type. Tr. Mosk. [Vin03] E. Mat. Obs., 64:54—89, 2003. [Yak06] Oksana Yakimova. Principal Gelfand pairs. Transform. Groups, 11(2):305—335, 2006.

Chapter 3

Groups with Atomic Regular Representation Keith F. Taylor To Larry, my guide

Abstract A locally compact group J is called an [AR] group if the left regular representation of J is the direct sum of irreducible representations. A number of results related to the construction of [AR] groups and their properties are surveyed. The relevance of [AR] groups in the occurrence of nontrivial projections in O1 (J) and in higher-dimensional continuous wavelet transforms is presented.

3.1 Introduction In 1976, Larry Baggett and I studied the vanishing of matrix coecients of representations of locally compact groups. This problem led us to consider groups with the property that their left regular representation decomposes as a direct sum of irreducible unitary representations (in this article, representation always means continuous unitary representation); that is, groups with atomic regular representation. Of course, compact groups have atomic regular representation, but we were concerned about the noncompact ones. One may feel that such groups are rare and, in a generic sense, they are. However, there are naturally occurring constructions that result in noncompact groups with atomic regular representations. Moreover, the resulting groups play a role in understanding two seemingly independent topics: construction of projections in the O1 -group algebra and generalizing the continuous wavelet transform to Rq or more general locally compact Abelian groups. This survey introduces groups with atomic regular representations, or [AR] groups, and describes a number of such topics that are of personal interest to me.

Keith F. Taylor Department of Mathematics and Statistics, Dalhousie University, Halifax, NS, B3H 4J1, Canada; e-mail: [email protected]

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Since many readers of this volume may not be familiar with all of the notation and terminology of abstract harmonic analysis, we begin with an introduction to the basic spaces and the theory of representations. In Section 3.3, the Fourier and Fourier—Stieltjes algebras are defined. Section 3.4 is devoted to square-integrable representations and the powerful theorem of Duflo and Moore. We then formally define [AR] groups and provide a recipe for cooking up a variety of [AR] groups in Section 3.5. The Riemann—Lebesgue Theorem tells us that the Fourier transform of an integrable function vanishes at infinity. This holds on any locally compact Abelian (LCA) group and even has useful generalizations to non-Abelian groups. However, on many LCA groups there exist finite Borel measures that are singular with respect to Haar measure, but whose Fourier—Stieltjes transform still vanishes at infinity. Think about the rotation invariant measure on the unit circle in the plane and what happens to a wave that scatters o a circular object and dissipates at infinity. In Section 3.6, this issue is formulated and the work with Baggett is introduced. The final three sections are devoted to other topics where the unique properties of [AR] groups turned out to be essential. To keep some statements clean, I make the assumption that any locally compact group considered is second countable. In most cases, this assumption is unnecessary. No proofs are provided.

3.2 Preliminaries There is a veritable menagerie of spaces that arise in abstract harmonic analysis and much interest centers around the interrelationships among the spaces. A good reference for the basic concepts introduced in this section is [11]. Let J be a locally compact group equipped with R left Haar measure. Integration with respect to this measure is denoted by J i ({)g{ for any function i on J for which the integration makes sense. As a locally compact space, J carries the following function spaces: F(J), the continuous complex-valued functions on J; Fe (J), the bounded elements of F(J); F0 (J), those that vanish at infinity; and F00 (J), the elements of F(J) with compact support. Both Fe (J) and F0 (J) are Banach spaces when equipped with the supremum norm || · ||4 , where ||i ||4 = sup{|i ({)| : { 5 J} for i : J $ C. Of in F0 (J) with respect to || · ||4 -convergence. course, F00 (J) is dense R R (J), i ({)g{ is well-defined and for any | 5 J, J i (|{)g{ = For i 5 F 00 J R i ({)g{. This is the left invariance of the Haar integral. There exists a J continuous homomorphism J : J $ C, called the modular function of J, such that Z Z i ({|)g{ = i ({)g{> (3.1) J (|) J

J

3 Groups with Atomic Regular Representation

35

for all i 5 F00 (J) and | 5 J. The modular function R also helps us with R inversion of the variable of integration: J i ({)g{ = J i ({1 )J ({1 )g{. The group J is called unimodular when J = 1. 1 s ? 4, define the Os -norm on F00 (J) by ||i ||s = R For any s ( J |i ({)| g{)1@s , for all i 5 F00 (J). Let Os (J) denote the completion of (F00 (J)> || · ||s ) as a normed linear space. As usual, the elements of k> n 5 O2 (J), define Os (J) are R treated as functions in their own right. For 2 hk> ni = J k({)n({)g{. With this inner product, O (J) is a Hilbert space. For i 5 O1 (J) and j 5 Os (J)> 1 s ? 4, the convolution i j of i and j is defined by Z i (|)j(| 1 {)g|> (3.2) i j({) = J

which converges for almost every { 5 J. Then i j 5 Os (J) and ||i j||s ||i ||1 ||j||s . In particular, O1 (J) is a Banach algebra under convolution. There is an isometric involution, i $ i on O1 (J) given by i ({) = J ({1 )i ({1 )> { 5 J.

A (continuous, unitary) representation of J is a pair (> H ), where H is a Hilbert space and is a homomorphism of J into the group of unitary operators on H that is continuous with respect to the weak operator topology. Often, the representation may be just named as with H then assumed. If is a representation of J and if > 5 H , define the matrix coecient function *> on J by (3.3) *> ({) = h({)> i> for { 5 J. The weak operator continuity requirement on representations simply means that each matrix coecient function *> is continuous. Since |h({)> i| |||| · ||||, it may be seen that *> 5 Fe (J) for all representations of J and any > 5 H . If is a representation of J and D is a subset of H , then D is called -invariant if ({)D D, for all { 5 J. For 5 H , (J) = {({) : { 5 J} is -invariant and so is K , the closed linear span of (J). If there exists a 5 H such that K = H , then is called a cyclic representation of J and a cyclic vector for . A representation of J is called irreducible if {0} and H are the only -invariant closed subspaces of H . It is easy to see that is irreducible if and only if every nonzero vector in H is a cyclic vector for . When expressed in terms of matrix coecients, this becomes a useful criterion to test for irreducibility of a given representation. We include it with two other standard characterizations in the next proposition that are essentially Schur’s Lemma. Note that B(H ) denotes the space of bounded linear operators on H , and L denotes the identity operator on H . As an algebra, the center of B(H ) is CL = {L : 5 C}. Proposition 3.1. Let be a representation of J. Then, the following are equivalent:

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Keith F. Taylor

(a) is irreducible. (b) The weak operator closure of the linear span of {({) : { 5 J} is B(H ). (c) {W 5 B(H ) : W ({) = ({)W> for all { 5 J} = CL. (d) *> 6= 0, for all > 5 H \ {0}. If and are two representations of J such that there exists a unitary map X : H $ H with X ({) = ({)X>

for all { 5 J, then and are called equivalent representations. Let [] denote the class of all representations equivalent to . Let b = {[] : is an irreducible representation of G}= J

b is called the dual space of J. If J is Abelian, then the equivalence The set J of (a), (b), and (c) in Proposition 3.1 forces any irreducible representation b consists of one-dimensional representations, to be one-dimensional. Thus, J or characters, and is a locally compact group in its own right under pointwise multiplication as the group product and equipped with an appropriate topology. If is any representation of J, it can be integrated to define a map, also denoted , of O1 (J) into B(H ), the space of bounded linear operators on H . That is, for i 5 O1 (J), Z i ({)h({)> ig{

(> ) $

J

is a bounded conjugate bilinear form on H , and, thus, there exists (i ) 5 B(H ) so that Z h(i )> i = i ({)h({)> ig{ (3.4) J

for all > 5 H . It is elementary to verify that i $ (i ) is a linear map of O1 (J) into B(H ) such that ||(i )|| ||i ||1 > (i j) = (i )(j)> and (i ) = (i ) , for all i> j 5 O1 (J). Therefore, to each representation of J there corresponds a continuous homomorphism, also denoted , of O1 (J) into B(H ) that respects the involution, a so-called -representation of O1 (J). For each i 5 O1 (J), define ||i || = sup {||(i )||}, where the supremum is over all representations of J. This defines a new norm || · || on O1 (J) that is dominated by || · ||1 . The group F -algebra of J is the normed -algebraic completion of (O1 (J)> || · || ) and is denoted F (J). The left regular representation of J is defined by translations of O2 (J). That is, for { 5 J, ({) is the unitary operator defined by ({)j(|) = j({1 |), for all | 5 J> j 5 O2 (J). One checks that is a representation of J. When integrated up to O1 (J), gives the module action of O1 (J) on O2 (J) by left convolution. That is, (i )j = i j, for all i 5 O1 (J) and j 5 O2 (J). The reduced F -algebra of J is F (J) = (O1 (J)), the closure of {(i ) : i 5 O1 (J)} in B(O2 (J)). In general, F (J) diers from F (J), but they agree when J is an amenable group [18].

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3.3 The Fourier and Fourier—Stieltjes Algebras In [9], Eymard laid the foundations for the study of two of the most important commutative Banach algebras associated with a, not necessarily commutative, locally compact group. The Fourier—Stieltjes Algebra of J is E(J) = {*> : is a representation of J> > 5 H }= Then E(J) is an algebra over C whenR equipped with pointwise defined operations. For * 5 E(J), i $ (*> i ) = J i ({)*({)g{ is a bounded linear functional when O1 (J) is equipped with the F -norm, || · || , so it extends to a continuous linear functional on F (J). This identifies E(J) with F (J) as a vector space and gives E(J) the norm ||*|| = sup{|(*> i )| : i 5 O1 (J)> ||i || 1}= With this norm, E(J) is a Banach algebra. The Fourier algebra of J is D(J) = {*j>k : j> k 5 O2 (J)}. Eymard [9] proved that D(J) is a closed ideal in E(J) and identified it with the predual of the von Neumann algebra generated by the left regular representation of J. More precisely, let Y Q (J) denote the weak operator topology closed subalgebra of B(O2 (J)) generated by {({) : { 5 J}. Each W 5 Y Q (J) defines a bounded linear functional on D(J) such that (W> *j>k ) = hW j> ki, for j> k 5 O2 (J). As an algebra of functions on J, D(J) is a uniformly dense subalgebra of F0 (J). Both D(J) and E(J) are Banach algebras with extremely complicated structure and they have been two of the motivating examples in the promising development of the theory of operator spaces [8]. b and b D(J) = {ib : i 5 O1 (J)} When J is Abelian, with dual group J, b b b and E(J) = {b : 5 P (J)}, where P (J) is the measure algebra of J, the Fourier (resp. Fourier—Stieltjes) transform is an isometric isomorphism. b $ O2 (J)> $ b is the Plancherel transform, then Moreover, if P : O2 (J) 4 b 1 Y Q (J) = PO (J)P .

3.4 Square-Integrable Representations Let be an irreducible representation of J. An element 5 H is called an admissible vector if there exists a nonzero 5 H such that *> 5 O2 (J). If there exists a nonzero admissible vector 5 H , then is called squareintegrable. In that case, the set of all admissible vectors is a dense subspace D of H and, for 5 D , *> 5 O2 (J), for all 5 H . The following significant theorem was established in [7].

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Theorem 3.2. (Duflo and Moore) Let be a square-integrable representation of J. Then there exists a unique operator N on H that is self-adjoint positive (N may be unbounded) and such that (i) domN 1@2 = D , (ii) ({)N({)1 = J ({)1 N, for all { 5 J, and (iii) for 1 > 2 5 D > 1 > 2 5 H, h*1 >1 > *2 >2 iO2 (J) = hN 1@2 1 > N 1@2 2 ih2 > 1 i= If J is unimodular, then D = H , N is bounded and (ii) implies that N is a scalar multiple of the identity. If J is compact, then that scalar is the dimension of H and (iii) is one of the orthogonality relations for irreducible representations of a compact group. Definition 3.3. The operator N in Theorem 3.2 is called the generalized dimension of . If is a square-integrable representation of J, select an admissible vector such that ||N 1@2 || = 1. Define a linear map Y : H $ O2 (J) by Y ({) = *> ({) = h> ({)i= Let V> = {Y : 5 H }. Then Theorem 3.2(iii) implies that V> is a closed subspace of O2 (J), and Y is a unitary map of H onto V> . Also, V> is -invariant and Y establishes the equivalence of with restricted to V> . Thus, any square-integrable representation sits as a subrepresentation of the regular representation. On the other hand, if is an irreducible subrepresentation of , then is square-integrable. Most noncompact groups have no square-integrable representations. However, there exist important noncompact groups, such as VO(2> R) and the ane group of R, which have some square-integrable representations. For VO(2> R), the set of square-integrable representations is known as the Discrete Series, and the regular representation of VO(2> R) is the direct sum of an atomic part (which is a sum of Discrete Series Representations) and a continuous part (which has no irreducible subrepresentations). In the case of the ane group, Ja = {(e> d) : d> e 5 R> d A 0} with group product given by (e1 > d1 )(e2 > d2 ) = (e1 + d1 e2 > d1 d2 ), there are two square-integrable representations + and , and is equivalent to C0 + C0 . This latter notation means that there are two families {Kl+ : l = 1> 2> 3> · · · } and {Kl : l = 1> 2> 3> · · · } of mutually orthogonal closed subspaces of O2 (Ja ), each -invariant, so that O2 (Ja ) =

4 X l=1

Kl+ +

and acting on Kl± is equivalent to ± .

4 X l=1

Kl

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3.5 [AR] Groups Definition 3.4. A locally compact group J is called an [AR] group if the left regular representation of J is the direct sum of irreducible representations. Of course, any compact group is [AR], and we just saw that Ja is an [AR] group. Actually, Ja is representative of a large class of examples of [AR] groups that can be constructed by the following procedure. Let D be an Abelian locally compact group and let K be another locally compact group such that there is a homomorphism : K $ Aut(D), where Aut(D) is the group of automorphisms of D. Further, assume that (d> k) $ k (d) is continuous from D×K into D. Form the semidirect product DoK = {(d> k) : d 5 D> k 5 K}, where (d1 > k1 )(d2 > k2 ) = (d1 k1 (d2 )> d1 d2 ) gives the group product. The so-called Mackey Machine can be used to parameterize D\ o K when a specific regularity assumption is satisfied. (See Section 6.6 of [11] for a readable introduction to the Mackey Machine.) We need to develop a little notation. b is the group of characters of D. The action of K on Since D is Abelian, D b defined by D generates an action (k> ") $ k · " of K on D k · "(d) = "(k1 (d))>

b the orbit of " in D b is O" = {k · " : k 5 K} and the for d 5 D. For " 5 D, stabilizer of " in K is K" = {k 5 K : k·" = "}, which is a closed subgroup of K. The regularity assumption that is needed has a simple formulation when we are assuming all locally compact groups considered are second countable. b so that O" _ is a singleton If there exists a Borel measurable subset D b then we say that the action of K on D b is regular. That for each orbit O" in D, b is, the action is regular when the orbit space D@K has a Borel cross-section. b is open Among the consequences of this assumption are that each orbit in D in its closure, and the map kK" $ k · " is a homeomorphism of K@K" with O" . To obtain [AR] groups, we consider two specific properties (let |V| denote b the Haar measure of a measurable V D): b such that (I) There exists a countable family {Om : m 5 M} of K-orbits in D b \ [^m5M Om ] | = 0= |Om | A 0> for m 5 M> and |D

(II) With (I) holding, for each m 5 M and " 5 Om , K" is [AR]. The following theorem is an easy application of standard techniques in the Mackey Machine and is contained in the first section of [3]. Theorem 3.5. (Baggett and Taylor) If properties (I) and (II) hold for the action of a locally compact group K on an Abelian group D, then D o K is an [AR] group.

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Using this construction, a variety of examples of noncompact [AR] groups are obtained in [3]. All of the [AR] groups highlighted there were nonunimodular and connected. In a similar manner, Mauceri and Picardello constructed families of unimodular [AR] groups in [16]; however, their examples were all totally disconnected. In [2], Baggett studied the interplay of unimodularity and [AR]. By combining his remark after Proposition 1.2 [2] with Theorem 2.3 [2], one obtains the following limitation. Theorem 3.6. (Baggett) If J is a connected unimodular [AR] group, then J is compact. Thus, if one wants to have a noncompact connected [AR] group, then one has to deal with the modular function. However, it is actually the modular function that plays a key role in many of the most interesting phenomena that occur on [AR] groups. Of course, Ja is an obvious example of the above construction. One already gets an interesting variety of additional examples by taking D = R2 and K to be almost any two-dimensional closed subgroup of the general linear group JO(2> R). One only has to be sure that the generic orbits of K do not collapse. An illustrative family of examples was studied in [19]. For 1 s 1, let Ks =

½µ

ds+1 { 0 d

¶

¾ : d> { 5 R> d A 0 =

Each group R2 o Ks has a normal subgroup isomorphic to the three-dimensional Heisenberg group H and is an extension of H by R acting on H as “dilations.” More precisely, realize H as {[{> |> }] : {> |> } 5 R} with product [{1 > |1 > }1 ] [{2 > |2 > }2 ] = [{1 + {2 > |1 + |2 > }1 + }2 + {1 |2 ] = A dilation action of R on H is one of the form h i u>v (w)[{> |> }] = huw {> hvw |> h(u+v)w } > where u and v are fixed real parameters and w 5 R. Groups of the form H ou>v R were studied and classified in [19]. Theorem 3.7. (Schulz and Taylor) Each Hou>v R is isomorphic to R2 oKs , for some 1 s 1. Moreover, the groups R2 oKs > 1 s 1 are mutually nonisomorphic and R2 o Ks is an [AR] group if and only if s 6= 1. c2 , which A good exercise for the reader is to calculate the action of Ks on R 2 can be identified with R , to verify that (I) and (II) hold when 1 ? s 1 and see how the open orbits collapse to lines when s = 1.

3 Groups with Atomic Regular Representation

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3.6 Representations Vanishing at Infinity It was observed earlier that D(J) F0 (J) _ E(J). The question of when equality holds has a long history, and its investigation leads to the study of [AR] groups. Indeed, it was the attempt to characterize J for which D(J) = F0 (J) _ E(J) that generated my initial interest in [AR]. In 1916, Mencho [17] showed that there exists a singular probability measure on T such that b(q) $ 0 as |q| $ 4. So b 5 [F0 (Z) _ E(Z)] \ D(Z). Hewitt and Zuckerman [12] proved that, for an Abelian locally compact group a-Talamanca J, J is compact if and only if D(J) = F0 (J)_E(J). In [10], Fig` showed that, if J is unimodular, then D(J) = F0 (J)_E(J) implies J is [AR]. Since compactness and [AR] coincide for Abelian groups, Fig`a-Talamanca’s result extends that of Mencho and Hewitt and Zuckerman. Larry Baggett and I turned our attention to extending Mencho’s theorem to general J in [4]. We showed that, for any non-[AR] group J, there exists a representation of J that has no subrepresentation in common with the regular representation and a 5 H > |||| = 1 such that *> 5 F0 (J). This is neatly stated as the following theorem. Theorem 3.8. (Baggett and Taylor) Let J be a second countable locally compact group. If D(J) = F0 (J) _ E(J), then J is an [AR] group. One may speculate that [AR] groups are characterized by D(J) = F0 (J)_ E(J). However, in [3], we constructed an [AR] group J and an irreducible representation of J that is not square-integrable but still vanishes at infinity; that is, for 5 H > *> 5 F0 (J). So, if 5 H \ {0}, then *> 5 [F0 (J) _ E(J)] \ D(J). The problem of characterizing groups for which D(J) = F0 (J) _ E(J) remains open.

3.7 Geometric Properties of A(G) and the [AR] Property b Moreover, If J is Abelian, then D(J) is isometrically isomorphic to O1 (J). b is an Abelian group J is [AR] if and only if J is compact, equivalently, J b discrete, equivalently, Haar measure on J is atomic. If ([> ) is a measure space, there are a number of geometric properties of Banach spaces that hold on O1 ([> ) if and only if is atomic. Essentially, these properties distinguish o1 from O1 [0> 1]. When J is non-Abelian, we cannot say that D(J) is O1 ([> ) for some measure space ([> ). However, many analogies hold. In particular, when J is [AR], D(J) shares characteristics with o1 . One striking property that O1 () has only when is atomic is that of being the dual of another Banach space.

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Theorem 3.9. (Taylor [21]) J is an [AR] group if and only if D(J) is a dual Banach space. Definition 3.10. A Banach space H has the Radon—Nikodym Property, RNP, if whenever ([> ) is a finite measure space and is a -continuous vectorvalued measure from [ into H of bounded variation, there exists a Bochner R integrable j from [ into H such that (d) = D jg for every measurable D [. The book by Diestel and Uhl [6] is a good source of information on RNP. There one will find a long list of properties of Banach spaces that turn out to be equivalent to RNP. For a measure space ([> ), O1 () has the RNP if and only if is an atomic measure.

Theorem 3.11. (Taylor [21]) J is an [AR] group if and only if D(J) has the RNP.

3.8 Constructing Projections in L1 (G) A self-adjoint idempotent in the Banach -algebra O1 (J) is called a projection. That is, i 5 O1 (J) is a projection if i i = i = i . This is the case if and only if (i ) is a projection operator on H , for every irreducible representation of J. Since the left regular representation is faithful as a representation of O1 (J), i is a projection if and only if (i ) is a projection operator on O2 (J). In [13] and [14], we developed methods for deciding whether certain groups J admit nonzero projections in O1 (J) and for explicit constructions of projections when they can exist. The most satisfying results apply for the type of [AR] groups constructed in Section 3.5. To keep things simple, we will assume that J = D o K with D Abelian b such that, for and K acting in such a manner that there are "1 > · · · > "q 5 D b 1 l q, K"l = {h}, O"l is open, and |D \ [^l O"l ]| = 0. Then clearly (I) and (II) of Section 3.5 hold and J is an [AR] group. Constructing projections in O1 (J) requires some preparation. Let K denote the modular function of K and let (k) denote the modulus of the automorphism (k) of D. Thus, Z Z j(k (d))gd = j(d)gd> (k) D

D

for any j 5 F00 (D), for example. There is a natural unitary representation of J on O2 (D) given by (d> k)j(e) = (k)1@2 j(k1 (d1 e))>

(3.5)

for all e 5 D, j 5 O2 (D), and (d> k) 5 J. To decompose into irreducibles, b denote the Fourier transform as a unitary map. let F : O2 (D) $ O2 (D)

3 Groups with Atomic Regular Representation

43

Define (d> k) = F(d> k)F 1 , for (d> k) 5 J. Then is a representation equivalent to , and a short calculation using standard properties of the b and (d> k) 5 J, b 5 O2 (D), Fourier transform shows that, for " 5 D, (d> k)(") = (k)1@2 "(d)(k1 · ")=

(3.6)

b to O"l and consider Since each O"l is open, we can restrict Haar measure on D 2 2 b O (O"l ) as a closed subspace of O (D) in the obvious way. Since O"l is an Kb one sees from (3.6) that O2 (O" ) is a -invariant subspace invariant set in D, l 2 b of O (D). Define l (d> k) = (d> k)|O2 (O"l ) , for (d> k) 5 J and for 1 l q. It can be shown that each l is irreducible and, in fact, equivalent to indJ D "l , the representation of J induced from "l . P 2 If Hl2 = {i 5 O2 (D) : F(i ) 5 O2 (O"l )}, then O2 (D) = l Hl and each 2 Hl2 . Hl is -invariant. Let l be the subrepresentation of associated Pwith Note that l is equivalent to l , for 1 l q. Not P only is = l l , but the left regular representation is equivalent to l C0 · l . In analogy with the projections that arise from the orthogonality relations associated with irreducible representations of compact groups, one can hope that there might be projections associated with the square-integrable representations, l , in light of Theorem 3.2. Investigating this potential in [13] and [14] led to defining, for z 5 O2 (D)> ¸1@2 (k) iz (d> k) = hz> l (d> k)zi= (3.7) K (k) Definition 3.12. A projection generating function (PGF) associated with O"l is a z 5 O2 (D) that satisfies: (i) supp( z) b O"l , R b 1 · "l )|2 gk = 1 , (ii) K |z(k (iii) iz 5 O1 (J), with iz defined by (3.7).

b is v(i ) = { 5 J b : For a projection i in O1 (J), the support of i in J 1 (i ) 6= 0}. For projections i and j in O (J), we write i j if i j = i . A projection i in O1 (J) is called minimal if i 6= 0 and, for any projection j in O1 (J), j i implies j = 0 or j = i . Theorem 3.13. (Kaniuth and Taylor [14]) With the above notation, the following three facts hold: (a) Let z be a PGF associated with the orbit O"l . Then iz , as defined in (3.7), is a minimal projection. (b) Every minimal projection in O1 (J) is of the form iz , with z a PGF associated with O"l , for some 1 l q. (c) Every projection in O1 (J) with v(i ) = l is the orthogonal direct sum of minimal projections. Many questions remain open on how completely one may describe the projections in O1 (J) for a general [AR] group J.

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3.9 Constructing Continuous Wavelet Transforms on Rn The final area to mention in which [AR] groups play a central role is the general construction of continuous wavelet transforms of O2 (Rn ) or O2 (D) for a general locally compact Abelian group D. Again, as in the previous section, assume that J = D o K with D Abelian and K acting in such a b such that, for 1 l q, K"l = {h}, manner that there are "1 > · · · > "q 5 D b \ [^l O"l ]| = 0. The results in this section are based on O"l is open, and |D [5]. There we assumed D = Rn , but the generalization is direct. Let be the representation of J on O2 (D) defined in (3.5). For z 5 O2 (D) and (d> k) 5 J, define zd>k = (d> k)z. For each l 5 {1> 2> · · · > q}, let l be the irreducible subrepresentation of associated with Hl2 . Since l is a square-integrable representation, the theorem of Duflo and Moore applies. A key point is to identify the operator N that arises in Theorem 3.2. This operator comes from the relationship between two natural measures on the orbit O"l . Since k $ k·"l is a homeomorphism of K with O"l , we can transfer the left Haar measure of K to O"l . Call this measure b restricted to O"l . Let l = [gl @gl ], l . Let l denote Haar measure on D the Radon—Nikodym derivative. Then l is a positive continuous function on O"l . Therefore, l operates on O2 (O"l ) by pointwise multiplication as an unbounded operator with domain { 5 O2 (O"l ) : 5 O2 (O"l )}. Now define Nl on Hl2 by Nl j = F 1 ( l jb), for any j 5 Hl2 such that l jb 5 O2 (O"l ) (these are the admissible vectors in the language of Section 3.4). Select zl 5 Hl2 such that ||Nl zl ||22 = || l zl ||22 = 1. The set Z = {z1 > · · · > zq } forms a multiwavelet for a continuous wavelet transform of O2 (D) in the sense expressed by (b) in the following theorem, which can be derived from [5] and Theorem 3.2. Theorem 3.14. (Bernier and Taylor) With the above notation, (a) Nl is the generalized dimension of l , for l = 1> · · · > q. (b) For any i 5 O2 (D), q Z Z X l l gk gd > hi> zd>k izd>k i= (k) l=1 D K weakly in O2 (D).

Acknowledgment Supported by an NSERC Canada Discovery Grant.

References 1. L. Baggett, A Separable Group Having a Discrete Dual is Compact, J. Func. Anal. 10 (1972), 131—148.

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2. L. Baggett, Unimodularity and Atomic Plancherel Measure, Math. Ann. 266 (1984), 513—518. 3. L. Baggett and K. F. Taylor, Groups with Completely Reducible Regular Representation, Proc. Amer. Math. Soc. 72 (1978), 593—600. 4. L. Baggett and K. F. Taylor, A Sucient Condition for the Complete Reducibility of the Regular Representation, J. Func. Anal. 34 (1979), 250—265. 5. D. Bernier and K. F. Taylor, Wavelets from Square-integrable Representations, SIAM J. Math. Anal. 27 (1996), 594—608. 6. J. Diestel and J. J. Uhl, Vector Measures, American Mathematical Society, Providence, 1977. 7. M. Duflo and C. C. Moore, On the Regular Representation of a Nonunimodular Locally Compact Group, J. Func. Anal. 21 (1976), 209—243. 8. E. G. Egros and Z.-J. Ruan, Operator Spaces, Clarendon Press, Oxford, 2000. 9. P. Eymard, L’alg` ebre de Fourier d’un Groupe Localement Compact, Bull. Soc. Math. France 92 (1964), 181—236. 10. A. Fig` a-Talamanca, Positive Definite Functions which Vanish at Infinity, Pacific J. Math. 69 (1977), 355—363. 11. G. Folland, A Course in Abstract Harmonic Analysis. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995. 12. E. Hewitt and H. Zuckerman, Singular Measures with Absolutely Continuous Convolution Squares, Proc. Camb. Phil. Soc. 62 (1966), 399—420. 13. K.-H. Gr¨ ochenig, E. Kaniuth and K. F. Taylor, Compact Open Sets in Duals and Projections in O1 -algebras of Certain Semi-direct Product Groups, Math. Proc. Camb. Phil. Soc. 111 (1992), 545—556. 14. E. Kaniuth and K. F. Taylor, Minimal Projections in O1 -algebras and Open Points in the Dual Spaces of Semi-direct Product Groups, J. London Math. Soc. 53 (1996), 141—157. 15. I. Khalil, Sur L’analyse Harmonique du Groupe Ane de la Droite, Studia Math. 51 (1974), 139—167. 16. G. Mauceri and M. Picardello, Noncompact Unimodular Groups with Purely Atomic Plancherel Measures, Proc. Amer. Math. Soc. 78 (1980), 77—84. 17. D. Menchog, Sur L’unicit´ e du D´ eveloppement Trigonom´ etrique, C. R. Acad. Sci. Paris 163 (1916), 433—436. 18. A. L. T. Paterson, Amenability, Mathematical Surveys and Monographs, 29, American Mathematical Society, Providence, RI, 1988. 19. E. Schulz and K. F. Taylor, Extensions of the Heisenberg Group and Wavelet Analysis in the Plane, CRM Proceedings and Lecture Notes 18 (1999), 217—225. 20. E. Schulz and K. F. Taylor, Projections in O1 -algebras and Tight Frames, Contemporary Math. 363 (2004), 313—319. 21. K. F. Taylor, Geometry of the Fourier Algebras and Locally Compact Groups with Atomic Unitary Representations, Math. Ann. 262 (1983), 183—190.

Chapter 4

Wavelet Transforms and Admissible Group Representations Eric Weber This paper is dedicated to Larry Baggett, my Advisor, Mentor, and Friend

Abstract Discrete wavelet transforms arise naturally from the sampling of a continuous wavelet transform, which in turn arises from a square-integrable (or more generally, admissible) representation of a locally compact group. We show that discrete wavelet transforms arising from groups with an ane structure possess an analogous admissibility condition. In particular, we show that the group performing the role of translations must satisfy an admissibility property. Finally, we relate these results to several notions of multiresolution.

4.1 Introduction Our motivation for understanding the role of admissibility in discrete wavelet transforms came from investigating three seemingly distinct problems in the theory of wavelets. Admissibility actually plays a central role in each of these three problems, and thus they are highly related. We describe them concisely here and intersperse more details of the problems throughout the paper. 1. The first problem is due to Larry Baggett: an orthonormal wavelet in O2 (R) can be associated to a nest of subspaces, called a Generalized Multiresolution Analysis (GMRA) [2], which is nearly an MRA, except that the core space Y0 does not necessarily contain a scaling function but is merely invariant under integer translations. This invariance gives rise to a group representation of the integers on Y0 . Using abstract harmonic analysis, this representation determines a measure and a multiplicity function on the dual group T1 [4]. The problem is whether this measure is always absolutely continuous with respect to Haar measure. It is true for wavelets Eric Weber Department of Mathematics, Iowa State University, 396 Carver Hall, Ames, IA 50011

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in O2 (R) because the integer translations acting on all of O2 (R) possesses a measure that is absolutely continuous [4]. We show in Theorem 4.37 that in general this measure is absolutely continuous. 2. The second problem is due to Guido Weiss: the GMRA that is associated to an orthonormal wavelet in O2 (R) need not have a corresponding scaling function (the Journ´e wavelet, appears in [26]), i.e., there is no single function in the core space Y0 whose integer translates form an orthonormal basis for Y0 . Is there a set of functions in the core space whose translates form a frame for Y0 ? Again, this is known to be true for wavelets in O2 (R) [29]; we will show in Theorem 4.34 that this is true in general. 3. The third problem is due to David Larson: the integer translation operator on O2 (R) is a bilateral shift of infinite multiplicity. Can wavelets arise from unitary operators that are not bilateral shifts of inifinite multiplicity? In Theorem 4.39, we show that the answer is no, that any unitary translation operator giving rise to an orthonormal wavelet must be a bilateral shift of infinite multiplicity. Discrete wavelet transforms generally are derived from discretizing a continuous wavelet transform obtained via a square-integrable representation of a locally compact group. If J is a locally compact group and is a squareintegrable representation of J on the (separable) Hilbert space H, then a discrete wavelet transform Z is constructed from two ingredients: (1) N J countable (subgroup or not), and (2) a wavelet # 5 H with the admissibility condition that the operator Z : H $ o2 (N) : i 7$ (hi> (n)#i)n is well-defined, bounded, and has a bounded left inverse. These conditions are typically expressed in terms of frame inequalities, which follow below. See Daubechies [10] for an exposition of this idea for the d{ + e group acting uhr [16] for an extensive list of on O2 (R); see Ali, Antoine, Gazeau [1] or F¨ other groups. The two most common groups used in the construction of discrete wavelet transforms are the d{ + e group acting on R and the Heisenberg group also acting on R (as well as generalizations to higher dimensions). In both cases, the unitary operators that appear in the representation, and hence form the building blocks for the discrete wavelet transforms, are in fact bilateral shifts of infinite multiplicity. Therefore, these unitary operators are quite simple in structure; in particular, they have purely continuous spectra. The first main result (Theorem 4.30) of the paper is that this is true in general, that for systems of unitary operators which resemble the d{ + e group in structure and which possess a wavelet transform, all of the operators have continuous spectra. We are motivated by the d{ + e group as a model for the discrete wavelet transforms we consider here, and an essential feature that the d{ + e group possesses versus the Heisenberg group is a multiresolution (time-scale) anal-

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ysis theory [22]. The original definition of a multiresolution analysis is due to Mallat [26] and has been generalized by replacing an orthonormal scaling function with frames [6, 31] or by shift invariance [2]. The second main result (Theorem 4.34) of the paper is that if there is a discrete wavelet transform, then shift invariance is enough to guarantee the existence of frames in the multiresolution analysis. The idea of an admissibility condition associated to discrete wavelet transforms connects the two main results and shows that they are actually equivalent (Theorem 4.41). Our notion of admissibility for representations of locally compact groups is meant to generalize square-integrability, though there are a number of ways to do so [1, 14, 28, 32].

4.1.1 Notation We establish some conventions and notation. H will always be a separable Hilbert space with scalar product h·> ·i; U(H) will denote the group of unitary operators on H; J will denote a finite or countable index set (each instance may be dierent than the previous, however). Definition 4.1. A sequence {{m }m5J H is Bessel if there exists a positive constant F2 such that for all y 5 H, X |hy> {m i|2 F2 kyk2 = m5J

The sequence is a frame if additionally there exists a positive constant F1 such that X |hy> {m i|2 F1 kyk2 m5J

holds. A frame is called Parseval if the constants can be chosen to be F1 = F2 = 1. ˜ are two frames, we say Definition 4.2. If {{m }m5J H and {|m }m5J H ˜ such that they are equivalent if there exists a unitary operator X : H $ H that X {m = |m . (The usual definition of similar frames allows X to be merely invertible.) For the Hilbert space O2 (Rg ), we make the following definitions. For 5 R , define the translation operators: g

W : O2 (Rg ) $ O2 (Rg ) : i (·) 7$ i (· )= If D 5 JO(g> R), define the dilation operator: GD : O2 (Rg ) $ O2 (Rg ) : i (·) 7$

p | det D|i (D·)=

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Finally, normalize the Fourier transform, for i 5 O1 (Rg ) by Z ˆ i ({)h2l{· g{= i () = Rg

4.1.2 Operator Theory In this subsection, we recall some basic operator theory, in order to introduce the terminology, make note of some important papers in the literature, and to develop the intuition of the reader. Definition 4.3. Let V be a bounded operator on H and let V˜ be a bounded operator on N. We say V and V˜ are unitarily equivalent if there exists a ˜ . unitary operator Y : H $ N such that V = Y 1 VY Definition 4.4. Let X be a bounded operator on H. We say X is a bilateral shift if there exists a cardinal number Q 5 {1> 2> = = = > 4} and there exists {{1 > = = = > {Q } H such that the set {X n {l : n 5 Z; l = 1> = = = > Q } is an orthonormal basis for H. The number Q is the multiplicity of the bilateral shift. Bilateral shift operators have been studied by a number of people. Two examples include Halmos [19], and, relevant to the theory of multiresolution analysis, Robertson [33] (see Example 4.7). Extensions of Robertson’s results appear in [17, 21, 24, 30]. Lemma 4.5. If X is a bounded operator on H and is a bilateral shift, then X is a unitary operator. Proposition 4.6. Suppose X is a bilateral shift of multiplicity Q on H and Y : H $ N is a unitary operator. Then W := Y X Y 1 is a bilateral shift of multiplicity Q on N. Conversely, if X , W are bilateral shifts of multiplicity Q on H, N, respectively, then X and W are unitarily equivalent. Example 4.7. Traditionally, in the setting of an MRA, P the space Y0 is identified with o2 (Z) via the identification of (fq )q ' q5Z fq !({ q), where ! is a (orthonormal) scaling function. Indeed, since {!({ q) : q 5 Z} is an orthonormal basis for Y0 , then the mapping X : o2 (Z) $ Y0 : (fq )q 7$ P q5Z fq !({ q) is a unitary operator. Then, W2 acting on Y0 is identified to the shift by 2 on o2 (Z); define V : 2 o (Z) $ o2 (Z) : fq 7$ fq2 . Note that V is a bilateral shift of multiplicity two, since {V n {h0 > h1 } : n 5 Z}, where hm is the point mass at m, is an orthonormal basis. Likewise, W2 acting on Y0 is a bilateral shift of multiplicity two since {W2n {G21 !> G21 #} : n 5 Z} is an orthonormal basis for Y0 , where # is a wavelet.

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A simple computation shows that W2 = X VX 1 . Moreover, if (dq )q 5 o2 (Z) P 1 is the mask of the scaling function, i.e., [G2 !]({) = q5Z dq !({ q) and eq = (1)q d1q , then the unitary operator X 1 maps the orthonormal basis {W2n {G21 !> G21 #} : n 5 Z} to the set {V n {(dq )q > (eq )q } : n 5 Z}. Therefore, corresponding wavelet bases in o2 (Z) are then the sequences given by the even shifts of (dq ) and (eq ). Lemma 4.8. Let 5 Rg ; D 5 JO(g> R) contractive (i.e., has norm strictly less than 1). Then W and GD are bilateral shifts of infinite multiplicity. Remark 4.9. It is shown in [23] that GD is a bilateral shift of infinite multiplicity if and only if D is not similar to a unitary matrix, where the unitary and the similarity matrix are both over the complex numbers. Remark 4.10. Some general comments about bilateral shifts. 1. Bilateral shifts have purely continuous spectra, which consists of the unit circle in the complex plane. In particular, they have no eigenvectors. (This fact is used in Balan et al. [5].) 2. Bilateral shifts have many invariant subspaces, all of which are infinite dimensional (by the previous comment). 3. It is sometimes convenient to consider the map : Z $ U(H) : } 7$ X } as a representation of the group Z. Natural extensions then occur by considering groups such as Zg .

4.2 Admissibility: Generalizing Square-integrability We assume J is a locally compact group. We denote the (left) invariant Haar measure on J by . A representation of J on H is a group homomorphism : J $ U(H) such that for every y> z 5 H, the function y>z : J $ C : j 7$ hy> (j)zi is continuous. Every group has a distinguished representation called the left regular representation, given by the following: let i 5 O2 (J> ) =: O2 (J) and let k 5 J. Define Ok i (j) = i (k1 j). The mapping O : J $ U(H) : k 7$ Ok is the left regular representation. Two ˜ are equivalent if there exists a unitary representations on H and on H ˜ operator X : H $ H such that (j) = X 1 (j)X for all j 5 J. Definition 4.11. A square-integrable representation of J is an irreducible representation of J on H such that there exists a nonzero vector # with the property that Z J

|h#> (j)#i|2 g ? 4=

A square-integrable group representation always generates a “continuous wavelet transform”:

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Theorem 4.12 (Duflo—Moore [11]). Suppose is an irreducible squareintegrable representation of J on H, with wavelet #, i.e., Z |h#> (j)#i|2 g ? 4= 0 6= J

Then for every y 5 H,

Z

J

|hy> (j)#i|2 g ? 4=

Moreover, the continuous wavelet transform Z# : H $ O2 (J) defined by Z# y(j) = hy> (j)#i is a constant times an isometry.

Both the usual continuous wavelet transform on O2 (Rg ) and the short time Fourier transform arise from square-integrable group representations: the continuous wavelet transform arises from the representation of the ane transformations of Rg acting on O2 (Rg ), and the short time Fourier transform comes from the representation of the Heisenberg group on O2 (Rg ) (after making the modification of factoring out the projective kernel). For a direct application to wavelets, see [25] for partial results on which subgroups of the ane group are admissible on O2 (Rg ).

4.2.1 Admissibility The more general definition of square-integrability that follows is due to Rieel [32]. Definition 4.13. The representation of J on H is square-integrable if the collection of vectors # 5 H that have a corresponding constant E# such that for all y 5 H, Z J

|hy> (j)#i|2 g E# kyk2

is a dense subspace of H.

Rieel proves that, using his definition of square-integrability, a representation is square-integrable if and only if it is equivalent to a subrepresentation of some multiple of the left regular representation. We introduce our own definition of generalized square-integrability via an admissibility condition. Definition 4.14. An admissible representation of J is a cyclic representation of J on H such that there exists a cyclic vector # and a positive constant E such that for every y 5 H, Z |hy> (j)#i|2 g Ekyk2 = (4.1) J

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Any vector # that satisfies the condition in Equation (4.1), whether it is a cyclic vector or not, will be called an admissible vector. Definition 4.15. We shall say is a frame representation of J if there exists a sequence {#n : n 5 Z} H such that XZ |hy> (j)#n i|2 g Ekyk2 = Dkyk2 n5Z

J

Remark 4.16. Our definition of a frame representation is more general than elsewhere in the literature. If instead of a sequence {#n }, there exists a single vector # that satisfies the property that there exist positive constants D> E such that for all y 5 H, Z |hy> (j)#i|2 g Ekyk2 Dkyk2 J

is called a continuous frame in [1], or a frame representation in [20]. Under this more strict definition, a frame representation is cyclic, hence is admissible. The converse of this does not hold, as the next example shows (see [15] for a description of which admissible representations are frame representations). Example 4.17. Consider the regular representation of R; that is () = W . This representation is admissible: let # 5 O2 (R) be such that #ˆ is bounded. Then, by virtue of Parseval’s identity and the Fourier inversion formula, we have: Z Z 2 2 |hi> W #i| g = |hiˆ> Wd #i| g R R Z Z 2 ˆ | iˆ()h2l #()g| g = ZR R 2 ˆ |[iˆ#(·)]ˇ()| g = R Z 2 ˆ |iˆ()#()| g = R

ˆ 2 ki k2 = k#k 4

Note that this computation demonstrates that any admissible vector must have bounded Fourier transform. This computation also shows that the regular representation of R is not a frame representation, since for any # 5 O2 (R), there is no D A 0 such that R 2 ˆ Dki k2 R |iˆ()#()| g for all i 5 O2 (R). Moreover, this computation shows that if “irreducible” is replaced by “cyclic” in the theorem of Duflo and Moore, the statement is no longer valid.

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ˆ 2 is still in O2 (R) but is not essentially Indeed, let # 5 O2 (R) be such that |#| bounded. Thus Z |h#> W #i|2 ? 4 J

but # is not an admissible vector. (We thank Larry Baggett for pointing this out.) It is unknown if an admissible representation is a frame representation as given in Definition 4.15 Lemma 4.18. Suppose is an admissible representation of J on H; then is equivalent to a subrepresentation of the left regular representation of J. Proof. Let # 5 H be an admissible vector. Denote the mapping { 7$ i (j) := h{> (j)#i by Y . Consider the following computation:

Y [(j0 ){](j) = h(j0 ){> (j)#i = h{> (j01 j)#i = [Y {](j01 j) = Oj0 [Y {](j)= Thus, we have Y (j0 ) = Oj0 Y=

(4.2)

Now, let X be the partial isometry of the polar decomposition of Y , i.e., Y = X S . By taking the adjoint of both sides of Equation (4.2), for every j0 5 J, (j0 )Y = Y Oj0 , whence Y Y (j0 ) = (j0 )Y Y . Moreover, S = s Y Y also commutes with (j0 ). Since, X S (j0 ) = Oj0 X S , we have that X (j0 ) = Oj0 X on the range of S . We claim that the range of S is dense in H. Indeed, we show that the range of Y Y is dense in H: suppose | 5 H and h|> Y Y {i = 0 for all { 5 H. Then, in particular, we have for all j 5 J: 0 = h|> Y Y (j)#i = hY Y |> (j)#i which implies that Y Y | = 0 since {(j)#} has dense span in H. Thus, 0 = hY |> Y |i Z Y |(j)Y |(j)g = ZJ |h|> (j)#i|2 g= = J

Again, since {(j)#} has dense span, if Y | = 0, then | = 0, whence Y Y has dense range. Therefore, by virtue of the fact that the range of S is dense in H, the commutation relation holds on all of H. Now, X : H $ O2 (J) is an isometry, hence a unitary from H onto its range, whence X is the required intertwining operator. t u

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Lemma 4.19. Let be a representation of J on H. Suppose for each n 5 N, , where the subrepresentation Hn H is an invariant cyclic subspace for P Hn = H, that is the span of of J on Hn is admissible. Finally, suppose ^n5Z Hn is dense in H. Then is equivalent to a subrepresentation of an infinite multiple of the left regular representation of J. Proof. Suppose #n 5 Hn is an admissible vector for . For any projection S in the commutant of , S #n is an admissible vector: Z Z 2 |hy> (j)S #n i| g = |hS y> (j)#n |2 g EkS yk2 Ekyk2 = J

J

Define !0 := #0 ; !1 = T0 #1 , where T0 is the projection onto the orthogonal complement of H0 ; !2 = T1 #2 , where T1 is the projection onto the orthogonal complement of the closed subspace spanned by H0 and H1 . Repeat for all n; let Pn be the cyclic subspace generated by !n . We have that !n is an admissible vector for , and H = n5Z Pn . Thus, H can be decomposed into a sequence of orthogonal subspaces, and the subrepresentation of on Pn is equivalent to a subrepresentation of the left regular representation by Lemma 4.18. t u

Remark 4.20. It may be that only a finite number of the subspaces Pn in the above lemma are nonzero. If this is the case, then is equivalent to a subrepresentation of a finite multiple of the left regular representation. Definition 4.21. If is a representation of J on H that satisfies the hypotheses of Lemma 4.19, then we say that is -admissible.

Remark 4.22. If is -admissible, with admissible vectors #n , then by scaling them appropriately, we obtain the following “generalized squareintegrability” condition: there exists a constant E such that for all y 5 H, XZ |hy> (j)#n i|2 g Ekyk2 = n5Z

J

Lemma 4.23. The definition of square-integrable due to Rieel and the definition of -admissible are equivalent. That is, a representation of J on H is square-integrable (in the sense of Definition 4.13) if and only if it is -admissible. Proof. First assume that is -admissible, with admissible vectors #n . For each j 5 J, (j)#n is also admissible, and any finite linear combination of admissible vectors is admissible. Thus, since {(j)#n : j 5 J; n 5 Z} has dense span in H, it follows that satisfies Definition 4.13. Conversely, suppose P = {# 5 H : # is admissible } is dense in H. Choose any countable dense subset of P , say {#n : n 5 Z}. Then, let Hn be the cyclic restricted to Hn is admissible in the subspace generated by #n ; by definition, P sense of Definition 4.14. Moreover, H = n5Z Hn , whence is -admissible by Lemma 4.19. t u

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4.2.2 Countable Groups The motivation for our study of admissibility is discrete wavelet transforms. In the classical discrete wavelet transforms, there is a countable group of unitary translation operators–a representation of a countable group. Thus, we now consider the structure of admissible representations of countable groups. Note that in contrast to the typical wavelet setting, we do not necessarily assume that the group is Abelian. Lemma 4.24. If J is a countable group, then the left regular representation is a frame representation. Therefore, any representation of J that is equivalent to a subrepresentation of the left regular representation is a frame representation. In particular, a representation of J is an admissible representation if and only if it is a frame representation. Proof. Since J is countable, Haar measure is counting measure, and the left regular representation acts on o2 (J). The left regular representation of J is a frame representation, since the vector # = "h , the characteristic function of the identity, has the property that {Oj # : j 5 J} is an orthonormal basis for o2 (J). Suppose is equivalent to a subrepresentation of the left regular representation, with intertwining operator X : H $ P o2 (J). Let T denote the orthogonal projection of o2 (J) onto X (H) = P ; note that X (H) is an invariant subspace under the left regular representation, whence T commutes with it. Thus, {Oj T# : j 5 J} is a (Parseval) frame for X (H), and {(j)X T# : j 5 J} is a (Parseval) frame for H. Finally, since an admissible representation of J is equivalent to a subrepresentation of the left regular representation, it is a frame representation. u t Proposition 4.25. Suppose is a unitary representation of J on H that is unitarily equivalent to some subrepresentation of some multiple of the left regular representation. Then there exists a sequence {im : m 5 J} such that {(j)im : j 5 J; m 5 J} forms a Parseval frame for H. Moreover, for any invariant subspace P H, {(j)SP im : j 5 J; m 5 J} forms a Parseval frame for P , where SP is the orthogonal projection of H onto P . Proof. This follows from the proof of Lemma 4.24.

t u

Corollary 4.26. If is a -admissible representation of J on H, then is a frame representation. Proof. Combine Lemma 4.19 and Proposition 4.25.

t u

Remark 4.27. Note that in the special case that J = Z, (1) = X , then is equivalent to some multiple of the regular representation if and only if X is a bilateral shift of some multiplicity.

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4.3 Wavelet Transforms and Admissibility We noted in the introduction that a continuous wavelet transform arises from a square-integrable representation. Our focus now rests on discrete wavelet transforms: for a unitary system U [9], we shall say that U possesses a discrete wavelet transform if there exists {#m : m 5 J} such that {X #m : X 5 U> m 5 J} is a frame for H. In the next subsection, we shall introduce some structure to U which resembles the ane properties of the usual wavelet system.

4.3.1 Ane Systems and Admissibility We continue to assume that J is countable. Definition 4.28. We say a triple (> J> G) is an ane system, where is a representation of J on H, and G is a unitary operator on H such that for every j 5 J, there exists a (unique) j 0 5 J such that G1 (j)G = (j 0 ); moreover the subgroup {G1 (j)G : j 5 J} is a normal subgroup of (J) of finite index. Normally, it is desirable that J is Abelian, and in many natural cases it is, but this need not hold. See [18] for an example of where J is anti-isomorphic to the discrete Heisenberg group. Additionally, one normally has that is injective, so that is an isomorphism onto its range. This is a reasonable assumption; first, in order to obtain a frame, the kernel of must have finite cardinality; second, one can then consider J modulo the kernel of , which only eliminates the repetitions in any sequence of the form {(j)#}. Note that for each n 5 Z, Gn (J)Gn is a group, which is isomorphic to (J), and hence J. Thus, for each n 5 Z, let this group be denoted by n (J) (we can naturally think of n as being another representation of J on H). We have n (J) n+1 (J) and thus the following commutation relations: given j 5 J, there exists some k> k0 5 J such that (j)G = G(k);

(j)G1 = G1 1 (k0 )=

Moreover, we have the following set relations: (J)Gn = Gn n (J) Gn (J) for n A 0;

(4.3)

Gn (J) = ^5(J)@n (J) Gn n (J) for n A 0;

(4.4)

(J)Gn = ^5n (J)@(J) Gn (J) for n ? 0=

(4.5)

Here and for the remainder of the paper, is understood to be coset representatives. Equation (4.3) follows from definition; Equation (4.4) follows since

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^5(J)@n (J) n (J) = (J). Equation (4.5) follows from the fact that by definition, (J)Gn = Gn n (J) and ^5n (J)@(J) (J) = n (J). Definition 4.29. We say {#1 > = = = > #q } H is a wavelet frame for (> J> G) if the collection {Gn (j)#l : n 5 Z; j 5 J; l = 1> = = = > q} is a frame for H. We say that the ane system (> J> G) possesses a wavelet transform (or discrete wavelet transform) if there exists a wavelet frame for (> J> G). Theorem 4.30. Let (> J> G) be an ane system on H that possesses a wavelet transform. Then is 1. -admissible; 2. a frame representation of J; 3. equivalent to a subrepresentation of an infinite multiple of the left regular representation of G. Proof. Suppose that # generates a wavelet frame under (> J> G). We will show that is -admissible by demonstrating that H can be “covered” by a countable number of admissible representations. For n = 0, let H0 = vsdq{(j)# : j 5 J}; clearly, restricted to H0 is admissible. For n A 0, let Hn> = vsdq{(j)Gn # : j 5 J}, where is a fixed coset representative in (J)@n (J). By Equations (4.3) and (4.4), {(j)Gn # : j 5 J; 5 (J)@n (J)}

= {Gn n (j)# : j 5 J; 5 (J)@n (J)}

= {Gn (j)#}

is a Bessel sequence, whence restricted to Hn> is -admissible. For n ? 0, let Hn = vsdq{(j)Gn # : j 5 J}; we claim that the subrepresentation of J on Hn is admissible. By Equation (4.5), {(j)Gn # : j 5 J} = ^5n (J)@(J) {Gn (j)# : j 5 J}. For each , {Gn (j)# : j 5 J} is a Bessel sequence, since {(J)#} is a Bessel sequence and Gn is a unitary operator. Moreover, the union above is over a finite index set, whence the union is a Bessel sequence, and thus restricted Hn is admissible. Since the span of ^n ^ Hn> contains the span of {Gn (j)# : j 5 J; n 5 Z}, which is dense in H, is -admissible. The above proof can be extended naturally for a wavelet frame with any number of generators. Item 2 now follows from Corollary 4.26, and item 3 follows from Lemma 4.19. t u

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4.3.2 Admissibility and Multiresolution Generalized Multiresolution Analyses (GMRAs) were introduced by Baggett, Carey, Moran, and Ohring [2]. Let (> J> G) be an ane system on H. A GMRA for (> J> G) is a sequence of closed subspaces {Ym : m 5 Z} such that the following hold: 1. 2. 3. 4. 5.

^m5Z Ym = H (density); _m5Z Ym = {0} (triviality); Ym Ym+1 (nestedness); GYm = Ym+1 (refinability); the core space Y0 is invariant under the action of (J) (reducibility).

If (> J> G) has a collection {#1 > = = = > #q } of cyclic vectors, i.e., {Gn (j)#l : n 5 Z; j 5 J; l = 1> = = = > q} has dense span in H, then define Ym (#1 > = = = > #q ) := vsdq{Gn (j)#l : n ? m; j 5 J; l = 1> = = = > q}. These subspaces satisfy the density, nestedness, and refinability properties. They may not satisfy the reducibility property [8], and they may not satisfy the triviality property, even if {#1 > = = = > #q } is a wavelet frame [7]. If {#1 > = = = > #q } is an orthonormal wavelet, then all conditions are satisfied [2]. A full understanding of these properties is still unknown: Problem 4.31. When is Y0 (#1 > = = = > #q ) invariant under ? (See [8] for partial results). When is _m5Z Ym (#1 > = = = > #q ) = {0}? The usual structure results from multiresolution analysis theory holds in our setting of ane systems [31]: Theorem 4.32. Suppose that {Ym : m 5 Z} is a GMRA for (> J> G). For each m, define Zm by Ym Zm = Ym+1 . The following hold: 1. for m L 0, Ym is invariant under ; 2. H = m5Z Zm ; 3. Zm+1 = GZm ; 4. for m 0, Zm is invariant under . Definition 4.33. If {Ym : m 5 Z} is a GMRA for (> J> G), then the invariance of Y0 under means that restricted to Y0 is a subrepresentation. This subrepresentation is called the core representation and is denoted by ˜. Frame Multiresolution Analyses were introduced by Benedetto and Li [6], in which Y0 contains a frame of the form {(j)! : j 5 J}, and Generalized Frame Multiresolution Analyses were introduced by Papadakis [31], where Y0 contains a frame of the form {(j)!m : j 5 J; m 5 J}. Problem 2 described in the introduction, due to Guido Weiss, can now be stated as, “Is every GMRA for an ane system (> J> G) a GFMRA?” The following corollary of Theorem 4.30 demonstrates that the answer is yes, under the assumption that the ane system possesses a wavelet transform. Example 4.35 demonstrates that without this assumption, the answer is in general no.

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Theorem 4.34. Suppose (> J> G) is an ane system and {#1 > = = = > #q } generates a wavelet frame such that Y0 (#1 > = = = > #q ) is invariant under . Then there exist scaling vectors {!m : m 5 J} Y0 (#1 > = = = > #q ) such that {(j)!m : j 5 J; m 5 J} is a frame for Y0 (#1 > = = = > #q ). In general, if an ane system possesses a wavelet transform, then every Generalized Multiresolution Analysis (GMRA) for (> J> G) is a Generalized Frame Multiresolution Analysis (GFMRA). Proof. If there exists a frame wavelet for the ane system, then is a frame representation of J by Theorem 4.30, and since Y0 (#1 > = = = > #q ) is invariant under , the restriction of to Y0 (#1 > = = = > #q ) is also a frame representation by Proposition 4.25 t u Example 4.35. It is possible for (> J> G) to be an ane system that has a GMRA, but no GFMRAs, and equivalently, no wavelet frames. For example, let H = O2 (R), let J be the group of dyadic rationals, let () = W , and c0 = {i 5 O2 (R) : let G be the usual dilation by 2. Let {Ym } be defined by Y ˆ vxss(i ) [1@2> 1@2]}, the MRA associated to the Shannon wavelet. This is a GMRA for the present ane system, but there are no frames of the form {W in : 5 J; n 5 Z} in either Y0 or Z0 . In this case, G1 (J)G = (J). See [2] for a similar example, where 1 G (J)G is a proper subgroup of (J) of index 2.

4.3.3 Absolutely Continuous GMRAs b denote the dual group of J; let denote Suppose that J is Abelian. Let J b normalized Haar measure on J. By Stone’s theorem, for any representation b such that of J, there exists a projection-valued measure s on J Z j()gs()= (j) = b J

b via Pontrjagin duality. Then, Here we are viewing j 5 J as a character on J by the theory of projection-valued measures, there exist a probability measure b a multiplicity function p : J b $ {0> 1> = = = > 4}, and a unitary operator on J, 2 X : H $ 4 m=1 O (Im > )>

b where Im = { 5 J|p() m}. The operator X intertwines the projectionb Thus valued measure on H and the canonical projection-valued measure on J. the representation is decomposed in terms of the multiplicity function p and the measure . The regular representation of J decomposes as p 1 and ; a representation is equivalent to a subrepresentation of a multiple of the regular representation if and only if ?? .

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Definition 4.36. A GMRA for (> J> G) is said to be absolutely continuous if in the decomposition of the core representation ˜ the measure ?? . Problem 1 described in the introduction, due to Larry Baggett, can now be stated as, “Is every GMRA for an ane system (> J> G) absolutely continuous?” The following theorem completely answers that question by characterizing the absolute continuity of a GMRA in terms of the ane system possessing a wavelet transform. Theorem 4.37. Let {Ym }m be a GMRA for (> J> G), where J is Abelian. The GMRA is absolutely continuous if and only if (> J> G) posseses a wavelet transform. Proof. If there exists a wavelet frame for (> J> G), then is -admissible, and hence is equivalent to a subrepresentation of a multiple of the regular representation, whence ˜ is also. Therefore, the GMRA is absolutely continuous. Conversely, if the GMRA is absolutely continuous, then ˜ is equivalent to a subrepresentation of a multiple of the regular representation. By Proposition 4.25, there exists a sequence {!m : m 5 J} Y0 such that {(j)!m : j 5 J> m 5 J} is a frame for Y0 . By the commutation relation of Equation (4.3), we have that {(j)G!m } ^ {(j)G!m } = {G(j)!m }> where the right-hand side is a Bessel sequence, whence both sets on the lefthand side are also Bessel sequences. Therefore, the subrepresentation of on vsdq{G(j)!m } = Y1 is also -admissible, and hence, the subrepresentation of on Z0 := Y1 ª Y0 is -admissible. Again by Proposition 4.25, there exists a sequence {#n } Z0 (possibly finite) such that {(j)#n } is a frame for Z0 . By items 2 and 3 of Theorem t u 4.32, it follows that {Gq (j)#n } is a wavelet frame for H. Remark 4.38. The wavelet frame in the above corollary may be generated by more than one vector. In fact, it is possible that the wavelet frame is infinitely generated. For fairly general methods of constructing wavelets from an absolutely continuous GMRA, see [3, 31, 36]. See [34] for a related idea of embedding GMRAs in MRAs. Problem 3 described in the introduction, due to Dave Larson, can be stated as “If (> Z> G) is an ane system that possesses a wavelet transform, is (1) a bilateral shift of infinite multiplicity?” The following theorem answers armatively, under the assumption that the wavelet transform corresponds to an orthonormal basis. Theorem 4.39. If J is Abelian and there exists a wavelet frame that is actually an orthonormal basis, then is equivalent to an infinite multiple of the regular representation. In particular, if J = Z, and (1) = W , then W is a bilateral shift of infinite multiplicity.

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Proof. Let # be such that {Gq (j)#} is an orthonormal basis, and let Ym = vsdq{Gq (j)# : j 5 J> q ? m}. Then {Ym } is a GMRA for (> J> G) [2]. By item 4 in Theorem 4.32 and the commutation relation in Equations (4.3) and (4.4), for each m 0, we have {Gm (j)#} = ^5(J)@n (J) {Gn n (J)#} = ^5(J)@n (J) {(J)Gn #}=

Since {Gm (j)#} is an orthonormal basis for Zm , the subrepresentation of on Zm is equivalent to a finite multiple of the regular representation (that multiple being the index of n (J) in (J)). Therefore, the subrepresentation of on Y0B is equivalent to an infinite multiple of the regular representation. Since the subrepresentation of on Y0 is equivalent to a subrepresentation of some multiple of the regular representation, together the representation is equivalent to an infinite multiple of the regular representation. t u Note that we have not answered the question entirely. It raises another question: Problem 4.40. Is it possible for (> J> G) to have a wavelet frame but no orthonormal wavelet bases? We encapsulate all of the results presented in this subsection and sum up the answers to the initial three questions posed in the introduction. Theorem 4.41. Let (> J> G) be an ane system, and let {Ym } be a GMRA for (> J> G). The following are equivalent: 1. there exists a frame wavelet {#m : m 5 J} Z0 ; 2. there exists frame scaling vectors {!o : o 5 L} Y0 , where L is some countable (perhaps finite) index set; 3. the representation of J on H is equivalent to a subrepresentation of some multiple of the left regular representation; 4. the core representation ˜ of J on Y0 is equivalent to a subrepresentation of some multiple of the left regular representation; 5. the representation of J on H is -admissible; 6. the core representation ˜ of J on Y0 is -admissible. In the case when J is Abelian, then the above are also equivalent to: 7. the GMRA {Ym } is absolutely continuous.

4.3.4 Admissibility and Equivalence ˜ Consider two ane systems (> J> G) and (> J> G)–note that the group is the same. If both ane systems possess an orthonormal wavelet basis,

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˜ are bilateral shifts of infinite multiplicity [20], hence are then both G and G equivalent to each other. Moreover, under the assumption that J is Abelian, we have seen that both and are equivalent to an infinite multiple of the regular representation (Theorem 4.39) and hence are equivalent to each other. A natural question then concerns the extent to which equivalence, in various guises, holds for ane systems. We present three possible definitions, in decreasing strength, for equivalence of ane systems. Definition 4.42. We say that two ane systems, (> J> G) on H and (> J> ˜ on H, ˜ are spatially equivalent (or jointly equivalent) if there exists a G) ˜ such that for all q 5 Z and j 5 J, unitary operator X : H $ H ˜ q (j)X= X Gq (j) = G Definition 4.43. We say that two ane systems, (> J> G) on H and (> J> ˜ on H, ˜ possess equivalent wavelet theories if for every {#1 > = = = > #n } H G) such that {Gq (j)#m : q 5 Z; j 5 J; m = 1> = = = > n} is a frame, there exists ˜ such that {G ˜ q (j)#˜m : q 5 Z; j 5 J; m = 1> = = = > n} is a {#˜1 > = = = > #˜n } H frame which is equivalent. Definition 4.44. We say that two ane systems, (> J> G) on H and (> J> ˜ possess equivalent wavelet transforms if there exist {#1 > = = = > #n } ˜ on H, G) ˜ such that {Gq (j)#m : q 5 Z; j 5 J; m = 1> = = = > n} H, {#˜1 > = = = > #˜n } H, ˜ q (j)#˜m : q 5 Z; j 5 J; m = 1> = = = > n} are both frames which are and {G equivalent. Clearly, we have the following relationship between two ane systems: spatially equivalent , possess equivalent wavelet theories , possess equivalent wavelet transforms.

(4.6)

˜ are two ane systems that Theorem 4.45. Suppose (> J> G) and (> J> G) ˜ possess equivalent wavelet transforms. Then the unitary operators G and G are equivalent. ˜ are the representation spaces for and , respecProof. Suppose H and H ˜ are such that the tively. Suppose {#1 > = = = > #n } H and {#˜1 > = = = > #˜n } H frames {Gq (j)#m : q 5 Z; j 5 J; m = 1> = = = > n}> and ˜ q (j)#˜m : q 5 Z; j 5 J; m = 1> = = = > n} {G

˜ such that are equivalent. Then there is a unitary operator X : H $ H ˜ q (j)#˜m = G ˜ q (j)X #m X Gq (j)#m = G

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for all q 5 Z, j 5 J, and m = 1> = = = > n. We claim that ˜ y X Gy = GX for every y 5 H. Indeed, let y 5 H, write y=

n XX X

fm>q>j Gq (j)#m =

m=1 q5Z j5J

We have X Gy = X G

n XX X

fm>q>j Gq (j)#m

m=1 q5Z j5J

=X

n X X

X

fm>q>j Gq+1 (j)#m

m=1 q5Z j5J

=

n X X

X

˜ q+1 (j)X #m fm>q>j G

m=1 q5Z j5J

˜ =G

n XX X

˜ q (j)X #m fm>q>j G

m=1 q5Z j5J

˜ y= = GX

t u

˜ to be equivalent, We conclude by noting that it is possible for G and G and and to be equivalent, without being jointly (spatially) equivalent. ˜ such that X G = GX ˜ , and That is to say, there are unitaries X> Y : H $ H Y (j) = (j)Y , but there is no unitary operator Z such that ˜ q (j)Z= Z Gq (j) = G If there were such a Z , the groups generated by the ane systems would be unitarily equivalent. The example below shows this need not be the case. In the special case of J = Z, the group generated by the ane system (> Z> G) is a Baumslag—Solitar group. Representations of such groups, and their relations to wavelets, are given in [27, 12] Example 4.46. We consider the group J = Z. We consider H = O2 (R) and ˜ = O2 (R> ), where is an extension of the Hausdor measure on the Cantor H set, extended to the entire real line by integer translations, and then extended further to maintain a certain triadic rescaling relationship [13]. (Palle Jorgensen refers to this space as “Cantor dust.”) In both cases, we will consider dilation by 3; also in both cases, we will have an orthonormal basis generated by two wavelets. For H, we may choose a scaling function !:

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ˆ = "[1@2>1@2) () !() and corresponding wavelets (with two generators) #1 and #2 : #ˆ1 () = "[3@2>1@2) ()>

#ˆ2 () = "[1@2>3@2) ()=

˜ we may choose a scaling function !: For H, !({) = "F ({); where F is the Cantor set. This scaling function satisfies the 3-scale relation: !({) = !(3{) + !(3{ 2); and has a corresponding low-pass filter 1 1 p() = s + s h2l(2) = 2 2 The corresponding wavelets (again, with two generators): #1 ({) = "F (3{) "F (3{ 2)>

#2 ({) = "F (3{ 1)=

Thus, for both ane systems, there exists an orthonormal wavelet basis. Therefore, both dilations are bilateral shifts of infinite multiplicity, and both representations of Z are equivalent to an infinite multiple of the regular representation. However, the representations of the corresponding Baumslag— Solitar groups are not equivalent. This is shown directly in [12]. We present here the key part of the argument. In the case of the Cantor dust, there exists the orthonormal scaling function "F with low-pass filter p(). However, it is known that no such scaling function exists in O2 (R) [35, 13]. This implies that the ane systems cannot be spatially equivalent, for if they were, say ˜ $ H, then ! := X "F 5 O2 (R) would be a such a scaling function. by X : H This example shows that possess equivalent wavelet transforms 6, spatially equivalent and thus, at least one of the directions in (4.6) is not reversible. Problem 4.47. Are either of the two implications in (4.6) reversible? Acknowledgments Most of the results contained here were initially announced at the Wavelet Workshop at the 150th Celebration of Mathematics at Washington University in St. Louis. We thank Larry Baggett, Gestur Olafsson, Manos Papadakis, Ed Wilson, and Guido Weiss for conversations regarding an earlier version of this paper. Support from NSF grant DMS-0355573 is acknowledged. We thank the anonymous referee for helping to clarify the presentation in several passages.

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References 1. S. T. Ali, J.-P. Antoine, and J.-P. Gazeau, Coherent States, Wavelets and Their Generalizations, Graduate Texts in Contemporary Physics, Springer-Verlag, New York, 2000. 2. L. Baggett, A. Carey, W. Moran, and P. Ohring, General existence theorems for orthonormal wavelets, an abstract approach, Pub. Res. Inst. Math. Sci. 31 (1995), no. 1, 95—111. 3. L. Baggett, J. Courter, and K. Merrill, Construction of wavelets from generalized conjugate mirror filters, Appl. Comput. Harmon. Anal. 13 (2002), no. 3, 201—223. 4. L. Baggett, H. Medina, and K. Merrill, Generalized multiresolution analyses, and a construction procedure for all wavelet sets in Rq , J. Fourier Anal. Appl. 5 (1999), no. 6, 563—573. 5. R. Balan, P. Casazza, C. Heil, and Z. Landau, Deficits and excesses of frames, Adv. Comput. Math. 18 (2003), no. 2-4, 93—116. 6. J. Benedetto and S. Li, The theory of multiresolution analysis frames and applications to filter banks, Appl. Comput. Harmon. Anal. 5 (1998), no. 4, 389—427. 7. M. Bownik and Z. Rzeszotnik, On the existence of multiresolution analysis of framelets, Math. Ann. 332 (2005), no. 4, 705—720. 8. M. Bownik and E. Weber, Ane frames, GMRA’s, and the canonical dual, Studia Math. 159 (2003), no. 3, 453—479. 9. X. Dai and D. Larson, Wandering vectors for unitary systems and orthogonal wavelets, vol. 134, Mem. Amer. Math. Soc., no. 640, AMS, Providence, RI, July 1998. 10. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF, SIAM, Philadelphia, 1992. 11. M. Duflo and C. Moore, On the regular representation of a nonunimodular locally compact group, J. Funct. Anal. 21 (1976), no. 2, 209—243. 12. D. Dutkay, Low-pass filters and representations of the Baumslag Solitar group, Trans. Amer. Math. Soc. 358 (2006), no. 12, 5271—5291 (electronic). 13. D. Dutkay and P. Jorgensen, Wavelets on fractals, Rev. Mat. Iberoamericana 22 (2006), no. 1, 131—180. ´ 14. R. Fabec and G. Olafsson, The continuous wavelet transform and symmetric spaces, Acta Appl. Math. 77 (2003), no. 1, 41—69. 15. H. F¨ uhr, Admissible vectors for the regular representation, Proc. Amer. Math. Soc. 130 (2002), no. 10, 2959—2970. 16. H. F¨ uhr, Abstract Harmonic Analysis of Continuous Wavelet Transforms, Lecture Notes in Mathematics, vol. 1863, Springer-Verlag, Berlin, 2005. 17. T.N.T. Goodman, S.L. Lee, and W.S. Tang, Wavelets in wandering subspaces, Trans. Amer. Math. Soc. 338 (1993), no. 2, 639—654. 18. K. Guo, D. Labate, W. Lim, G. Weiss, and E. Wilson, Wavelets with composite dilations, Electron. Res. Announc. Amer. Math. Soc. 10 (2004), 78—87 (electronic). 19. P. Halmos, Shifts on Hilbert spaces, J. Reine Angew. Math. 208 (1961), 102—112. 20. D. Han and D. Larson, Frames, bases and group representations, vol. 147, Mem. Amer. Math. Soc., no. 697, AMS, Providence, RI, September 2000. 21. D. Han, D. Larson, M. Papadakis, and T. Stavropoulos, Multiresolution analyses of abstract Hilbert spaces and wandering subspaces, The Functional and Harmonic Analysis of Wavelets and Frames (San Antonio, TX, 1999), Contemporary Mathematics, vol. 247, American Mathematical Society, Providence, RI, 1999, pp. 259—284. 22. C. Heil, J. Ramanathan, and P. Topiwala, Linear independence of time-frequency translates, Proc. Amer. Math. Soc. 124 (1996), no. 9, 2787—2795. 23. E. Ionascu, D. Larson, and C. Pearcy, On wavelet sets, J. Fourier Anal. Appl. 4 (1998), no. 6, 711—721. 24. D. Larson, W. S. Tang, and E. Weber, Multiwavelets associated with countable abelian groups of unitary operators in Hilbert spaces, Int. J. Pure Appl. Math. 6 (2003), no. 2, 123—144.

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25. R. Laugesen, N. Weaver, G. Weiss, and E. Wilson, A characterization of the higher dimensional groups associated with continuous wavelets, J. Geom. Anal. 12 (2002), no. 1, 89—102. 26. S. Mallat, Multiresolution approximations and wavelet orthonormal bases of O2 (R), Trans. Amer. Math. Soc. 315 (1989), no. 1, 69—87. 27. F. Martin and A. Valette, Markov operators on the solvable Baumslag-Solitar groups, Experiment. Math. 9 (2000), no. 2, 291—300. ´ 28. G. Olafsson and D. Speegle, Wavelets, wavelet sets, and linear actions on Rq , Wavelets, Frames and Operator Theory, Contemporary Mathematics, vol. 345, American Mathematical Society, Providence, RI, 2004, pp. 253—281. 29. M. Papadakis, On the dimension function of a wavelet, Proc. Amer. Math. Soc. 128 (2000), no. 7, 2043—2049. , Frames of translates in abstract hilbert spaces and the generalized frame 30. multiresolution analysis, Trends in Approximation Theory (Kopotun, Lynche, and Neamtu, eds.), Vanderbilt University Press, 2001, pp. 353—362. , Generalized frame multiresolution analysis of abstract Hilbert spaces, Sam31. pling, Wavelets, and Tomography, Appl. Numer. Harmon. Anal., Birkh¨ auser Boston, Boston, MA, 2004, pp. 179—223. 32. M. Riegel, Integrable and proper actions on C*-algebras, and square-integrable representations of groups, preprint, available at arxiv.org: Math OA/9809098, 1998. 33. J. Robertson, On wandering subspaces for unitary operators, Proc. Amer. Math. Soc. 16 (1965), 233—236. 34. S. Schager, Generalized multiresolution analyses and applications of their multiplicity functions to wavelets, Ph.D. thesis, University of Colorado, Boulder, 2000. 35. G. Strang and T. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, Wellesley, MA, 1996. 36. E. Weber, Frames and single wavelets for unitary groups, Canad. J. Math. 54 (2002), no. 3, 634—647.

Chapter 5

The Density Theorem and the Homogeneous Approximation Property for Gabor Frames Christopher Heil To Larry, a wonderful mathematician and friend

Abstract The Density Theorem for Gabor frames is a fundamental result in time-frequency analysis. Beginning with Baggett’s proof that a rectangular lattice Gabor system {h2lqw j(w n)}q>n5Z must be incomplete in O2 (R) whenever A 1, the necessary conditions for a Gabor system to be complete, a frame, a Riesz basis, or a Riesz sequence have been extended to arbitrary lattices and beyond. The first partial proofs of the Density Theorem for irregular Gabor frames were given by Landau in 1993 and by Ramanathan and Steger in 1995. A key fact proved by Ramanathan and Steger is that irregular Gabor frames possess a certain Homogeneous Approximation Property (HAP), and that the Density Theorem is a consequence of this HAP. This chapter provides a brief history of the Density Theorem and a detailed account of the proofs of Ramanathan and Steger. Furthermore, we show that the techniques of Ramanathan and Steger can be used to give a full proof of a general version of Density Theorem for irregular Gabor frames in higher dimensions and with finitely many generators.

5.1 Introduction 5.1.1 Frames and Gabor Systems Frames provide basis-like but generally nonunique representations of vectors in a Hilbert space. Moreover, these expansions possess a variety of desirable stability properties. One of these is inherent in the definition itself: A sequence {iq }q5N is a frame if the c2 -norm of the frame coecients {hi> iq i}q5N of i 5 Christopher Heil School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332 e-mail: [email protected]

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K is an equivalent norm for K. Thus a perturbation of i is directly reflected in the size of the frame coecients, and vice versa. Moreover, it follows Pfrom this equivalence of norms that there exist expansions of the form i = fq (i ) iq for each i 5 K. The coecients fq (i ) are not unique in general, but there is a canonical choice, and for this choice the expansions converge unconditionally (regardless of ordering). A frame for which the coecients are unique for each i is called a Riesz basis and is the image of an orthonormal basis under a continuous bijection of K onto itself. Precise definitions of all terminology are given in Section 5.2 below. Frames were first introduced by Dun and Schaeer [19] in the context of nonharmonic Fourier expansions, and today they have applications in a wide range of areas. We will be especially interested in a particular class of frames for O2 (Rg ) whose elements are simply generated from a single generator in O2 (Rg ). Given j 5 O2 (Rg ), called a window function or an atom, and given a sequence of points in the time-frequency plane Rg × Rg = R2g , the Gabor system generated by j and is G(j> ) = {h2l·w j(w {)}({>)5 = {P W{ j}({>)5 > where W{ j(w) = j(w{) is translation and P j(w) = h2lw· j(w) is modulation. That is, the Gabor system is generated by applying a discrete collection of time-frequency shift operators P W{ to the window j. The Gabor system is a frame if there exist constants D, E A 0 such that X |hi> P W{ ji|2 E ki k22 = ; i 5 O2 (Rg )> D ki k22 ({>)5

We will be concerned in this chapter with the question of when a Gabor system can be complete, a frame, a Riesz basis, or a Riesz sequence (a Riesz basis for its closed span) in K. There is a rich literature on this subject, which we will only attempt to briefly review, while the main body of the chapter will be concerned with an approach to this question introduced by Ramanathan and Steger in [47]. For a detailed survey of the history, context, and evolution of the Density Theorem, we refer to [30].

5.1.2 Gaussian Windows We begin our history with the special case of the Gaussian window *(w) = 2 21@4 hw in one dimension. Von Neumann [43, p. 406] claimed without proof that if is the unit lattice = Z2 , then the Gabor system G(j> Z2 ) = {Pq Wn *}n>q5Z is complete in O2 (R). This claim was proved to be correct in [6], [46], and [1]. However, completeness is quite a weak property. It only means that the finite linear span is dense, so every vector i 5 O2 (R) can

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be approximated as well as we like by some finite linear combination of the elements of G(*> Z2 ), but it gives us no information about what those linear combinations are. Gabor [23] conjectured more: he wrote that each i 5 O2 (R) can actually be represented in the form X fnq (i ) Pq Wn * (5.1) i= n>q5Z

for some scalars fnq (i ). This is not a consequence of completeness. Gabor’s claim was proved by Janssen [34] to be correct–but only in a weak sense, as he proved that the series in (5.1) converge only in the sense of tempered distributions and not in the norm of O2 . G(*> Z2 ) is neither a frame nor a Riesz basis for O2 (R).

5.1.3 The Density Theorem for Rectangular Lattice Gabor Systems As we have seen, the Gabor system generated by the Gaussian window with respect to the unit lattice is not particularly useful. However, we can try other windows and other index sets, such as a rectangular lattice of the form = Z×Z. Indeed, Daubechies, Grossmann, and Morlet sparked a revitalization of interest in frame theory when they proved in [16] that for each , A 0 with 1 it is possible to construct a compactly supported j such that G(j> Z × Z) forms a frame for O2 (R). When ? 1, we can even do this with j 5 Ff4 (R). For = 1, if we let j be the characteristic function of the interval [0> ], then G(j> Z × 1 Z) is an orthonormal basis for O2 (R). Thus, at least when ? 1, we can form Gabor frames using windows that are very well localized in the time-frequency plane. Two later fundamental results for Gabor systems on rectangular lattices show that the properties of the “Painless Nonorthogonal Expansions” of [16] are actually typical of Gabor systems. The first is the following Density Theorem, which states that the value of distinguishes between the cases where G(j> Z × Z) can be complete, a frame, a Riesz basis, or a Riesz sequence. Only the value of the product is relevant since by applying the unitary dilation operator Gu j(w) = u1@2 j(uw), G(j> Z × Z) is complete, a frame, a Riesz basis, or a Riesz sequence if and only if the same is true of G(Gu j> u Z×uZ). Theorem 5.1 (Density Theorem for Rectangular Lattices). Let j 5 O2 (R) and let = Z × Z where , A 0. Then the following statements hold. (a) If A 1, then G(j> Z × Z) is incomplete in O2 (R). (b) If G(j> Z × Z) is a frame for O2 (R), then 0 ? 1.

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(c) G(j> Z × Z) is a Riesz basis for O2 (R) if and only if it is a frame for O2 (R) and = 1. (d) If G(j> Z × Z) is a Riesz sequence in O2 (R), then 1. The Density Theorem and its extensions will be the focus of this chapter, but we also mention here a second result that explains why there is such a dierence in quality between the constructions of [16] for the cases ? 1 and = 1. This is the Balian—Low Theorem, which states that the generator of any rectangular lattice Gabor Riesz basis must be poorly localized in the time-frequency plane. Theorem 5.2 (Classical Balian—Low Theorem). If j 5 O2 (R) is such that G(j> Z × Z) is a Riesz basis for O2 (R), then µZ ¶ µZ ¶ |wj(w)|2 gw |ˆ j ()|2 g = 4= R

R

As a consequence, it is “redundant” (non-Riesz basis) Gabor frames that are usually used in practice.

5.1.4 Brief History of the Density Theorem Rectangular lattice Gabor systems are especially attractive because there are so many tools available for studying them. We will mention some of these while giving a brief review of the history of the Density Theorem. Part (a) of Theorem 5.1 was proved by Baggett in [2]. The time-frequency operators Pq Wn corresponding to the rectangular lattice Z × Z generate a von Neumann algebra, and Baggett made use of the representation theory of the discrete Heisenberg group to derive his proof. For an exposition of Baggett’s operator-theoretic proof, see [21]. Part (a) was also proved, for the special case that the product is rational by Daubechies [13]. Her proof relied on the Zak transform, which is another “algebraic” tool in the sense that it is dependent on the fact that the index set is a rectangular lattice. This proof is constructive in the sense that if j and a rational A 1 are given, then it constructs a nonzero function that is orthogonal to all the elements of G(j> Z × Z). Daubechies also noted in [13] that a proof for general can be inferred from results of Rieel [48] on coupling constants of F -algebras (however, no part of the Density Theorem is explicitly stated in [48]). Another proof of part (a), also based on coupling constants, was given by Daubechies, H. Landau, and Z. Landau in [17], and other proofs can be found in [22] and [9]. Speaking of the operator-theoretic approaches by Baggett and Rieel, Gr¨ochenig [25, p. 139] remarks:

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The power of this abstract mathematical theory seems to have been slightly suspicious to applied mathematicians . . .

and as a consequence they inspired a wide range of new and dierent approaches to issues connected with the Density Theorem. In particular, whereas part (b) is a special case of part (a), several dierent and revealing proofs of part (b) have been found (i.e., proofs which show that G(j> Z×Z) cannot be a frame when A 1, but which do not show that G(j> Z × Z) must be incomplete). For example, Janssen [35] gave a “simple” proof of parts (b) and (c) based on the algebraic structure of the rectangular lattice Z × Z and the remarkable Wexler—Raz biorthogonality relations for Gabor frames G(j> Z × Z). Before stating the Wexler—Raz relations, let us recall a few basic facts. If G(j> Z × Z) is a frame, then the frame operator X hi> Pq Wn ji Pq Wn j Vi = n>q5Z

is a positive, invertible mapping of O2 (R) onto itself. The canonical dual frame of G(j> Z × Z) is V 1 (G(j> Z × Z)). However, it is easy to see that V commutes with Pq Wn for n, q 5 Z, hence V 1 commutes with them as well, and so this dual frame actually is the Gabor frame G(˜ j > Z × Z) where j˜ = V 1 j. By expanding the equalities i = VV 1 i = V 1 Vi , we obtain the frame expansions X X hi> Pq Wn j˜i Pq Wn j = hi> Pq Wn ji Pq Wn j˜> (5.2) i= n>q5Z

n>q5Z

for i 5 O2 (R), where these series converge unconditionally in the norm of O2 (R). In general, however, the coecients in these expansions need not be unique. It is important to note that the structure of the index set = Z×Z is critical to these remarks–for an arbitrary sequence the dual frame of G(j> ) need not be another Gabor frame, cf. [5]. The next result is actually just a special case of the Wexler—Raz relations, cf. [25, Thm. 7.3.1], and see [30] for more detailed references. Theorem 5.3 (Wexler—Raz Biorthogonality Relations). Let j 5 O2 (R) and , A 0 be such that G(j> Z × Z) is a frame for O2 (R), with canonij > 1 Z × 1 Z) are cal dual frame G(˜ j > Z × Z). Then G(j> 1 Z × 1 Z) and G(˜ biorthogonal, specifically, ® 1 P q W n j> P q0 W n0 j˜ = nn0 qq0 =

(5.3)

Note that a frame has a biorthogonal system if and only if it is a Riesz basis. However, in the Wexler—Raz relations, it is not the frame G(j> Z×Z) that has a biorthogonal system, but rather the Gabor system G(j> 1 Z × 1 Z)

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defined on the adjoint lattice 1 Z × 1 Z. Part (b) of the Density Theorem follows directly from the Wexler—Raz relations. Using the notation given above, if G(j> Z × Z) is a frame, then from equation (5.3) we have that = hj> j˜i. However, we know that V 1 is a positive operator that commutes with Pq Wn , so by applying the frame expansions in (5.2) to the function i = j (and recalling that j˜ = V 1 j), we obtain the following: = hj> j˜i À ¿ X hj> Pq Wn V 1 ji Pq Wn j> V 1 j = n>q5Z

=

X

n>q5Z

=

X

n>q5Z

hj> Pq Wn V 1 ji hV 1 Pq Wn j> ji |hj> Pq Wn V 1 ji|2

|hj> V 1 ji|2

(5.4)

= |hj> j˜i|2 = ()2 = Consequently, 1, which proves part (b) of Theorem 5.1. Further, if G(j> Z × Z) is a Riesz basis, then it follows from basic frame principles that it is biorthogonal to its canonical dual frame G(˜ j > Z × Z), specifically, ®

Pq Wn j> Pq0 Wn0 j˜ = nn0 qq0 =

Hence equality holds in line (5.4) above, so = ()2 , and therefore = 1. This proves part (c) of Theorem 5.1. Part (d) of Theorem 5.1, dealing with Riesz sequences, is also related to Wexler—Raz, and more specifically to a stronger result known as the Duality Principle (sometimes called the Ron—Shen Duality Principle). A restricted form of the Duality Principle is as follows, cf. [25, Sec. 7.4], and see [30] for more detailed discussion and references.

Theorem 5.4 (Duality Principle). Let j 5 O2 (R) and let = Z × Z where , A 0. Then the following statements are equivalent. (a) G(j> Z × Z) is a frame for O2 (R). (b) G(j> 1 Z × 1 Z) is a Riesz sequence in O2 (R). If we accept the Duality Principle, then by interchanging the roles of , and 1 , 1 in Theorem 5.4, we see that if G(j> Z × Z) is a Riesz sequence in O2 (R), then G(j> 1 Z × 1 Z) must be a frame for O2 (R). But then by Theorem 5.1(b), we must have 1 1 1, which implies 1.

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5.1.5 The Density Theorem for Arbitrary Lattices Theorem 5.1 extends easily to rectangular lattices of the form = Zg ×Zg in higher dimensions. We can also immediately extend the Density Theorem to certain more general lattices, specifically those of the form = D(Z2g ) where D is a symplectic matrix and A 0. This is because if has this form, then there exists a unitary operator D : O2 (Rg ) $ O2 (Rg ) and scalars fD (n> q) with unit modulus such that ¡ © ¢ª (5.5) G(j> D(Z2g )) = fD (n> q) D Pq Wn k n>q5Zg >

where k = 1 D (j). The operator D is called a metaplectic transform. The reason that symplectic matrices and metaplectic transforms arise in this context is directly related to the representation theory of the Heisenberg group, see [25, Sec. 9.4] for definitions and details. If we remove the scalars fD (n>¢ q) from the right-hand set in (5.5), then ¡ the resulting set is D G(k> Z2g ) , which is the image of a rectangular lattice Gabor system under a unitary map. Yet although the right-hand set in (5.5) is not precisely a Gabor system, since the property of being complete, a frame, a Riesz basis, or a Riesz sequence is preserved both by unitary mappings and by multiplication of the elements by scalars of unit modulus, we see that if D is a symplectic matrix, then G(j> D(Z2g )) is complete, a frame, g a Riesz basis, or a Riesz sequence in O2 (R ¢ only if the same is true ¡ ) if and of the rectangular lattice Gabor system G k> Z2g . This allows us to easily extend the Density Theorem to the particular case where = D(Z2g ) with D symplectic. The surprise is that the Density Theorem actually extends to arbitrary lattices = D(Z2g ), where D can be any invertible 2g × 2g matrix. This is nontrivial when D is not a multiple of a symplectic matrix. Define the volume of a lattice D(Z2g ) to be the area of a fundamental domain for the lattice: ¢ ¡ vol D(Z2g ) = | det(D)|= ¡ In ¢particular, for a one-dimensional (g = 1) rectangular lattice, vol Z × Z = . With this notation, the Density Theorem with respect to arbitrary lattices is as follows. Theorem 5.5 (Density Theorem for Lattices). Let j 5 O2 (Rg ) and let = D(Z2g ) where D is an invertible 2g × 2g matrix. Then the following statements hold. (a) If vol(D) A 1, then G(j> D(Z2g )) is incomplete in O2 (Rg ). (b) If G(j> D(Z2g )) is a frame for O2 (Rg ), then 0 ? vol(D) 1. (c) G(j> D(Z2g )) is a Riesz basis for O2 (Rg ) if and only if it is a frame for O2 (Rg ) and vol(D) = 1.

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(d) If G(j> D(Z2g )) is a Riesz sequence in O2 (Rg ), then vol(D) 1. We refer to [30] for a complete discussion of references. Parts (b) and (c) of the Density Theorem are immediate consequences of Ramanathan and Steger’s results on the Density Theorem for irregular Gabor frames in [47], which we will discuss in more detail below. More precisely, [47] applies to g = 1; the extension to higher dimensions was made in [11] and is also proved in this chapter. Parts (b), (c), and (d) are consequences of the extension by Feichtinger and Kozek of the Wexler—Raz relations and the Duality Principle to arbitrary lattices [20]. In that paper, they introduced the notion of the adjoint of a general lattice (which is distinct from the better-known dual lattice that plays a role in many formulas in Fourier analysis). It appears that part (a) of the Density Theorem was only established in its full generality recently, by Bekka. The following is [7, Thm. 4], and is only a special case of the more general results obtained in that paper (Bekka himself attributes this result to Feichtinger and Kozek [20], but while, as we have mentioned, that paper does contain many results for Gabor systems on arbitrary lattices, it does not contain Theorem 5.6). Theorem 5.6 (Existence of Lattice Gabor Frames). Let = D(Z2g ) be a lattice in R2g . Then the following statements are equivalent. (a) vol() 1. (b) There exists a j 5 O2 (Rg ) such that G(j> ) is complete in O2 (Rg ). (c) There exists a j 5 O2 (Rg ) such that G(j> ) is a frame for O2 (Rg ). The following statements are also equivalent. (a’) vol() = 1. (b’) There exists a j 5 O2 (Rg ) such that G(j> ) is a Riesz basis for O2 (Rg ). (c’) There exists a j 5 O2 (Rg ) such that G(j> ) is an orthonormal basis for O2 (Rg ). For the case of separable lattices, i.e., = D(Zg )×E(Zg ), the equivalences in Theorem 5.6 were earlier proved in [28], [22]. We feel that Theorem 5.6 is quite surprising. For the one-dimensional case, 2 Daubechies and Grossmann [15] conjectured that if *(w) = 21@4 hw is the Gaussian window, then G(*> Z × Z) is a frame if and only if 0 ? ? 1. Evidence supporting this conjecture was given in [13], and the full conjecture was proved by Lyubarskii [42] and by Seip and Wallst´en [50], [51] (see also the simple proof and additional references in [35]). It is therefore tempting to expect that if we let (w) = 2g@4 hw·w be the g-dimensional Gaussian function, then G(> D(Z2g )) will be a frame whenever vol(D) ? 1. However, this is false. Even if we consider more general rectangular lattices of the form

5 The Homogeneous Approximation Property

=

g Y

l=1

l Z ×

79 g Y

l Z>

l=1

i.e., a diagonal matrix D, it follows from the Lyubarskii and Seip/Wallst´en characterization for g = 1 that G(> ) is complete in O2 (Rg ) if and only if l l 1 for each l. Hence if 0 ? vol() ? 1 but l l A 1 for some l, then G(> ) will be incomplete. In particular, whenever g A 1, there exist lattices with arbitrarily small volume such that G(> ) is incomplete in O2 (Rg ). Yet by Theorem 5.6, there must exist some j such that G(j> ) is a frame. However, Bekka’s proof of Theorem 5.6 is not constructive. On the other hand, the proof in [28] of Theorem 5.6 for the special case of separable lattices = D(Zg ) × E(Zg ) is constructive. The window constructed in [28] is the characteristic function of a set. In some cases, this set will be compact; e.g., this is the case if D, E have all rational entries.

5.1.6 The Density Theorem for Irregular Gabor Systems All of the previous discussion relied in one way or another on the structural properties of the index set . However, consider what happens if we take a lattice D(Z2g ) and perturb even one single point of the lattice–the resulting set is no longer a lattice. Every “algebraic” tool that has so far been mentioned, including the Duality Principle, the Wexler—Raz biorthogonality relations, the Zak transform, and von Neumann algebra techniques (and other techniques that we have not described such as the Walnut representation and the Janssen representation), is rendered inapplicable to the study of G(j> ). Yet necessary conditions for G(j> ) to be a frame or a Riesz basis for O2 (Rg ) are known even for arbitrary sequences . These require entirely new tools, which were first supplied by H. Landau [40], Ramanathan and Steger [47], and Janssen [36], who provided the first partial extensions of the Density Theorem to irregular Gabor frames (and, in the case of [36], to more general systems). In Theorems 5.1 and 5.5, the value that distinguishes between the various cases is the volume of the lattice, which is the area of a fundamental domain for the lattice. In the irregular setting there is no analogue of a fundamental domain, and instead it is the Beurling density of that distinguishes between the various cases. Beurling density measures in some sense the average number of points inside unit cubes. However, because the points are not uniformly distributed, there is not a single definition, but rather lower and upper limits to the average density, which we denote by G () and G+ (), respectively. The precise definition is given in Section 5.2.4 below.

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The Density Theorem for Gabor systems with arbitrary index sets is as follows (we say that is uniformly separated if there exists a A 0 such that | | for all 6= 5 ). Theorem 5.7 (Density Theorem for Gabor Frames). Let j 5 O2 (Rg ) be given and let be a sequence of points in R2g . (a) If G(j> ) is complete in O2 (Rg ), then 0 G () G+ () 4. (b) If G(j> ) is a frame for O2 (Rg ), then 1 G () G+ () ? 4. (c) If G(j> ) is a Riesz basis for O2 (Rg ), then G () = G+ () = 1. Moreover, is uniformly separated. (d) If G(j> ) is a Riesz sequence in O2 (Rg ), then 0 G () G+ () 1. Moreover, is uniformly separated. The critical value G± () = 1 is sometimes called the Nyquist density. To compare with the previous Density Theorems, we note that the Beurling density of a rectangular lattice is G (Zg × Zg ) = G+ (Zg × Zg ) =

1 > ()g

and the density of a general lattice D(Z2g ) is G (D(Z2g )) = G+ (D(Z2g )) =

1 1 = > | det(D)| vol(D)

i.e., high density corresponds to small volume and conversely. There is a rich literature on ideas and results closely related to the Density Theorem in the settings of sampling and interpolation of band-limited functions, density conditions for systems of windowed exponentials, sampling in the Bargmann—Fock space of entire functions, and density conditions for abstract localized frames. Indeed, the precise formulation of the Nyquist density is due to Landau [38], [39], in the context of sampling and interpolation of band-limited functions. We will not attempt to describe that literature here.

5.1.7 A Brief History Note that in contrast with the Density Theorem for the case of lattices, part (a) of Theorem 5.7 has an empty conclusion (i.e., it gives no information), for by definition of Beurling density we always have that 0 G () G+ () 4. Ramanathan and Steger conjectured in [47] that Theorem 5.7(a) should be improvable to say that if G+ () ? 1, then G(j> ) is incomplete in O2 (R). This was shown in [8] to be false: for any % A 0, there exists a function j 5 O2 (R) and a sequence R2 with 0 ? G+ () ? % such that G(j> ) is complete. Indeed, it has even been shown that there exist j and

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such that G+ () = 0 yet G(j> ) is complete [44], [45], [49]. Thus there is a stark contrast between lattice and nonlattice Gabor systems with regard to completeness. For the one-dimensional case (g = 1), part (b) of Theorem 5.7 was first proved, but with extra hypotheses on j, by H. Landau [40]. Landau’s technique is related to the energy concentration result for rectangular lattice Gabor frames proved by Daubechies [14, Thm. 3.5.2]. Parts (b) and (c) were proved for arbitrary j 5 O2 (R), but with extra hypotheses on , by Ramanathan and Steger [47]. Specifically, Ramanathan and Steger only considered the case where is uniformly separated. However, not all irregular Gabor frames are uniformly separated. For example, if we let s j = "[0>1] , then G(j> Z2 ^ 2Z2 ) is a frame, but the index set is not uniformly separated. The fact a Gabor Riesz sequence must be uniformly separated was proved in [18]. Janssen [36] proved part (b) for “half irregular” R2 , i.e., of the form = Z × with irregular. Furthermore, Janssen’s result actually applies to certain more general systems whose elements need not be exact timefrequency shifts of a single generator. In their paper, Ramanathan and Steger introduced a fundamental new concept in the study of Gabor frames. Namely, they showed that all Gabor frames satisfy a certain Homogeneous Approximation Property (HAP). Gr¨ochenig and Razafinjatovo [27] modified and improved this technique to derive density conditions in the context of sampling of band-limited functions. Inspired by [27], a complete proof of Theorem 5.7, without restrictions on j or , was given in [11]. Additionally, the Density Theorem was extended in [11] to the case of higher dimensions and finitely many generators, and some other applications were made. Further results related to the density of systems of windowed exponentials appear in [31], and the role of the HAP in wavelet theory is explored in [32]. The HAP was one of the inspirations for the study of abstract localized frames in the papers [4], [5]. Among other results, it was shown there that the HAP and the Density Theorem are consequences of more general considerations rather than the particular rigid structure of Gabor systems. In particular, part (d) of Theorem 5.7 was first proved in [5]. Localized frames were independently introduced and published by Gr¨ochenig [26], for entirely dierent purposes.

5.1.8 Goals of This Paper The introduction of the HAP by Ramanathan and Steger was a fundamental advance in the understanding of arbitrary (“irregular”) Gabor frames. Unfortunately, the proofs in [47] were mostly given in “sketch” form, and the significance of this paper has not generally been well appreciated. Our goal in

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the remainder of this chapter is to give a more complete exposition of some of the proofs of the results of Ramanathan and Steger. In the process, we also show that their approach can in fact be applied to completely arbitrary Gabor frames G(j> ), without restrictions on j or (although the modified technique introduced by Gr¨ ochenig and Razafinjatovo in [27] clearly remains more “elegant”). We also extend to the case of finitely many generators and to higher dimensions. The theorems we obtain are not new and indeed are contained in full generality in [11]. However, we hope this chapter will inspire the reader to further consider the Homogeneous Approximation Property and to pursue the most recent results on localized frames. Of course, [47] is not the only significant advance in the theory of irregular Gabor frames. An incomplete list of other references not already mentioned above includes [24], [12], [52], [3], [41], [53], [37]. Our chapter is organized as follows. In Section 5.2, we present our notation, background, and technical results on the short-time Fourier transform, frames, density, and weak convergence of sequences of sets. The drawback of the Ramanathan and Steger approach is evident in this section–a great deal of technicality is required to deal with the weak convergence of sequences of sequences and their corresponding Gabor systems. The modified HAP introduced by Gr¨ochenig and Razafinjatovo allows a much “cleaner” approach, and in particular, the use of weak convergence is avoided in [27]. In Section 5.3, we define several versions of the HAP and use this to derive the Density Theorem in higher dimensions with finitely many generators.

5.2 Notation and Preliminaries In this section, we define our terminology and provide some background and discussion related to our results.

5.2.1 General Notation Let = {l }l5L be a sequence of points in R2g , with a countable index set L. For simplicity of notation, we will write R2g , but we always mean that is a sequence and not merely a subset of R2g . In particular, repetitions of elements are allowed. A sequence is a lattice if = D(Z2g ) where D is a 2g × 2g invertible matrix. g Often we will deal SQ with several sequences 1 > = = = > Q R , and will use the notation = n=1 n to denote the disjoint union of these sequences. In particular, if each n is indexed as n = {mn }m5N , then is the sequence = {11 > = = = > 1Q > 21 > = = = > 2Q > = = = }.

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Given } 5 R2g and k A 0, we let Tk (}) denote the closed cube in R2g centered at } with side lengths k, specifically, Tk (}) =

2g Y

[}m k2 > }m + k2 ]=

m=1

If K is a Hilbert space and il 5 K for l 5 L, then span{il }l5L will denote the finite linear span of {il }l5L , and span{il }l5L will denote the closure of this set in K. The distance from a vector i 5 K to a closed subspace Y K is dist(i> Y ) = inf{ki yk : y 5 Y } = ki SY i k, where SY is the orthogonal projection onto Y . The translation of a function j by { 5 Rg is W{ j(w) = j(w {), and the modulation of j by 5 Rg is P j(w) = h2l·w j(w). Using this notation, G(j> ) = {P W{ j}({>)5 . The family of translation operators is strongly continuous, as is the family of modulation operators. This implies the following. Lemma 5.8. Let i 5 O2 (Rg ) and % A 0 be given. Then there exists A 0 such that ; x> 5 Rg >

|x|> || ? =, kP Wx i i k2 ? %=

Corollary 5.9. Let N Rg be compact, and let i 5 O2 (Rg ) and % A 0 be given. Then there exists A 0 such that ; ({> ) 5 N>

; (x> ) 5 R2g > |x|> || ? =, kP Wx P W{ i P W{ i k2 ? %=

5.2.2 Amalgam Space Properties of the Short-Time Fourier Transform We will need the following facts regarding properties of the short-time Fourier transform (STFT). We refer to [25] for a detailed discussion of the STFT, the modulation spaces, and the amalgam spaces, and for references to the original literature. Also, a survey of amalgam spaces appears in [29]. Definition 5.10. The short-time Fourier transform (STFT) of a function i 5 O2 (Rg ) with respect to a window j 5 O2 (Rg ) is

® Yj i ({> ) = i> P W{ j Z i (w) h2l·w j(w {) gw> ({> ) 5 R2g = = Rg

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We have that Yj i is continuous and bounded on R2g , and kYj i k2 = ki k2 kjk2 > see [25, Cor. 3.2.2]. We will need some finer properties of the STFT, which we recall next. The modulation spaces were invented by Feichtinger, and extensively investigated by Feichtinger and Gr¨ ochenig. They are now recognized as the appropriate function spaces for time-frequency analysis, and they occur naturally in mathematical problems involving time-frequency shifts P W{ . For our purposes, the following particular modulation space will be sucient. Definition 5.11. The modulation space P 1 (Rg ) consists of all i 5 O2 (Rg ) such that Z Z |Y i ({> )| g{ g ? 4> ki kP 1 = kY i k1 = Rg

Rg

where (w) = 2g@4 hw·w . P 1 is a Banach space, and its definition is independent of the choice of window, i.e., may be replaced by any nonzero function in the Schwartz class, or indeed by any function in P 1 , in the sense of equivalent norms. P 1 contains the Schwartz class, and hence is dense in O2 . The space P 1 is also called the Feichtinger algebra, and is sometimes denoted V0 . We will need the following amalgam space property of the STFT, see [25, Thm. 12.2.1]. Lemma 5.12. If j 5 O2 (Rg ) and i 5 P 1 (Rg ), then Yj i belongs to the amalgam space Z (O4 > c2 ), i.e., X kYj i kZ (O4 >c2 ) = kYj i · "T1 (m) k24 ? 4= (5.6) m5Z2g

Any size cubes may be used to define the amalgam space (in the sense of equivalent norms), i.e., we may replace the unit cubes T1 (m) in (5.6) by the cubes T (m) where A 0. The first amalgam spaces were introduced by Wiener in his study of generalized harmonic analysis. A comprehensive general theory of amalgam spaces on locally compact groups was introduced by Feichtinger, and extensively studied by Feichtinger and Gr¨ochenig. The amalgam space properties of the STFT were not available when Ramanathan and Steger wrote their paper. Instead, they made use of the fact 2 that if is the Gaussian function (w) = hw , then the STFT Y i is very closely related to the Bargmann transform Ei , which is an analytic function. Our use of amalgam spaces is perhaps a little more straightforward, but is not an essential change to their argument.

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5.2.3 Bases and Frames We use the standard notations for frames and Riesz bases as found in the texts [10], [14], [25], [54] or the research-tutorial [33]. Definition/Facts 5.13. Let F = {il }l5N be a sequence in a Hilbert space K. (a) F is complete if its finite linear span is dense in K.

(b) F is minimal if there exists a sequence {i˜l }l5N in K that is biorthogonal to F, i.e., hil > i˜m i = lm for l, m 5 N. Equivalently, {il }l5N is minimal @ span{il }l6=m for each m 5 N. We say that F is exact if it is both if im 5 minimal and complete. In this case, the biorthogonal sequence is unique. (c) A Riesz basis is the image of an orthonormal basis under a continuous invertible mapping of K onto itself. (d) F is a frame for K if there exist constants D, E A 0, called frame bounds, such that ; i 5 K>

D ki k2

4 X l=1

|hi> il i|2 E ki k2 =

(5.7)

All Riesz bases are frames, but not conversely. (e) If F satisfies at least the second inequality in (5.7), then we say that it is a Bessel sequence or that it possesses an upper frame bound, and we call E a Bessel bound. We have that F is a Bessel sequence if and only if the analysis operator Fi = {hi> il i}l5N is a bounded mapping F : K $ c2 . In this case, the adjointPof F is the synthesis operator F : c2 $ K fl il (the series converges unconditionally in given by F ({fl }l5N ) = the norm of K). In particular, if E is a Bessel bound, then 2

; {fl }l5N 5 c >

°2 °X X ° ° ° fl il ° |fl |2 = ° E ° l5N

(5.8)

l5N

Hence kil k2 E for every l, so every Bessel sequence is uniformly bounded above in norm. P (f) If F is a frame, then the frame operator Vi = F Fi = hi> il i il is a bounded, positive definite, invertible map of K onto itself.

(g) Every frame F has a canonical dual frame F˜ = {i˜l }l5N given by i˜l = V 1 il where V is the frame operator. Writing out and rearranging the equalities i = VV 1 i = V 1 Vi gives the frame expansions ; i 5 K>

i=

4 X l=1

hi> i˜l i il =

4 X l=1

hi> il i i˜l >

(5.9)

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and furthermore the series in (5.9) converge unconditionally for every i (so any countable index set can be used to index a frame). In general, the coecients in (5.9) need not be unique. (h) If F is a frame and F˜ is its canonical dual frame, then the following statements are equivalent: i. F is a Riesz basis, ii. F is exact, iii. the coecients in (5.9) are unique for each i 5 K, iv. F and F˜ are biorthogonal. (i) We say that F is a frame sequence or a Riesz sequence if it is a frame or a Riesz basis for its closed linear span in K, respectively. For the case of a Gabor frame G(j> ), if is a rectangular lattice of the form = Zg × Zg , then the canonical dual frame is also a Gabor frame with respect to the same lattice, i.e., it has the form G(˜ j > ) where j˜ = V 1 j [25, Prop. 5.2.1]. However, if is not a lattice, then the dual frame will not be a Gabor frame in general. There will exist a canonical dual frame, but this dual frame need not consist of time-frequency shifts of a single function.

5.2.4 Beurling Density Beurling density measures in some sense the average number of points contained in unit cubes. The precise definition is as follows. Definition 5.14. Let = {l }l5L be a sequence of points in R2g . The upper and lower Beurling densities of are, respectively, ¡ ¢ # _ Tk (}) > G+ () = lim sup sup k2g k$4 }5Rg ¡ ¢ # _ Tk (}) G () = lim inf inf = k$4 }5Rg k2g In general Q X

n=1

G (n ) G

³S Q

n=1

Q ´ ³S ´ X Q n G+ n G+ (n )> n=1

n=1

but these inequalities may be strict, e.g., consider 1 = {(n1 > n2 ) 5 Z2 : n1 0> n2 5 Z}> 2 = {(n1 > n2 ) 5 Z2 : n1 ? 0> n2 5 Z}=

5 The Homogeneous Approximation Property

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We say that a sequence is -uniformly separated if | | for all 6= 5 . A sequence satisfies G+ () ? 4 if and only if = 1 ^ · · · ^ Q with each n being n -uniformly separated for some n A 0 [11, Lem. 2.3]. It was shown in [11] that if G(j> ) is a Bessel sequence, then must have finite density. Specifically, we have the following. Proposition 5.15. If j 5 O2 (Rg ) \ {0} and R2g are such that G(j> ) is a Bessel sequence, then G+ () ? 4. Ramanathan and Steger only considered uniformly separated sequences, which automatically have finite density.

5.2.5 Weak Convergence of Sequences In this section, we develop some machinery concerning the weak convergence of a sequence of subsets of R2g . Definition 5.16. (a) Given a set H R2g , for each w 0 define Hw = {{ 5 R2g : dist({> H) ? w}= (b) The Fr´echet distance between two closed sets H, I R2g is ª © [H> I ] = inf w 0 : H Iw and I Hw =

(c) Given closed sets Hq R2g and given a closed set I R2g , we say that Hq converges weakly to I if ; compact N R2g >

lim [Hq _ N> I _ N] = 0=

q$4

w

In this case, we write Hq $ I . Lemma 5.17. Let 1 > = = = > Q R2g be countable sequences such that each n is n -uniformly separated for some n A 0. Then given any sequence of points {}q }q5N in R2g , there exists a subsequence {zq }q5N of {}q }q5N and there exist sequences 0n R2g such that w

n zq $ 0n as q $ 4>

n = 1> = = = > Q=

Further, if for some n we have G (n ) A 0, then we can construct 0n so that G (0n ) A 0 as well. Proof. Since a subsequence of a weakly convergent sequence of sets is still weakly convergent to the same limit set, it suces to consider the case Q = 1.

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Thus, assume that R2g is -uniformly separated and that {}q }q5N is given. If G () A 0, there will exist some u A 0 such that every cube Tu (}) in 2g R will contain at least one element of . If G () = 0, define u = 1. Write S Um > R2g = m5N

where the Um are closed cubes of sidelength u with disjoint interiors. Since is uniformly separated, there is a finite integer P such that ; m> q 5 N>

#( }q _ Um ) P=

(5.10)

Furthermore, in the case that G () A 0 we also have ; m> q 5 N>

1 #( }q _ Um )=

(5.11)

Because of equation (5.10), there must exist some integer 0 P1 P and some subsequence {}q1 }q5N of {}q }q5N such that ; q 5 N>

#( }q1 _ U1 ) = P1 =

In the case that G () A 0, equation (5.11) implies that P1 1. Similarly, again because of equation (5.10), there must exist some integer 0 P2 P and some subsequence {}q2 }q5N of {}q1 }q5N such that ; q 5 N>

#( }q2 _ U2 ) = P2 >

and again P2 1 if G () A 0. Continue in this way to create a subsequence {}qm+1 }q5N of {}qm }q5N for each m 5 N. Set zq = }qq . Then {zq }q5N is a subsequence of {}qm }qm for each m 5 N, so ; m 5 N>

; q m>

#( zq _ Um ) = Pm >

where 0 Pm P . Let M = {m 5 N : Pm A 0}. For m 5 M and q m, write zq _ Um = {qm>1 > = = = > qm>Pm }= Fix m 5 M and 1 p Pm . The sequence {qm>p }qm is contained in the compact set Um , so it must have a convergent subsequence. By using another diagonalization argument, passing to a subsequence of {zq }q5N if necessary, we can assume that {qm>p }q5N converges for each m 5 M and p = 1> = = = > Pm . Set m>p = lim qm>p q$4

and define

5 The Homogeneous Approximation Property

89

0 = {m>p }m5M> p=1>===>Pm =

Then for any compact set N R2g and any % A 0, we have that for all q large enough, every point of 0 _ T is within % of a point in zq and conversely. w That is, [ zq _ N> 0 _ N] ? % for all q large enough, so zq $ 0 . Finally, if G () A 0, then Pm 1 for every m, so each Um contains at least one element of 0 . Thus, in this case we have G (0 ) A 0. Remark 5.18. 0 could be empty. For example, consider = {0} × Z and w }q = (q> 0). Then }q $ >. However, if G () A 0, then Lemma 5.17 implies that we can construct a nonempty 0 .

5.2.6 Weak Convergence of Gabor Frames We will need the following technical result. Lemma 5.19. Let j1 > = = = > jQ 5 O2 (Rg ) \ {0} and 1 > = = = > Q R2g be such that: (a) each n is n -uniformly separated for some n A 0, and S 2 g (b) G = Q n=1 G(jn > n ) is a frame for O (R ) with frame bounds D, E.

Suppose that }q 5 R2g and 01 > = = = > 0Q R2g are such that w

n }q $ 0n as q $ 4> Then G 0 =

SQ

n=1

n = 1> = = = > Q=

G(jn > 0n ) is a frame for O2 (Rg ) with frame bounds D, E.

Proof. Set = min{1 @4> = = = > Q @4}, so each n is 2-uniformly separated. In particular, any cube T (}) can contain at most one point of any n . Choose any % A 0. Fix any nonzero i 5 P 1 (Rg ). Then by Lemma 5.12, we have Yjn i 5 Z (O4 > c2 ). Hence we can find an p 5 N such that X

m5Z2g \Tp (0)

% > kYjn i · "T (m) k24 ? 2Q

n = 1> = = = > Q=

(5.12)

Set U = (2p + 1). Since each translated set n }q has the same density, we have ¡ ¢ # n }q _ TU (0) ? 4= G = sup q5N> n=1>===>Q w

Fix ? @2. Since n }q $ 0n , for compact set N R2g we can find an q such that [n }q _ N> 0 _ N] ? >

n = 1> = = = > Q=

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Hence each point of n }q _ N is within of a point in 0 _ N, and conversely. Since n }q is 2-uniformly separated and ? @2, it follows that each point of n }q _ N is within of a unique point in 0 _ N, and conversely. Hence, if 6= 5 0 _N, then there exist unique points 6= 5 n }q _N such that | | ? and | | ? . Since n }q _ N is 2-uniformly separated, we therefore have 2 | | | | + | | + | | ? | | + + ? | | + = Thus | | A , and hence we have that 0n _ N is -uniformly separated. Since this is true for all compact sets N, we conclude that 0n is -uniformly separated. Thus each cube T (}) can contain at most one point of each 0n . Now, if ({> ) 5 R2g \ TU (0), then there is a unique m 5 Z2g \ Tp (0) such that ({> ) 5 T (m). Hence, Q X

n=1

X

({>)50n \TU (0)

|hi> P W{ ji|2 =

Q X

n=1 Q X

X

({>)50n \TU (0)

X

n=1 m5Z2g \Tp (0)

?

% > 2

|Yjn i ({> )|2 sup

(x>)5T (m)

|Yjn i (x> )|2 (5.13)

the last inequality following from equation (5.12). Similarly, for each q 5 N we have Q X

X

n=1 ({>)5n }q \TU (0)

|hi> P W{ jn i|2 ?

% = 2

Consider for the moment the case G A 0. Since TU (0) is compact, we have by Corollary 5.9 that there exists A 0 such that ; ({> ) 5 TU (0)> ; |x|> || ? >

kP Wx P W{ i P W{ i k2 ?

% = 2(Q G)1@2 ki k2

Let q be large enough that [n }q _ TU (0)> 0n _ TU (0)] ? = points |x({> > n)|, Then for each ({> ) 5 n¡ }q _ TU (0), there exist unique ¢ |({> > n)| ? such that { + x({> > n)> + ({> > n) 5 0n . Furthermore, ¡ ¢ ({> ) 7$ { + x({> > n)> + ({> > n)

5 The Homogeneous Approximation Property

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is a bijection of n }q _ TU (0) onto 0n _ TU (0). Hence, ¯µX ¯ Q ¯ ¯

X

n=1 ({>)50n _TU (0)

|hi> P W{ ji|2

µX Q

¶1@2

X

n=1 ({>)5n }q _TU (0)

µX Q

µX Q

|hi> P+({>>n) W{+x({>>n) jn i hi> P W{ jn i|2

X

|hi> P+({>>n) W{+x({>>n) jn P W{ jn i|2

X

ki k22

n=1 ({>)5n }q _TU (0)

µX Q

n=1 ({>)5n }q _TU (0)

?

µ Q G ki k22

¶1@2 ¯ ¯ ¯ ¯

X

n=1 ({>)5n }q _TU (0)

=

|hi> P W{ jn i|2

¶1@2

%2 4Q Gki k22

=

kP+({>>n) W{+x({>>n) jn

¶1@2

P W{ jn k22

% = 2

¶1@2

¶1@2 (5.14)

If G = 0, then the quantity being estimated above is zero. In any case, if we write }q = (yq > q ), then it follows that µX Q

X

n=1 ({>)50n

µX Q

|hi> P W{ ji|2 X

n=1 ({>)50n _TU (0)

+

µX Q

¶1@2

|hi> P W{ ji|2 X

n=1 ({>)50n \TU (0)

µX Q

X

n=1 ({>)5n }q _TU (0)

¶1@2 2

|hi> P W{ ji| 2

|hi> P W{ ji|

¶1@2

¶1@2 +

by (5.13) and (5.14)

µX Q

X

n=1 ({>)5n

2

|hPq Wyq i> P W{ ji|

¶1@2

+ %

E 1@2 kPq Wyq i k2 + % = E 1@2 ki k2 + %=

% % + 2 2

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Christopher Heil

Similarly, µX Q

X

2

n=1 ({>)50n

µX Q

|hi> P W{ ji| X

n=1 ({>)50n _TU (0)

µX Q

|hi> P W{ ji|2

X

n=1 ({>)5n }q _TU (0)

=

µX Q

X

n=1 ({>)5n }q

Q X

¶1@2

|hi> P W{ ji|2

X

X

2

n=1 ({>)5n }q

|hi> P W{ ji|

µX Q

% 2

X

|hi> P W{ ji|2

by (5.13)

¶1@2

% 2

¶1@2

n=1 ({>)5n }q \TU (0)

D1@2 ki k2

¶1@2

|hi> P W{ ji|2

n=1 ({>)5n }q \TU (0)

µX Q

¶1@2

|hi> P W{ ji|

2

¶1@2

% 2

% % = D1@2 ki k2 %= 2 2

Since % is arbitrary, we conclude that 1

g

; i 5 P (R )>

D ki k22

Q X

n=1

X

({>)50n

|hi> P W{ ji|2 E ki k22 =

Since P 1 (Rg ) is dense in O2 (Rg ), this inequality extends to all i 5 O2 (Rg ), which completes the proof.

5.3 Density of Gabor Frames We now develop the main results on the Homogeneous Approximation Property and the density of Gabor frames. First we define the Homogeneous Approximation Property (HAP) introduced by Ramanathan and Steger and some variations on this theme. The

5 The Homogeneous Approximation Property

93

particular HAP introduced in [47] will here be called the Ramanathan—Steger Weak HAP. We also introduce a slightly stronger property that we call the Ramanathan—Steger Strong HAP. In [27], Gr¨ochenig and Razafinjatovo introduced a modified version of the Ramanathan—Steger Weak HAP. The analogue of their definition for the case of Gabor systems will be called the Weak HAP. This Weak HAP was used in [11] to give a complete proof of the Density Theorem for Gabor frames. We also define a slightly stronger version that we call the Strong HAP. Remark 5.20. The terminology for Weak and Strong HAP used here is consistent with that used in [4], but diers from that used in [11]. Specifically, the definition of what was called “Strong HAP” in [11] was equivalent to the definition of the “Weak HAP” in [11], and both of those coincide with the definition of the Weak HAP used in this chapter.

5.3.1 Definition of the HAP Definition 5.21. Let j1 > = = = > jQ 5 O2 (Rg ) \ {0} and 1 > = = = > Q R2g be such that Q Q S S G(jn > n ) = {P W{ jn }({>)5n G= n=1

n=1

is a frame O2 (Rg ). Let

Q S {˜ j{>>n }({>)5n G˜ = n=1

denote the canonical dual frame of G (in general, G˜ need not itself be a Gabor frame). For each k A 0 and (x> ) 5 R2g , set ª © (5.15) Z (k> x> ) = span j˜{>>n : ({> ) 5 n _ Tk (x> )> n = 1> = = = > Q > ª © ˜ (k> x> ) = span P W{ jn : ({> ) 5 n _ Tk (x> )> n = 1> = = = > Q = (5.16) Z (a) We say that G possesses the Ramanathan—Steger Weak Homogeneous Approximation Property (R—S Weak HAP) if

; i 5 O2 (Rg )> ; % A 0> < U A 0 such that ; (x> ) 5 R2g > ¢ ¡ ˜ (U> x> ) ? %= dist P Wx i> Z (5.17) (b) We say that G possesses the Weak Homogeneous Approximation Property (Weak HAP) if

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Christopher Heil

; i 5 O2 (Rg )> ; % A 0> < U A 0 such that ; (x> ) 5 R2g > ¢ ¡ dist P Wx i> Z (U> x> ) ? %= (5.18) (c) We say that G possesses the Ramanathan—Steger Strong Homogeneous Approximation Property (R—S Strong HAP) if ; i 5 O2 (Rg )> ; % A 0> < U A 0 such that ; (x> ) 5 R2g > ° ° Q X X ° ° °P Wx i hP Wx i> j˜{>>n i P W{ jn ° ° ? %= ° n=1 ({>)5n _TU (x>)

2

(5.19) (d) We say that G possesses the Strong Homogeneous Approximation Property (Strong HAP) if ; i 5 O2 (Rg )> ; % A 0> < U A 0 such that ; (x> ) 5 R2g > ° ° Q X X ° ° ° °P Wx i hP W i> P W j i j ˜ x { n {>>n ° ? %= ° n=1 ({>)5n _TU (x>)

2

(5.20)

Note that since the function Q X

X

hP Wx i> P W{ jn i j˜{>>n

n=1 ({>)5n _TU (x>)

is one element of the space Z (U> x> ), the Strong HAP implies the Weak HAP. A argument similar to [5, Thm. 5.1(e)] can be used to show that if G is a Riesz basis, then the Weak HAP implies the Strong HAP. Similar remarks apply to the R—S Strong and Weak HAPs.

5.3.2 The HAP for Gabor Frames Now we establish that every Gabor frame with finitely many generators satisfies the R—S Weak HAP (in comparison, the argument in [11] shows that every Gabor frame with finitely many generators satisfies the Strong HAP). Theorem 5.22. If j1 > = = = > jQ 5 O2 (Rg ) \ {0} and 1 > = = = > Q R2g are S 2 g such that G = Q n=1 G(jn > n ) is a frame for O (R ), then G possesses the R—S Weak HAP. Proof. By Proposition 5.15, we have G+ (n ) ? 4 for each n. Therefore each n is the union of finitely many uniformly separated subsequences. Hence,

5 The Homogeneous Approximation Property

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by passing to subsequences if necessary, we may assume that each n is n uniformly separated for some n A 0. Set = min{1 @4> = = = > Q @4}. Then each n is 2-uniformly separated. Let Q S {˜ j{>>n }({>)5 G˜ = n

n=1

denote the canonical dual frame of G, and let D, E be frame bounds for G. Suppose that the R—S Weak HAP fails. Then there exists an i 5 O2 (Rg ) and an % A 0 such that for each q 5 N we can find a point }q = (xq > q ) 5 R2g such that ¢ ¡ ˜ (q> xq > q ) %> dist Pq Wxq i> Z or, equivalently, ³ © ª´ dist Pq Wxq i> span P W{ jn : ({> ) 5 n _ Tq (}q )> n = 1> = = = > Q %=

Hence, ³ ª´ © %= (5.21) dist i> span P W{ jn : ({> ) 5 n }q _Tq (0)> n = 1> = = = > Q

By Lemma 5.17, there exists a subsequence {zq }q5N of {}q }q5N and there exist 0n R2g such that for each n = 1> = = = > Q we have w

n zq $ 0n

as q $ 4=

SQ

Therefore, by Lemma 5.19, G 0 = n=1 G(jn > 0n ) is a frame for O2 (Rg ) with frame bounds D, E. We claim now that, for any U A 0, ³ ª´ % © dist i> span P W{ jn : ({> ) 5 0n _ TU (0)> n = 1> = = = > Q = (5.22) 2 To see this, choose any scalars {fn>{> }({>)50n _TU (0)> n=1>===>Q . Let G=

Q X

X

n=1 ({>)50n _TU (0)

|fn>{> |=

We already know from equation (5.21) that ki 0k2 %, so we may assume that G 6= 0. By Lemma 5.8, there exists ? @2 such that ; |{|> || ? >

; x> 5 Rg > ° ° °P+ W{+x jn P Wx jn ° ? % > 2 2G

n = 1> = = = > Q=

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Christopher Heil

As in the proof of Lemma 5.19, we can find q large enough that each point of 0n _TU (0) is within of a unique point in n zq _TU (0), and conversely. So, we can write n zq _ TU (0) ©¡ ¢ ª = { + x({> > n)> + ({> > n) : ({> ) 5 0n _ TU (0)> n = 1> = = = > Q >

with |{ + x({> > n)|, | + ({> > n)| ? . Hence ° ° ° ° Q X X ° ° ° °i f P W j n>{> { n° ° ° ° n=1 ({>)50 _TU (0) n

° ° Q X ° =° i ° ° n=1 +

2

X

fn>{> P W{ jn

({>)5n zq _TU (0)

Q X

X

fn>{> P+({>>n) W{+x({>>n) jn

n=1 ({>)50n _TU (0)

Q X

n=1

° ° ° fn>{> P W{ jn ° ° ° ({>)50 _TU (0) X n

2

° ° ° ° Q X X ° ° ° °i fn>{> P W{ jn ° ° ° ° n=1 ({>)5n zq _TU (0) 2 ° ° °X ° Q X ° ¡ ¢° ° P ° f W j P W j n>{> { n ° +({>>n) {+x({>>n) n ° °n=1 ({>)50 _TU (0) ° n

2

³ © ª´ dist i> span P W{ jn : ({> ) 5 n zq _ TU (0)> n = 1> = = = > Q

Q X

X

n=1 ({>)50n _TU (0)

A%

Q X

X

n=1 ({>)50n _TU (0)

=

% = 2

° ° |fn>{> | °P+({>>n) W{+x({>>n) jn P W{ jn °2

|fn>{> |

% 2G

Since this is true for every choice of scalars, we conclude that equation (5.22) holds. But, since U is arbitrary, this implies that i 5 @ span(G), which contradicts the fact that G is complete (since it is a frame).

5 The Homogeneous Approximation Property

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5.3.3 The Comparison Theorem We saw in Theorem 5.22 that all Gabor frames possess the R—S Weak HAP. We will show in this section that the R—S Weak HAP implies that certain density conditions must be satisfied in comparison to any other Gabor Riesz sequence. Theorem 5.23 (Comparison Theorem). Assume that (a) j1 > = = = > jQ 5 O2 (Rg ) \ {0} and 1 > = = = > Q R2g are such that G= is a frame for O2 (Rg ), and

Q S

G(jn > n )

n=1

(b) !1 > = = = > !P 5 O2 (Rg ) and 1 > = = = > P Rg are such that =

P S

n=1

is a Riesz sequence in O2 (Rg ). Set =

SQ

n=1

n and =

SP

n=1

G(!n > n )

n . Then

G () G ()

and

G+ () G+ ()=

Proof. Note that, by Theorem 5.22, we have that G possesses the R—S Weak HAP. We are given that is a Riesz basis for its closed span within O2 (Rg ). Let P S ˜ {!{>>n }({>)5n ˜ = n=1

denote the dual frame within that closed span. Given k A 0 and (x> ) 5 R2g , set ª © ˜ (k> x> ) = span P W{ jn : ({> ) 5 n _ Tk (x> )> n = 1> = = = > Q > Z © ª Y (k> x> ) = span P W{ !n : ({> ) 5 n _ Tk (x> )> n = 1> = = = > P =

By Proposition 5.15, we have G+ (n ), G+ (n ) ? 4 for each n, so these are finite-dimensional spaces. Fix any % A 0. Applying the definition of the R—S Weak HAP to the functions i = !n , we see that there exists an U A 0 such that ; (x> ) 5 R2g >

¡ ¢ ˜ (U> x> ) ? % > dist P Wx !n > Z G

n = 1> = = = > P> (5.23)

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Christopher Heil

where

ª © G = sup k!˜x>>n k : (x> ) 5 n > n = 1> = = = > P =

Fix any k A 0 and (x> ) 5 R2g . Let SY and SZ denote the orthogo˜ (U + k> x> ), nal projections of O2 (Rg ) onto Y = Y (k> x> ) and Z = Z respectively. Define W : Y $ Y by W = SY SZ = SY SZ SY . Note that W is self-adjoint and Y is finite-dimensional, so W has a finite, real trace. We will estimate the trace of W . First, every eigenvalue of W satisfies || kW k kSY k kSZ k = 1. Since the trace is the sum of the nonzero eigenvalues, this provides us with an upper bound for the trace: ¢ ¡ (5.24) trace(W ) rank(W ) dim(Z ) = # _ TU+k (x> ) = For a lower estimate, note that

{P W{ !n : ({> ) 5 n _ Tk (x> )> n = 1> = = = > P } is a basis for the finite-dimensional space Y (since is a Riesz sequence and hence is linearly independent). The dual basis in Y is the biorthogonal system in Y , which is ª © SY !˜{>>n : ({> ) 5 n _ Tk (x> )> n = 1> = = = > P = Therefore,

trace(W ) =

P X

X

W (P W{ !n )> SY !˜{>>n

n=1 ({>)5n _Tk (x>)

=

P X

X

®

SY SZ (P W{ !n )> SY !˜{>>n

n=1 ({>)5n _Tk (x>)

=

P X

X

SZ (P W{ !n )> SY !˜{>>n

n=1 ({>)5n _Tk (x>)

=

P X

X

³

n=1 ({>)5n _Tk (x>)

P W{ !n > SY !˜{>>n

®

®

®

®´

+ (SZ L)(P W{ !n )> SY !˜{>>n = (5.25)

˜ By the biorthogonality of and , ®

P W{ !n > SY !˜{>>n = 1=

(5.26)

Additionally, if ({> ) 5 Tk (x> ), then we have TU ({> ) TU+k (x> ), so Z (U> {> ) Z (U + k> x> ) and therefore

5 The Homogeneous Approximation Property

99

¯ ®¯ ¯ (SZ L)(P W{ !n )> SY !˜{>>n ¯ k(SZ L)(P W{ !n )k2 kSY !˜{>>n k ¡ ¢ dist P W{ !n > Z (U + k> x> ) k!˜{>>n k2 ¡ ¢ dist P W{ !n > Z (U> {> ) G % G = %= (5.27) G Combining (5.25)—(5.27) yields the lower bound trace(W )

P X

X

¡ ¢ (1 %) = (1 %) # _ Tk (x> ) =

(5.28)

n=1 ({>)5n _Tk ({>)

Finally, combining the upper estimate (5.24) with the lower estimate (5.28), we see that ¡ ¢ ¡ ¢ ; (x> ) 5 R2g > ; k A 0> (1 %) # _ Tk (x> ) # _ TU+k (x> ) > and so

¡ ¢ # _ Tk (x> ) G () = lim inf inf k$4 (x>)5R2g k2g ¡ ¢ # _ TU+k (x> ) (U + k)2g 1 lim inf inf 1 % k$4 (x>)5R2g (U + k)2g k2g

=

1 G ()= 1%

Since % is arbitrary, we conclude that G () G (), and a similar calculation shows that G+ () G+ ().

5.3.4 The Density Theorem Combining our previous results on the HAP and the Comparison Theorem yields a proof of an extended version of the Density Theorem. Theorem 5.24 (Density Theorem). Let j1 > = = = > jQ 5 O2 (Rg ) \ {0} and 1 > = = = > Q R2g be given. Let G=

Q S

n=1

G(jn > n )

Then the following statements hold.

and

=

Q S

n =

n=1

(a) If G is a frame for O2 (Rg ), then 1 G () G+ () ? 4.

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Christopher Heil

(b) If G is a Riesz sequence in O2 (Rg ), then 0 G () G+ () 1. (c) If G is a Riesz basis for O2 (Rg ), then G () = G+ () = 1.

Proof. Define ! = "T1(0) and = Zg . Note that = G(!> ) is an orthonormal basis for O2 (Rg ) and hence is both a frame and a Riesz basis. Also, G± () = 1. (a) Assume that G is a frame for O2 (Rg ). Then G+ (n ) ? 4 for each n by Lemma 5.15, so G+ () ? 4 as well. Further, applying Theorem 5.23 to G and implies that G () G () = 1. (b) Suppose that G is a Riesz sequence in O2 (Rg ). Then applying Theorem 5.23 to and G yields 1 = G () G ().

(c) If G is a Riesz basis, then it is both a frame and a Riesz sequence, so parts (a) and (b) together imply that G () = G+ () = 1. Acknowledgments During the years 2002—2006, I had the honor of being part of an NSF-sponsored Focused Research Group (FRG) along with seven other wonderful people, including Larry. This was my first nontrivial interaction with Larry, and I quickly grew to greatly respect him both as a mathematician and a friend. Of all the many good things that came out of that FRG, I value the fact that it brought me into contact with Larry among the highest. Larry’s incompleteness result in [2] was one of the explicit beginnings of the Density Theorem for Gabor frames, and it was the inspiration for the talk that I gave at the conference held in honor of Larry at the University of Colorado in May 2006. The history of the Density Theorem given in this chapter is adapted from that talk. The accounting of the Ramanathan—Steger proof given in the body of this chapter is inspired by discussions of the author with Gerard Ascensi in Spring 2005. We also thank Gitta Kutyniok for valuable discussions, and we are indebted to Jay Ramanathan for many collaborations and insights. The support of NSF Grant DMS-0139261 is gratefully acknowledged.

References 1. H. Bacry, A. Grossmann, and J. Zak, Proof of completeness of lattice states in the kq representation, Phys. Rev. B, 12 (1975), 1118—1120. 2. L. Baggett, Processing a radar signal and representations of the discrete Heisenberg group, Colloq. Math., 60/61 (1990), 195—203. 3. R. Balan, P. G. Casazza, C. Heil, and Z. Landau, Excesses of Gabor frames, Appl. Comput. Harmon. Anal., 14 (2003), 87—106. 4. R. Balan, P. G. Casazza, C. Heil, and Z. Landau, Density, overcompleteness, and localization of frames, I. Theory, J. Fourier Anal. Appl., 12 (2006), 105—143. 5. R. Balan, P. G. Casazza, C. Heil, and Z. Landau, Density, overcompleteness, and localization of frames, II. Gabor systems, J. Fourier Anal. Appl., 12 (2006), 307—344. 6. V. Bargmann, P. Butera, L. Girardello, and J. R. Klauder, On the completeness of coherent states, Rep. Math. Phys., 2 (1971), 221—228. 7. B. Bekka, Square integrable representations, von Neumann algebras and an application to Gabor analysis, J. Fourier Anal. Appl., 10 (2004), 325—349.

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8. J. J. Benedetto, C. Heil, and D. F. Walnut, Digerentiation and the Balian—Low theorem, J. Fourier Anal. Appl., 1 (1995), 355—402. 9. M. Bownik and Z. Rzeszotnik, The spectral function of shift-invariant spaces, Michigan Math. J., 51 (2003), 387—414. 10. O. Christensen, An Introduction to Frames and Riesz Bases, Birkh¨ auser, Boston, 2003. 11. O. Christensen, B. Deng, and C. Heil, Density of Gabor frames, Appl. Comput. Harmon. Anal., 7 (1999), 292—304. 12. O. Christensen, S. Favier, and Z. Felipe, Irregular wavelet frames and Gabor frames, Approx. Theory Appl. (N.S.), 17 (2001), 90—101. 13. I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory, 39 (1990), 961—1005. 14. I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992. 15. I. Daubechies and A. Grossmann, Frames in the Bargmann space of entire functions, Comm. Pure Appl. Math., 41 (1988), 151—164. 16. I. Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys., 27 (1986), 1271—1283. 17. I. Daubechies, H. Landau, and Z. Landau, Gabor time-frequency lattices and the Wexler—Raz identity, J. Fourier Anal. Appl., 1 (1995), 437—478. 18. B. Deng and C. Heil, Density of Gabor Schauder bases, in: Wavelet Applications in Signal and Image Processing VIII (San Diego, CA, 2000), A. Aldroubi et al., eds., Proc. SPIE Vol. 4119, SPIE, Bellingham, WA, 2000, 153—164. 19. R. J. Dun and A. C. Schaeger, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), 341—366. 20. H. G. Feichtinger and W. Kozek, Quantization of TF lattice-invariant operators on elementary LCA groups, in: Gabor Analysis and Algorithms: Theory and Applications, H. G. Feichtinger and T. Strohmer, eds., Birkh¨ auser, Boston, 1998, 233—266. 21. G. B. Folland, The abstruse meets the applicable: Some aspects of time-frequency analysis, Proc. Indian Acad. Sci. Math. Sci., 116 (2006), 121—136. 22. J.-P. Gabardo and D. Han, Frame representations for group-like unitary operator systems, J. Operator Theory, 49 (2003), 223—244. 23. D. Gabor, Theory of communications, J. Inst. Elec. Eng. (London), 93 (1946), 429— 457. 24. K. Gr¨ ochenig, Irregular sampling of wavelet and short-time Fourier transforms, Constr. Approx., 9 (1993), 283—297. 25. K. Gr¨ ochenig, Foundations of Time-Frequency Analysis, Birkh¨ auser, Boston, 2001. 26. K. Gr¨ ochenig, Localization of frames, Banach frames, and the invertibility of the frame operator, J. Fourier Anal. Appl., 10 (2004), 105—132. 27. K. Gr¨ ochenig and H. Razafinjatovo, On Landau’s necessary density conditions for sampling and interpolation of band-limited functions, J. London Math. Soc. (2), 54, 557—565 (1996). 28. D. Han and Y. Wang, Lattice tiling and the Weyl—Heisenberg frames, Geom. Funct. Anal., 11 (2001), 742—758. 29. C. Heil, An introduction to weighted Wiener amalgams, in: Wavelets and their Applications (Chennai, January 2002), M. Krishna, R. Radha and S. Thangavelu, eds., Allied Publishers, New Delhi, 2003, 183—216. 30. C. Heil, History and evolution of the density theorem for Gabor frames, J. Fourier Anal. Appl., 13 (2007), 113—166. 31. C. Heil and G. Kutyniok, Density of frames and Schauder bases of windowed exponentials, preprint (2006). 32. C. Heil and G. Kutyniok, The homogeneous approximation property for wavelet frames, J. Approx. Theory, 147 (2007), 28—46. 33. C. E. Heil and D. F. Walnut, Continuous and discrete wavelet transforms, SIAM Review, 31 (1989), 628—666. 34. A. J. E. M. Janssen, Gabor representation of generalized functions, J. Math. Anal. Appl., 83 (1981), 377—394.

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35. A. J. E. M. Janssen, Signal analytic proofs of two basic results on lattice expansions, Appl. Comput. Harmon. Anal., 1 (1994), 350—354. 36. A. J. E. M. Janssen, A density theorem for time-continuous filter banks, in: Signal and Image Representation in Combined Spaces, Y. Y. Zeevi and R. R. Coifman, eds., Wavelet Anal. Appl., Vol. 7, Academic Press, San Diego, 1998, 513—523. 37. G. Kutyniok, Beurling density and shift-invariant weighted irregular Gabor systems, Sampl. Theory Signal Image Process., 5 (2006), 163—181. 38. H. Landau, Sampling, data transmission, and the Nyquist rate, Proc. IEEE, 55, 1701— 1706 (1967). 39. H. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math., 117, 37—52 (1967). 40. H. Landau, On the density of phase-space expansions, IEEE Trans. Inform. Theory, 39 (1993), 1152—1156. 41. Y. Liu and Y. Wang, The uniformity of non-uniform Gabor bases, Adv. Comput. Math., 18 (2003), 345—355. 42. Yu. I. Lyubarski˘i, Frames in the Bargmann space of entire functions, in: Entire and subharmonic functions, Amer. Math. Soc., Providence, RI, 1992, 167—180. 43. J. von Neumann, Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932. English translation: Mathematical foundations of quantum mechanics, Princeton University Press, Princeton, NJ, 1955. 44. A. Olevskii, Completeness in O2 (R) of almost integer translates, C. R. Acad. Sci. Paris, 324 (1997), 98—991. 45. A. Olevskii and A. Ulanovskii, Almost integer translates. Do nice generators exist?, J. Fourier Anal. Appl., 10 (2004), 93—104. 46. A. M. Perelomov, On the completeness of a system of coherent states (English translation), Theoret. Math. Phys., 6 (1971), 156—164. 47. J. Ramanathan and T. Steger, Incompleteness of sparse coherent states, Appl. Comput. Harmon. Anal., 2, 148—153 (1995). 48. M. Riegel, Von Neumann algebras associated with pairs of lattices in Lie groups, Math. Ann., 257 (1981), 403—418. 49. E. Romero, A complete Gabor system of zero Beurling density, Sampl. Theory Signal Image Process., 3 (2004), 299—304. 50. K. Seip, Density theorems for sampling and interpolation in the Bargmann—Fock space I, J. Reine Angew. Math., 429 (1992), 91—106. 51. K. Seip and R. Wallst´ en, Sampling and interpolation in the Bargmann—Fock space II, J. Reine Angew. Math., 429 (1992), 107—113. 52. W. Sun and X. Zhou, Irregular wavelet/Gabor frames, Appl. Comput. Harmon. Anal., 13 (2002), 63—76. 53. W. Sun and X. Zhou, Irregular Gabor frames and their stability, Proc. Amer. Math. Soc., 131 (2003), 2883—2893. 54. R. Young, An Introduction to Nonharmonic Fourier Series, Revised First Edition, Academic Press, San Diego, 2001.

Chapter 6

Recent Developments on Dual Wavelet Frames Bin Han Dedicated to Professor Larry Baggett

Abstract We discuss some recent developments on pairs of dual wavelet frames in Sobolev spaces constructed by the Oblique Extension Principle (OEP) from refinable function vectors. Our investigation sheds a new light on understanding the OEP method for constructing MRA (multiresolution analysis) dual wavelet frames.

6.1 Introduction and Motivation In this chapter, we are concerned with pairs of dual wavelet frames in Sobolev spaces that are constructed by the Oblique Extension Principle (OEP) from refinable function vectors. Classical wavelet theory deals with systems of O2 (Rg ) that are generated by integer shifts and dilates of a finite set of functions in O2 (Rg ). As an important part of wavelet theory, wavelet frames have been extensively investigated and largely constructed in the function space O2 (Rg ). To only mention a few references here, see [1]—[42] and numerous related articles therein. In particular, many tight and dual wavelet frames in O2 (R) have been constructed via the (mixed) Unitary Extension Principle (UEP) from scalar refinable functions in [40, 41]. In Section 6.2, we shall present some background and motivations about the OEP for constructing pairs of dual wavelet frames in O2 (Rg ). Wavelets in Sobolev spaces are motivated by applications of wavelets in numerical algorithms and image processing, since the solution spaces of many partial dierential equations and the classes of images are modeled by various Sobolev spaces. One key feature of wavelets is the norm equivalence for Bin Han Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 e-mail: [email protected]

103

104

Bin Han

characterizing a function space in terms of the weighted norm of a wavelet coecient sequence ([10]). Quite often, wavelets are constructed in O2 (Rg ) and then their norm equivalence is established for other Sobolev spaces. In Section 6.3, we shall recall a notion from [35] of a pair of dual wavelet frames in a pair of Sobolev spaces (K v (Rg )> K v (Rg )) with v 5 R, where the Sobolev space K v (Rg ) consists of all tempered distributions i on Rg such that Z 1 2 |iˆ()|2 (1 + kk2 )v g ? 4= (6.1) ki kK v (Rg ) := (2)g Rg R The Fourier transform used in this chapter is iˆ() := Rg i ({)hl{· g{, where i 5 O1 (Rg ) and { · denotes the inner product of the two vectors { and in Rg . As we shall see in Section 6.3, the notion of dual wavelet frames in a pair of Sobolev spaces has several consequences. First of all, most results on dual wavelet frames in O2 (Rg ) (that is, the case v = 0 by K 0 (Rg ) = O2 (Rg )) can be naturally (but nontrivially) generalized to the new setting. In particular, in Section 6.3 we generalize the OEP, which is proposed in [13, 15] (and independently in [7]) for constructing dual wavelet frames in O2 (Rg ) from refinable functions, to dual wavelet frames in Sobolev spaces from refinable function vectors. Second, the notion makes the construction of wavelet frames in Sobolev spaces relatively easy and systematic. Third, characterization of Sobolev spaces by wavelets has a natural reinterpretation under the new notion. We shall address these issues in Section 6.3. In Section 6.4, we shall briefly discuss how to construct dual wavelet frames in Sobolev spaces by the projection method. The projection method, first introduced in [17] and further developed in [19, 20, 23, 25, 26, 34], is very useful for analyzing various optimal properties of multivariate refinable function vectors. It turns out that it is also an interesting tool for constructing low-dimensional dual wavelet frames from high-dimensional ones in Sobolev spaces. Moreover, the projection method works well with the OEP by preserving the nice structure of OEP and refinable function vectors. As an example, all the spline tight wavelet frames in O2 (R) obtained in [40] can be obtained easily by applying the projection method to the high-dimensional tensorproduct Haar orthonormal wavelets. The projection method is sort of an inverse operation to the well-known tensor product method for constructing wavelets, and in fact it includes the oversampling theorems in [9] on wavelet frames as a special case. As the core part of this chapter, in Section 6.5, we shall address several fundamental issues on MRA dual wavelet frames obtained by OEP to have a better understanding of the OEP. Our approach starts at the understanding of the implementation of the fast frame transform associated with dual wavelet frames constructed by the OEP. Despite the great flexibility and popularity of OEP for constructing compactly supported MRA wavelet frames in the literature, however, the associated fast frame transform is generally not compact, and a deconvolution appears in the frame transform (see [15]).

6 Recent Developments on Dual Wavelet Frames

105

Here we say that a frame transform is compact if it can be implemented by convolutions, coupled with upsampling and downsampling, using only finiteimpulse-response (FIR) filters. In this chapter, we shall understand and apply the OEP to scalar refinable functions (that is, refinable function vectors with multiplicity one) and truly refinable function vectors (that is, multiplicity is at least two). Our investigation leads to a surprising dierence between dual wavelet frames derived from scalar refinable functions and dual multiwavelet frames derived from truly refinable function vectors via OEP if a compact fast frame transform is required. More precisely, we present two complementary results on dual wavelet frames that are obtained via OEP from scalar refinable functions and from truly refinable function vectors with multiplicity greater than one. On one hand, by a nontrivial argument, we show that from any pair of compactly supported spline refinable functions ! and !˜ (not necessarily having stable integer shifts) with finitely supported masks, if we require that the associated frame transform be compact, then any compactly supported dual wavelet frames derived via OEP from ! and !˜ can have vanishing moments at most one and the frame approximation order at most two. On the other hand, we prove in a constructive way that from any pair of compactly supported refinable function vectors ! and !˜ with multiplicity at least two and with finitely supported masks, then we can always build a pair of compactly supported dual wavelet frames in O2 (R) with the following properties: (i) The associated fast frame transform is compact; therefore, no deconvolution appears in the frame transform. (ii) All the frame generators achieve the highest possible order of vanishing moments; the pair of dual multiwavelet frames has the highest possible frame approximation order. (iii) The pair of dual multiwavelet frames and its fast frame transform have the highest possible balancing order; therefore, the diculty of approximation ineciency facing most multiwavelet transforms does not appear here in the associated fast frame transform. In short, the two desirable properties (i) and (ii) of dual wavelet frames obtained via OEP from scalar refinable functions are generally mutually conflicting, whereas they coexist very well for truly refinable function vectors and multiwavelets with the additional property of high balancing orders in (iii). One of the key ingredients in our study of MRA multiwavelet frames is an interesting canonical form of a matrix mask that greatly facilitates the investigation of refinable function vectors and multiwavelets. Namely, the OEP has advantages for constructing dual multiwavelet frames from truly refinable function vectors instead of from the commonly used scalar refinable functions. We point out that most results in this chapter have already been or could be extended to multiwavelets and refinable function vectors with a general dilation matrix. The notion and results on dual wavelet frames in Sobolev

106

Bin Han

spaces can also be generalized to other function spaces such as Besov spaces. For simplicity of presentation, we only discuss dual wavelet frames in Sobolev spaces or in O2 (Rg ) with the dilation matrix 2Lg . Due to the page limit, in this chapter we shall aim at explaining the background, motivation, and interpretation of results as clear and self-contained as possible, while referring the interested readers to the surveyed articles for the details of proofs of the surveyed results and for more detailed treatment of the topics. We hope that through this chapter, we will provide the readers a framework picture of some recent developments on wavelet frames obtained via OEP. Though unfortunately we have to restrict ourselves in this chapter for surveying recent developments on dual wavelet frames only from our research group, it is important to point out that there are many other recent exciting developments on wavelet frames obtained by many researchers and groups; for example, by the research group led by Larry Baggett on construction of Parseval frames from redundant filter systems and generalized MRAs [1, 2].

6.2 Some Background on Oblique Extension Principle In this section, we present some background and motivations about the OEP for constructing pairs of dual wavelet frames in O2 (Rg ). In order to do so, let us recall some necessary definitions and notations. For # 1 > = = = > # O 5 O2 (Rg ), we denote ª c : m 5 Z> n 5 Zg > c = 1> = = = > O > (6.2) [(# 1 > = = = > # O ) := {#m>n

c where #m>n := 2mg@2 # c (2m · n). We say that [(# 1 > = = = > # O ) is a wavelet frame in O2 (Rg ) if there exist two positive constants F1 and F2 such that

F1 ki k2O2 (Rg )

6

O X X X

c=1 m5Z n5Zg

c |hi> #m>n i|2

6 F2 ki k2O2 (Rg ) >

i 5 O2 (Rg )>

(6.3)

R where h·> ·i denotes the inner product in O2 (Rg ): hi> ji := Rg i ({)j({) g{. For # 1 > = = = > # O > #˜1 > = = = > #˜O 5 O2 (Rg ), ([(# 1 > = = = > # O )> [(#˜1 > = = = > #˜O )) is a pair of dual wavelet frames in O2 (Rg ) if 1. each of [(# 1 > = = = > # O ) and [(#˜1 > = = = > #˜O ) is a wavelet frame in O2 (Rg ). 2. the following identity holds hi> ji =

O X X X

c=1 m5Z n5Zg

c c hi> #˜m>n ih#m>n > ji>

i> j 5 O2 (Rg )=

6 Recent Developments on Dual Wavelet Frames

107

A pair of dual wavelet frames in O2 (Rg ) is also called a bi-framelet in the literature ([15, 41]). For a pair ([(# 1 > = = = > # O )> [(#˜1 > = = = > #˜O )) of dual wavelet frames in O2 (Rg ), one has the wavelet representations for functions in O2 (Rg ): i=

O X X X

c=1 m5Z n5Zg

c c hi> #˜m>n i#m>n =

O X c c hi> #m>n i#˜m>n >

i 5 O2 (Rg )=

c=1

A pair of biorthogonal wavelets in O2 (Rg ) consists of a special family of pairs of dual wavelet frames in O2 (Rg ). For # 1 > = = = > # O > #˜1 > = = = > #˜O 5 O2 (Rg ), we say that ([(# 1 > = = = > # O )> [(#˜1 > = = = > #˜O )) is a pair of biorthogonal wavelets in O2 (Rg ) if it is a pair of dual wavelet frames in O2 (Rg ) and it satisfies the biorthogonality relation: 0 c > #˜mc0 >n0 i = cc0 mm 0 nn0 > h#m>n

m> m 0 5 Z> n> n0 5 Zg > c> c0 = 1> = = = > O> (6.4)

where denotes the Dirac sequence such that 0 = 1 and = 0 for all 6= 0. The wavelet functions # c > #˜c are generally obtained from refinable functions via a multiresolution analysis. Let ! and !˜ be compactly supported ˆ˜, that ˆ and d functions in O2 (Rg ) such that ! and !˜ are refinable with masks d is, ˆ˜ ˆ˜ ˆ ˆ ˆ˜()!()> (6.5) !(2) =d ˆ()!() and !(2) =d 5 Rg > ˆ where d ˆ and d ˜ are 2-periodic trigonometric polynomials in g-variables. E-splines consist of an important family of refinable functions. Let E1 := "[0>1] , the characteristic function of the interval [0> 1]. The E-spline Ep of order p is obtained via the following recursive formula: Z 1 Ep1 (· w) gw= Ep := Ep1 E1 = 0

Then Ep 5 F p2 (R) is a function of piecewise polynomials, and one can p d d easily verify that Ep is refinable since E (1 + hl )p E p (2) = 2 p (), 5 R. Generally, the wavelet functions # c and the dual wavelet functions #˜c are obtained from the two refinable functions ! and !˜ via c ˆ˜ cc (2) = ebc ()!() ˆ ˜c (2) = ˜ebc ()!()> and # #

5 Rg > c = 1> = = = > O> (6.6)

where all ebc and ˜ebc , c = 1> = = = > O, are 2-periodic trigonometric polynomials. If the following two conditions are satisfied:

(1) the two compactly supported refinable functions ! and !˜ with masks d ˆ ˆ and d ˜ belong to O2 (Rg ) and satisfy

108

Bin Han

˜ n)i = n > h!> !(·

n 5 Zg

ˆ˜ !(0) ˆ = 1> and !(0)

ˆ˜, ebc > ˜ebc , ˆ, d (2) O = 2g 1 and all the 2-periodic trigonometric polynomials d c = 1> = = = > 2g 1, satisfy ˆ d ˜()ˆ d( + ) +

g 2X 1

c=1

˜ebc ()ebc ( + ) = >

5 Rg > 5 {0> 1}g >

g g then it is well-known ([12]) that ([(# 1 > = = = > # 2 1 )> [(#˜1 > = = = > #˜2 1 )) is a pair of biorthogonal wavelets in O2 (Rg ). The mixed unitary extension principle in [41] nicely extends the above construction recipe for biorthogonal wavelets to dual wavelet frames, and it says that if the following two conditions are satisfied: ˆ (10 ) the two compactly supported refinable functions ! and !˜ with masks d ˆ ˜ˆ !(0) ˆ = 1, and d ˜ belong to O (Rg ) and satisfy !(0)

2

ˆ˜, ebc > ˜ebc , c = 1> = = = > O, ˆ, d (2 ) all the 2-periodic trigonometric polynomials d satisfy 0

ˆ d ˜()ˆ d( + ) +

O X ˜ebc ()ebc ( + ) = > c=1

5 Rg > 5 {0> 1}g >

then ([(# 1 > = = = > # O )> [(#˜1 > = = = > #˜O )) is a pair of dual wavelet frames in ˆ˜ = d ˆ, ˜ebc = ebc for all c = 1> = = = > O, O2 (Rg ). For the special case !˜ = !> #˜c = # c , d the mixed unitary extension principle becomes the unitary extension principle (UEP) in [40] and one obtains a tight wavelet frame [(# 1 > = = = > # O ) in O2 (Rg ), that is, ([(# 1 > = = = > # O )> [(# 1 > = = = > # O )) is a pair of dual wavelet frames in O2 (Rg ), or more concisely, the following identity holds ki k2O2 (Rg ) =

O X X X c=1 m5Z

n5Zg

c |hi> #m>n i|2

; i 5 O2 (Rg )=

One of the most important properties of a wavelet system [(# 1 > = = = > # O ) is the order of vanishing moments. We say that [(# 1 > = = = > # O ) has the vanishing moments of order p if h# c > S i = 0 for all c = 1> = = = > O and for all polynomials S in g-variables of (total) degree less than p. Many interesting examples of tight and dual wavelet frames in O2 (R) have been constructed from various refinable functions such as the Espline refinable functions via the (mixed) unitary extension principle, see [7, 13, 15, 40, 41] and many other references therein. However, for all compactly supported tight wavelet frames or dual wavelet frames in O2 (R) derived from any E-spline refinable function, it is known [7, 13, 15, 40, 41] that such tight and dual wavelet frames can have vanishing moments at most one, even

6 Recent Developments on Dual Wavelet Frames

109

though the E-spline function can provide a high approximation order and a high order of vanishing moments of a wavelet system is a very much desired property in many applications. This consideration is the main motivation, or at least one of the major motivations, for extending the mixed unitary extension principle in [41] to the (mixed) oblique extension principle in [7, 13, 15] to achieve high vanishing moments for compactly supported tight and dual wavelet frames in O2 (Rg ). More precisely, if one assumes that there exists a ˆ such that 2-periodic trigonometric polynomial (100 ) the two compactly supported refinable functions ! and !˜ with masks d ˆ ˆ ˆ ˜ !(0) ˆ !(0) ˆ = 1, and d ˜ belong to O2 (Rg ) and satisfy () 00 ˆ ˆ, d ˜, ebc > ˜ebc , c = 1> = = = > O (2 ) all the 2-periodic trigonometric polynomials d satisfy

O X ˜ebc ()ebc (+) = ()> ˆ ˆ ˆ d ˜()ˆ d(+)+ (2) c=1

5 Rg > 5 {0> 1}g >

then the (mixed) OEP in [7, 13, 15] says that ([(# 1 > = = = > # O )> [(#˜1 > = = = > #˜O )) is a pair of dual wavelet frames in O2 (Rg ). With the additional freedom in the ˆ now from any E-spline choice of a 2-periodic trigonometric polynomial , refinable function of order p, one can always obtain a compactly supported tight wavelet frame or a pair of compactly supported dual wavelet frames in O2 (R) with the vanishing moments of the highest possible order p. See [7, 13, 15] for more details on the motivations and developments of the oblique extension principle. We shall present and generalize the oblique extension principle for Sobolev spaces in Section 6.3. It is far from trivial to generalize the oblique extension principle for constructing pairs of dual wavelet frames from O2 (Rg ) to Sobolev spaces. Such a generalization of OEP in Section 6.3 enables us to construct pairs of dual wavelet frames and wavelet frames in Sobolev spaces in a systematic and relatively straightforward way. Interested readers should consult [7, 13, 15, 35, 40, 41] for more detailed discussions on the unitary extension principle and the oblique extension principle.

6.3 Dual Wavelet Frames in Sobolev Spaces by OEP In this section, we shall recall the notion from [35] of pairs of dual wavelet frames in a pair of Sobolev spaces (K v (Rg )> K v (Rg )) and discuss the generalization of the OEP from the function space O2 (Rg ) to Sobolev spaces. Note that K v (Rg ) is a Hilbert space under the inner product: Z W 1 j () (1 + kk2 )v g> i> j 5 K v (Rg )= (6.7) hi> jiK v (Rg ) := iˆ()ˆ (2)g Rg

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Moreover, for each j 5 K v (Rg ), Z W 1 j () g> hi> ji := iˆ()ˆ g (2) Rg

i 5 K v (Rg )

(6.8)

defines a linear functional on K v (Rg ). The spaces K v (Rg ) and K v (Rg ) form a pair of dual spaces. For (K v (Rg ))p×q we denote the Banach (or Hilbert) space, equipped with a norm induced by k · kK v (Rg ) , of all p × q matrices with all entries in K v (Rg ). For i 5 (K v (Rg ))c×p and j 5 (K v (Rg ))q×p , we can naturally define the inner product hi> ji as an c × q matrix given in (6.8). Similarly, we define hi> jiK v (Rg ) in (6.7) for i 5 (K v (Rg ))c×p and j 5 W

(K v (Rg ))q×p . Note that jˆ() in both (6.7) and (6.8) denotes the transpose of the complex conjugate of the matrix function jˆ(). Denote N0 := N ^ {0}. For given !> # 1 > = = = > # O 5 (K v (Rg ))u×1 , the corresponding properly normalized wavelet system in the Sobolev space K v (Rg ) is defined as: [ v (!; # 1 > = = = > # O ) := {!0>n := !(· n) : n 5 Zg } ^ © c>v ª #m>n := 2m(g@2v) # c (2m · n) : m 5 N0 > n 5 Zg > c = 1> = = = > O = (6.9)

We say that [ v (!; # 1 > = = = > # O ) is a wavelet frame in K v (Rg ) if there exist positive constants F1 and F2 such that F1 ki k2K v (Rg )

6

X

n5Zg

6

2

|hi> !0>n iK v (Rg ) | +

F2 ki k2K v (Rg ) >

4 X O X X

c=1 m=0 n5Zg

c>v |hi> #m>n iK v (Rg ) |2

i 5 K v (Rg )>

(6.10)

where | · | denotes the Euclidean norm on Ru . For a real number v, let !> # 1 > = = = > # O be functions (or tempered distribu˜ #˜1 > = = = > #˜O belong to (K v (Rg ))u×1 . We say tions) in (K v (Rg ))u×1 and let !> v 1 O v ˜ ˜1 that ([ (!; # > = = = > # )> [ (!; # > = = = > #˜O )) is a pair of dual wavelet frames in (K v (Rg )> K v (Rg )) when the following two conditions are satisfied: ˜ #˜1 > = = = > #˜O ) is 1. [ v (!; # 1 > = = = > # O ) is a wavelet frame in K v (Rg ) and [ v (!; v g a wavelet frame in K (R ). 2. For all i 5 K v (Rg ) and j 5 K v (Rg ), the following identity holds: hi> ji =

X

n5Zg

hi> !˜0>n ih!0>n > ji +

4 X O X X

c=1 m=0 n5Zg

c>v c>v hi> #˜m>n ih#m>n > ji=

(6.11)

c>v c>v By the definition of #˜m>n and #m>n in (6.9), it is evident that c>v c>v c>0 c>0 c>0 c hi> #˜m>n ih#m>n > ji = hi> #˜m>n ih#m>n > ji and #m>n = #m>n = 2mg@2 # c (2m · n)=

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˜ #˜1 > = = = > #˜O )) is a pair of dual wavelet frames in If ([ v (!; # 1 > = = = > # O )> [ v (!; v g v g (K (R )> K (R )), then we have the wavelet representations in the Sobolev spaces K v (Rg ) and K v (Rg ) as follows: i=

X

n5Zg

j=

X

n5Zg

hi> !˜0>n i!0>n + hj> !0>n i!˜0>n +

O X 4 X X

c=1 m=0 n5Zg

4 X O X X

c=1 m=0 n5Zg

c>v c>v hi> #˜m>n i#m>n >

i 5 K v (Rg )> (6.12)

c>v ˜c>v hj> #m>n i#m>n >

j5K

v

g

(R )

with the series converging unconditionally in K v (Rg ) and K v (Rg ), respectively. Notice the dierences between [(# 1> = = = > # O ) in (6.2) and [ 0 (!; # 1> = = = > # O ) in (6.9). In most wavelet-based applications, the half system [ v (!; # 1> = = = > # O ) with nonnegative scales is employed instead of the commonly studied whole system [(# 1 > = = = > # O ) with all scales in (6.2), which is often derived from the scaled half system [ 0 (!; # 1 > = = = > # O ) by letting the negative scale go to 4. For an u × 1 vector i of tempered distributions defined on Rg , we denote n o 2 (i ) := sup v 5 R : i 5 (K v (Rg ))u×1 = (6.13) If the set in (6.13) is the empty set, then we simply set 2 (i ) := 4. In order to facilitate our discussion of OEP for refinable function vectors in Section 6.5, in the following we present the OEP for constructing dual wavelet frames from refinable function vectors, though dual wavelet frames in Sobolev spaces have only been discussed and proved for scalar refinable functions in [35, Corollary 2.5]. The arguments in [35] apply to the general case of refinable function vectors. For convenience of later discussions in Section 6.5, in the following, let us state the Oblique Extension Principle in a symmetric form. Theorem 6.1. Let ! and !˜ be two u × 1 refinable function vectors of compactly supported tempered distributions such that ˆ ˆ !(2) =d ˆ()!()

and

ˆ˜ ˜ˆ ˆ˜()!()> !(2) =d

5 Rg >

(6.14)

ˆ˜ of 2-periodic trigonometric polynomials in for some u × u matrices d ˆ and d ˆ˜ ebc > ˜ebc , c = 1> = = = > O, ˆ , g-variables. Suppose that there exist u × u matrices > of 2-periodic trigonometric polynomials in g-variables such that W

ˆ˜ ˆ˜ !(0) ˆ ˆ !(0) (0) (0) =1 and for 5 {0> 1}g ,

(6.15)

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ˆ ˆ ˜ ˆ (2) d ˜() (2)ˆ d( + ) +

O W W X ˆ˜ ˜ebc () ebc ( + ) = () ˆ ()> (6.16) c=1

where denotes the Dirac sequence with 0 = 1 and = 0 for 6= 0. Define W

W

ˆ˜ ˆ˜ ˆ˜ ˆ˜ ˆ !()> ˆ ˆ = () ˆ ˆ() := () ˆ˜() := () ()> (6.17) () !() with () ˜ are u × u matrices of 2-periodic trigonometric polynomials. where ˆ and ˆ Denote cc (2) := ebc ()!() ˆ #

and

For a real number v, if

c ˆ ˜c (2) := ˜ebc ()!()> ˜ #

2 (!) A v

and

˜ A v> 2 (!)

c = 1> = = = > O=

(6.18)

(6.19)

and there exist nonnegative numbers and , ˜ with A v and ˜ A v, such that the following vanishing moment conditions hold: cc () = R(kk ) #

and

c ˜c () = R(kk˜ )> #

$ 0> c = 1> = = = > O> (6.20)

; #˜1 > = = = > #˜O )) is a pair of dual wavelet frames then ([ v (; # 1 > = = = > # O )> [ v (˜ in the pair of Sobolev spaces (K v (Rg )> K v (Rg )). Moreover, if v = 0, then ([(# 1 > = = = > # O )> [(#˜1 > = = = > #˜O )) is also a pair of dual wavelet frames in O2 (Rg ). ˆ˜ ˆ The condition in (6.15) is a normalization condition on !(0) and !(0). If v A 0, then one can take = 0 and so, no vanishing moment is needed for # 1 > = = = > # O in (6.20). Equation (6.19) basically says that ! 5 K v+% (Rg ) and !˜ 5 K v+% (Rg ) for any % A 0. The condition in (6.16) is the same set of equations appearing in the OEP for constructing dual wavelet frames in O2 (Rg ) (see [7, 13, 15] and Section 6.2). The key ingredient in the OEP in Theorem 6.1 is the idea of changing from the given generators ! and !˜ to W new generators and ˜ such that 1 ˆ() ˆ˜() = 1 + R(||p ), $ 0, for a suitable large integer p. See Section 6.5 for detail. As a related concept, let us introduce the definition of sum rules here. For an u × u matrix d ˆ of 2-periodic trigonometric polynomials in g-variables, we say that d ˆ satisfies the sum rules of order p if there exists a 1 × u vector |ˆ of 2-periodic trigonometric polynomials such that |ˆ(0) 6= 0 and |ˆ(2)ˆ d( + ) = |ˆ() + R(kkp )> $ 0> 5 {0> 1}g = (6.21) Note that only the values |ˆ(m) (0), m = 0> = = = > p 1, are needed in (6.21). In other words, (6.21) is equivalent to |ˆ(0) 6= 0 and [ˆ | (2·)ˆ d](m) () = |ˆ(m) (0) g d) the for all m = 0> = = = > p 1 and 5 {0> 1} . In particular, we denote vu(ˆ

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largest nonnegative integer p such that d ˆ satisfies the sum rules of order p. d) associated with the matrix mask d ˆ. There is a very important quantity 2 (ˆ Define °1@q ° ° ° q1 d> |ˆ) := sup lim sup °ˆ ·) · · · d ˆ(2·)ˆ d(·)ˆ y (·)° > (6.22) d(2 p (ˆ g O2 ([>] )

y ˆ5Vp>|ˆ q$4

where Vp>ˆ| denotes the linear space of all u × 1 vectors yˆ of 2-periodic trigonometric polynomials such that |ˆ()ˆ y() = R(kkp ) as $ 0. We d) as in [22] by define the following important quantity 2 (ˆ d) := g@2 log2 (ˆ d)> 2 (ˆ

(6.23)

d> |ˆ) such that (6.21) holds for some where (ˆ d) denotes the infimum of all p (ˆ p 5 N0 and some 1×u vector |ˆ of 2-periodic trigonometric polynomials with d) plays a very important role in characterizing |ˆ(0) 6= 0. The quantity 2 (ˆ the convergence of a vector cascade algorithm in a Sobolev space and in characterizing the Sobolev smoothness of a refinable function vector. For example, for a nonnegative integer v, the vector cascade algorithm associated with a 2-periodic trigonometric polynomial mask d ˆ converges in K v (Rg ) if d) A v ([22, Theorem 4.3]). Let ! be a compactly supported and only if 2 (ˆ ˆ ˆ ˆ refinable function vector satisfying !(2) =d ˆ()!() and !(0) 6= 0. Then we always have 2 (!) > 2 (ˆ d). If the shifts of ! are stable, then we further have d). See [22, 24] and references therein for more details. 2 (!) = 2 (ˆ In the following, let us present the following simple example (see [35, Theorem 1.2]) to illustrate Theorem 6.1. Example 6.2. Let d ˆ be a 2-periodic trigonometric polynomial in g-variables Q ˆ ˆ(2m ) for 5 Rg . Assume that with d ˆ(0) = 1. Define ! by !() := 4 m=1 d ! 5 O2 (Rg ) and vu(ˆ d) > 1. By [21, Theorem 2.2] or [24, Theorem 4.1], we d)), have 2 (!) A 0. Then for any v such that 0 ? v ? min(2 (!)> vu(ˆ © m(g@2v) ª 2 (6.24) !(2m · n) : m 5 N0 > n 5 Zg is a wavelet frame in K v (Rg ).

ˆ˜ (see [35]) such that Proof. Find a 2-periodic trigonometric polynomial d ˆ˜(0) = 1> d

ˆ˜) > 2> vu(d

ˆ˜) A 0> 2 (d

ˆ˜() = R(kkp )> 1d ˆ()d

$ 0>

with p = vu(ˆ d). Let ebc () := 2vg@2 hlc · >

c = 1> = = = > 2g

ˆ˜ ˆ with {1 > = = = > 2g } = {0> 1}g . Take () = () = 1 and O = 2g in Theorem 6.1. Then all filters ˜ebc > c = 1> = = = > 2g are uniquely determined by (6.16) and all the conditions in Theorem 6.1 are satisfied for 0 ? v ?

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min(2 (!)> vu(ˆ d)). Thus, [ v (!; # 1 > = = = > # 2 ), which is the same system as in u t (6.24), is a wavelet frame in K v (Rg ). Riesz wavelets are closely related to wavelet frames and biorthogonal wave˜ #˜1 > = = = > #˜O )) is a pair of dual lets. We say that ([ v (!; # 1 > = = = > # O )> [ v (!; v g v g Riesz wavelet bases in (K (R )> K (R )) if it is a pair of dual wavelet frames in (K v (Rg )> K v (Rg )) and for all m> m 0 5 N0 , n> n0 5 Zg , c> c0 = 1> = = = > O, h!0>n > !˜0>n0 i = nn0 > 0

h!0>n > #˜mc0 >v >n0 i = 0>

0

c>v ˜c >v h#m>n > #m 0 >n0 i = mm 0 nn0 cc0 c>v ˜ h#m>n > !0>n0 i = 0=

(6.25)

By Theorem 6.1, the following is a general version of [35, Corollary 1.4]. 2g 1 and d 2g 1 be u × u matrices of ˆ˜> ˜eb1 > = = = > ˜e[ Theorem 6.3. Let d ˆ> eb1 > = = = > e[ 2-periodic trigonometric polynomials in g-variables. Let ! and !˜ be two u×1 W ˆ˜ ˆ !(0) =1 vectors of compactly supported tempered distributions such that !(0) v 1 O v ˜ ˜1 O ˜ and (6.14) holds. Then ([ (!; # > = = = > # )> [ (!; # > = = = > # )) is a pair g of dual Riesz wavelet bases in (K v (Rg )> K v (Rg )), where # 1 > = = = > # 2 1 , g #˜1 > = = = > #˜2 1 are defined in (6.18), if and only if W

ˆ˜() d ˆ(+)+ d

g 2X 1

c=1

and

W

˜ebc () ebc (+) = Lu >

d) A v 2 (ˆ

and

5 Rg > 5 {0> 1}g (6.26)

ˆ˜) A v= 2 (d

(6.27)

In fact, by a technical argument, one can further show that Theorem 6.3 still holds for masks and filters with exponential decay. For more detail, see [27] for the case v = 0. ˆ˜() = 1 and denote Example 6.4. Let d d ˆ() := cos2p (@2)

p1 X m=0

(p + m 1)! sin2m (@2)> m!(p 1)!

p 5 N=

Then d ˆ() + d ˆ( + ) = 1. Let ! and !˜ be the compactly supported reˆ˜, respectively. Then ! is interfinable distributions associated with d ˆ and d polating ([12]): ! is continuous and !(n) = n for all n 5 Z. Moreover, ˆ˜) = 1@2, by !˜ = is the Dirac distribution. Since 2 (ˆ d) A 1@2 and 2 (d ˜ #)) ˜ is a pair of dual Riesz wavelet bases in Theorem 6.3, ([ v (!; #)> [ v (!; ˆ ˆ (K v (R)> K v (R)) for any 1@2 ? v ? 2 (ˆ d), where #(2) := hl !() and ˆ ˆ l l ˜ =h d ˜ ˆ( + )!() ˆ( + ). #(2) := h d According to the following simple observation and (6.12), the norm equivalence in a Sobolev space is just a wavelet frame in (K v (Rg )> K v (Rg )):

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Proposition 6.5. ([35, Proposition 2.1]) (6.10) is equivalent to for all j 5 K v (Rg ), 4 X O X X X c>0 2 2 2 |hj> !0>n i| + 22vm |hj> #m>n i| F1 kjkK v (Rg ) 6 c=1 m=0 n5Zg

n5Zg

6

F2 kjk2K v (Rg ) =

6.4 Dual Wavelet Frames by the Projection Method In this section, we shall briefly discuss the projection method and its application to the OEP for constructing dual wavelet frames. The idea of the projection method is quite simple. Let S be a g˜ × g real˜ Let ! be a matrix of valued matrix with g˜ 6 g such that S is of rank g. g ˆ be a matrix of compactly supported tempered distributions on R and d 2-periodic trigonometric polynomials in g-variables. Then !ˆ is a continuous function, and we can define a projected function S ! and a projected mask S d in the frequency domain by ˆ W ) and Scd() := d ˆ(S W )> Sc!() := !(S

˜

5 Rg =

(6.28) ˜

Then S ! is a matrix of compactly supported tempered distributions on Rg , and if S is also an integer matrix, then Scd is a matrix of 2-periodic trigonometric polynomials in g˜ variables. To understand the projection method better, let us look at the projection method in the time domain with the simplest g˜ × g projection matrix S = where Lg˜ denotes the g˜ × g˜ identity matrix. For i 5 O1 (Rg ) and [Lg˜> 0], P d ˆ() = n5Zg dn hln· , it is known from [19, Theorem 2.2] that Z X ˜ ˜ [S i ]({) = i ({> w) gw> { 5 Rg and [S d]m = d(m>n) > m 5 Zg > Rgg˜

n5Zgg˜

P lm· and l denotes the imaginary unit. This where Scd() = n5Zg˜[S d]m h justifies the name of the projection method. The projection method is closely related to box splines. Box splines are obtained by projecting a high-dimensional unit cube [0> 1]g into a low-dimensional space. For a given g˜ × g (direction) integer matrix of rank g˜ with g˜ 6 g, the Fourier transform of its associated box spline P is given by d P () :=

Y 1 hln· > ln ·

˜

5 Rg >

(6.29)

n5

where n 5 means that n is a column vector of and n goes through all the columns of once and only once. Let "[0>1]g denote the characteristic function

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of the unit cube [0> 1]g . From (6.29), it is evident that the box spline P is a projection of "[0>1]g under the projection matrix since "[0>1]g = P by g˜ W \ \ " [0>1]g () = " [0>1]g ( )> 5 R . To measure the smoothness of ! and to study subdivision schemes in d), see Os (Rg ) for 1 6 s 6 4, we can similarly define quantities s (!) and s (ˆ [22] for more detail. The simple key observation of the projection method in wavelet analysis is the following result ([17, Theorem 4.4], [19, Theorem 2.2], [25, Theorem 4.1] and [34, Theorem 3.1]): Theorem 6.6. Let ! be a nontrivial compactly supported refinable function ˆ ˆ vector satisfying !(2) =d ˆ()!() with a 2-periodic trigonometric polyno˜ For mial matrix mask d ˆ. Let S be a g˜ × g integer matrix with full rank g. g˜ c c c every 1 6 s 6 4, then S !(2) = S d()S !(), 5 R and s (!) 6 s (S !). ˜ If in addition S Zg = Zg , then s (ˆ d) 6 s (Scd) and vu(ˆ d) 6 vu(Scd).

In general, it is much more dicult to investigate or compute various properties of multivariate refinable function vectors and multiwavelets than those of univariate or low-dimensional refinable function vectors and multiwavelets. One of the purposes of the projection method is to understand various properties of refinable function vectors and matrix masks in high dimensions through the analysis of the projected refinable function vectors and projected masks in low dimensions. Since it is much easier to analyze low-dimensional refinable function vectors, s (Scd) and s (S !) provide us d) and s (!) that we can expect. See upper bounds for the quantities s (ˆ [17, 19, 26, 23, 25, 34] for details. It turns out that the projection method can be applied not only to refinable function vectors, but also to wavelet frame systems ([25, 26]). The following result is a general version of [26, Theorem 4] and [25, Theorems 2.3 and 4.7]: Theorem 6.7. Let S be a g˜ × g integer matrix of rank g˜ such that ˜

˜

S W (Zg \[2Zg ]) Zg \[2Zg ]= ˜ ˜> #˜1 > = = = > #˜O be u × 1 vectors of compactly supported Let !> > # 1 > = = = > # O > !> tempered distributions on Rg such that min(2 (!)> 2 ()> 2 (# 1 )> = = = > 2 (# O )) A v> ˜ 2 (˜ )> 2 (#˜1 )> = = = > 2 (#˜O )) A v min(2 (!)>

(6.30)

; #˜1 > = = = > #˜O )) is a pair of for a real number v. If ([ v (; # 1 > = = = > # O )> [ v (˜ v g v g dual wavelet frames in (K (R )> K (R )), then ¡ v ¢ [ (S ; S # 1 > = = = > S # O )> [ v (S ˜; S #˜1 > = = = > S #˜O ) ˜

˜

is a pair of dual wavelet frames in (K v (Rg )> K v (Rg )). For v = 0, a similar result holds: if ([(# 1 > = = = > # O )> [(#˜1 > = = = > #˜O )) is a pair of dual wavelet

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117

frames in O2 (Rg ), then ([(S # 1 > = = = > S # O )> [(S #˜1 > = = = > S #˜O )) is a pair of ˜ dual wavelet frames in O2 (Rg ). In addition, if > # 1 > = = = > # O , ˜> #˜1 > = = = > #˜O are obtained via the OEP in Theorem 6.1 from the refinable function vectors ˜ then so are the projected ones S > S # 1 > = = = > S # O > S ˜> S˜ # 1 > = = = > S˜ # O ! and !, from the projected refinable function vectors S ! and S !˜ by replacing all the filters in Theorem 6.1 with their corresponding projected ones. g

Example 6.8. Let ! = "[0>1]g and # 1 > = = = > # 2 1 be the g-dimensional tensorg product Haar orthonormal wavelet. Then [ 0 (!; # 1 > = = = > # 2 1 ) is an org thonormal basis in O2 (R ). Thus, it is a tight wavelet frame in O2 (Rg ) and it is ˆ˜ = 1 (that is, the ˆ= obviously generated by the OEP in Theorem 6.1 with unitary extension principle in [40]). Let S := [1> = = = > 1]. Then by Theorem 6.7, g [ 0 (S !; S # 1 > = = = > S # 2 1 ) is a tight wavelet frame in O2 (R). Note that S ! is exactly the E-spline of order g satisfying Sc!(2) = 2g (1 + hl )g Sc!(). g Though there are 2g 1 functions in S # 1 > = = = > S # 2 1 , a closer look reveals that up to a multiplicative constant, this set has in fact only g distinct functions. It is a simple fact that if two generators f1 # and f2 # with f1 > f2 5 C appear in the generating set of p a wavelet frame in O2 (Rg ), then they can be replaced by only one generator |f1 |2 + |f2 |2 #. Now it is easy to see that all the spline tight wavelet frames constructed in [40] can be easily obtained by the projection method from the tensor-product Haar orthonormal wavelets. For the projection method on refinable function vectors, see [17, 19, 20, 21, 23, 26], and on wavelet and frame systems, see [25, 26]. The oversampling results on wavelet frames in [9, 37] correspond to the special case of the projection method with g˜ = g.

6.5 Properties of OEP and Fast Frame Transforms We concentrate in this section on the understanding of the OEP for constructing dual wavelet frames, from the point of view of fast frame transforms, by surveying some results from [28]. For simplicity of presentation, we only consider the one-dimensional case in O2 (R), that is, g = 1 and v = 0. We only deal with the dilation factor 2 here; interested readers can check [28] for detailed treatments on dual multiwavelet frames with a general dilation factor.

6.5.1 The New Canonical Form of a Matrix Mask ˆ () of 2-periodic trigonometric polynomials, we say For a square matrix X ˆ ˆ () is invertible for all 5 R and the inverse that X is strongly invertible if X

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ˆ () is also a matrix of 2-periodic trigonometric polynomials. When X ˆ () of X ˆ ˆ ˆ ˆ (2)ˆ ˆ ()1 is strongly invertible, for !(2) =d ˆ()!(), setting ˚ d() := X d()X ˆ ˆ ˆ ˚ ˆ ˚ ˆ !(). ˚ ˆ ()!(), and !() := X one can deduce that !(2) = ˚ d() It is evi˚ is compactly supported. dent that ! is compactly supported if and only if ! ˆ is a Similarly, d ˆ is a 2-periodic trigonometric polynomial if and only if ˚ d 2-periodic trigonometric polynomial. As demonstrated in [22, 30], the canonical form of a matrix mask is a very useful tool in the study of dual multiwavelet frames in [30] and of vector subdivision schemes and refinable function vectors in [22]. Based on the results in [22, 30] on the canonical form of a matrix mask, in [28] we further improved the canonical form of a matrix mask that plays a critical role in our understanding of the fast frame transforms associated with OEP. Now we have the following result on the canonical form of a matrix mask with multiplicity greater than one. Theorem 6.9. ([28, Theorem 2.1]) Let d ˆ be an u × u matrix of 2-periodic trigonometric polynomials with multiplicity u A 1. Suppose that d ˆ satisfies the sum rules of order p in (6.21) with a 1 × u vector |ˆ of 2-periodic trigonometric polynomials. If there is an u×1 vector ! of compactly supported tempered distributions such that ˆ ˆ !(2) =d ˆ()!()

and

ˆ |ˆ(0)!(0) 6= 0>

(6.31)

then for any nonnegative integer q, there exists a strongly invertible u × u ˆ () of 2-periodic trigonometric polynomials such that X ˆ can be matrix X ˆ 1 is a matrix of 2-periodic trigonometric polyobtained constructively, X ˆ (2)ˆ ˆ ()1 takes the form nomials, and X d()X ¸ (1 + hl )p S1>1 () (1 hl2 )p S1>2 () (6.32) S2>2 () (1 hl )q S2>1 () with (1 + hl )p S1>1 () = 1 + R(||q )>

$ 0>

(6.33)

where S1>1 > S1>2 > S2>1 and S2>2 are some 1 × 1, 1 × (u 1), (u 1) × 1 and (u 1) × (u 1) matrices of 2-periodic trigonometric polynomials. Moreover, ˆ ˚u ]W is a refinable distribution ˚ ˆ ˚ = [! ˚1 > = = = ! ˆ ()!(), letting !() := X then ! vector with the mask given in (6.32) and c b c q q ˚ ˚ ˚ ! 1 () = !1 (0) + R(|| ) and !c () = R(|| )> $ 0> c = 2> = = = > u= (6.34)

˚1 has the coiflet property of order q, and all other components That is, ! ˚ ˚ !2 > = = = > !u have the vanishing moments of order q.

The essential improvement of the canonical form of a matrix mask in the above Theorem 6.9 of [28] over that in [22, 30] lies in the extra and critical

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property in (6.33) and (6.34), which hold only if u A 1. Note that if 1 is a ˆ ˆ with !(0) ˆ simple eigenvalue of d ˆ(0) and ! satisfies !(2) =d ˆ()!() 6= 0, then (6.31) must hold. The essence of the proof of Theorem 6.9 in [28] largely lies on the following simple observation, which is proved in [28, Lemma 5.1]. Lemma 6.10. ([28, Lemma 5.1]) Let |ˆ and |˜ˆ be two 1 × u vectors of F 4 (R) functions such that |ˆ(0) 6= 0 and |ˆ˜(0) 6= 0. If u A 1, then for any nonnegaˆ () of 2-periodic tive integer q, there is a strongly invertible u × u matrix X trigonometric polynomials such that ˆ () + R(||q )> |ˆ() = |ˆ˜()X

$ 0=

(6.35)

In other words, if a matrix mask satisfies the sum rules of order p with a vector |ˆ, as long as the multiplicity of the mask is greater than one, then by Theorem 6.9 one can transform, without loss of any desirable properties, the matrix mask so that the new mask satisfies the sum rules of order p but with a simple vector |ˆqhz := [1> 0> = = = > 0], and the new mask has a well-organized form in (6.32) and (6.33). In certain sense, Theorem 6.9 tells us that the study of truly multiwavelets and refinable function vectors could be “easier” than that of scalar wavelets and scalar refinable functions. There is also a similar canonical form of a matrix mask in high dimensions ([22, 29]). For more details on the canonical form of a matrix mask, see [22, 28, 29, 30].

6.5.2 Approximation Order of Dual Wavelet Frames by OEP Frame approximation order is an important indicator for the performance of a dual wavelet frame. Following [15], we say that a pair of dual wavelet frames ([(# 1 > = = = > # O )> [(#˜1 > = = = > #˜O )) in O2 (R) provides frame approximation order v if there exists a positive constant F, independent of i and q, such that ki Tq (i )kO2 (R) 6 F2vq |i |K v (R) where the seminorm |i |2K v (R) := operator Tq is defined to be Tq (i ) :=

q1 X

1 2

R

R

; i 5 K v (R) and q 5 N> (6.36) ||2v |iˆ()|2 g and the truncated frame

O X X c c hi> #˜m>n i#m>n >

i 5 O2 (R)> q 5 Z=

(6.37)

m=4 c=1 n5Z

Now we have the following result (see [28, Theorem 3.4]) on the frame approximation order for a pair of dual multiwavelet frames obtained via OEP in Theorem 6.1 from refinable function vectors.

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Theorem 6.11. ([28, Theorem 3.4]) Let ! and !˜ be two compactly supported u × 1 refinable function vectors in O2 (R) such that (6.14) holds with g = 1 ˆ˜ of 2-periodic trigonometric polynomials. for some u × u matrices d ˆ and d Suppose that d ˆ satisfies the sum rules of order p in (6.21) with a 1 × u ˆ vector |ˆ of 2-periodic trigonometric polynomials such that |ˆ(0)!(0) = 1. ˆ˜ ˆ Assume that (6.15) and (6.16) are satisfied for some u × u matrices > , b ebc > ˜ec , c = 1> = = = > O, of 2-periodic trigonometric polynomials. Let # c and #˜c , c = 1> = = = > O, be defined in (6.18). If W

W

ˆ ˆ˜ ˜ ˆ () !() () = |ˆ() + R(||p )>

$ 0>

(6.38)

then the pair ([(# 1 > = = = > # O )> [(#˜1 > = = = > #˜O )) of dual wavelet frames in O2 (R) has the frame approximation order p. Moreover, (6.38) is a necessary conˆ dition for the frame approximation order p if span{!(2n) : n 5 Z} = Cu . The above result on the frame approximation order for dual multiwavelet frames is based on the approach of sum rules of a matrix mask instead of the approach in [15] on approximation order provided by the shift-invariant spaces generated by the corresponding refinable function. Thus, our approach here is more direct and in certain sense improves the corresponding result in [15, Theorem 2.8] even for the scalar case.

6.5.3 Fast Frame Transforms Associated with OEP In order to discuss the fast frame transform, let us recall some concepts first. By (c(Z))p×q we denote the linear space of all sequences x : Z 7$ Cp×q . For a sequence x : Z 7$ Cu×u , the associated subdivision operator Vx and the transition operator Wx are defined to be: For y 5 (c(Z))p×u , s X s X [Vx y]m := 2 yn xm2n and [Wx y]m := 2 yn xn2m W > m 5 Z= (6.39) n5Z

n5Z

P For y : Z 7$ Cp×u , we denote yˆ() := n5Z yn hln . In the frequency domain, one can easily verify that (6.39) is equivalent to s d 2ˆ y (2)ˆ x()> V x y() = (6.40) £ W W¤ s ˆ(@2)ˆ x(@2) + yˆ(@2 + )ˆ x(@2 + ) @ 2= Wd x y() = y

Basically, s Vx y is obtained by an upsampling of y followedsby a convolution with 2x while Wx y is obtained by a convolution of y with 2x followed by a downsampling. We say that a (wavelet or frame) transform is compact if it can be implemented by the subdivision operators Vxm and the transition operators

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Wxm , m = 1> = = = > M, with all xm being finitely supported sequences on Z and M being a finite number. More generally, we say that a transform is compact if it can be implemented using only convolutions, coupled with downsampling and upsampling, with finite-impulse-response (FIR) filters. The compactness of a transform is a very much desired property in practical applications due to its eciency, speed, and simple implementation. As already pointed out in [15, Proposition 4.5], the fast frame transform associated with tight wavelet frames obtained via OEP is generally not compact, and a deconvolution appears in the frame reconstruction transform. For the convenience of the reader, let us generalize the frame transform from the setting of tight wavelet frames in [15] to the general case of dual wavelet frames obtained via OEP in Theorem 6.1. In order to understand the key ingredient of the OEP in Theorem 6.1, it is helpful to have a look at its proof. We assume that all the conditions in Theorem 6.1 are satisfied with g = 1. Denote and ˜ in (6.17). By (6.16), one can easily obtain the frame decomposition formula in the setting of functions ([13]): for m 5 Z and i 5 O2 (R), Sm i = Sm1 i +

O X X X c c hi> #˜m1>n i#m1>n with Sm i := hi> ˜m>n im>n > (6.41) c=1 n5Z

n5Z

which can be proved by observing the following identity: For i> j 5 O2 (R), Z X W 1 jˆ() hSm i> ji = iˆ( + 2gm n)ˆ˜(gm + 2n) ˆ(gm ) g= (6.42) 2 R n5Z

Note that by (6.42), the operators Sm are independent of the choices of ˆ and ˆ˜ in (6.17). Now it follows from the formula in (6.41) that SM2 i = SM1 i +

MX O X 2 1 X

c c hi> #˜m>n i#m>n >

M1 ? M2 > i 5 O2 (R)=

(6.43)

m=M1 c=1 n5Z

Since all the operators Sm : O2 (R) 7$ O2 (R) are well-defined and uniformly bounded with kSm k = kS0 k for all m 5 Z, one can easily show ([28]) that limm$+4 hSm i> ji = hi> ji and limm$4 hSm i> ji = 0 for all i> j 5 O2 (R). By [21, Theorems 2.2 and 2.3], each of [(# 1 > = = = > # O ) and [(#˜1 > = = = > #˜O ) is a Bessel wavelet sequence in O2 (R), that is, the right-side inequality in (6.3) holds. In other words, we proved that for every M1 5 Z, c : m > M1 > n 5 Z> c = 1> = = = > O} and {M1 >n : n 5 Z} ^ {#m>n c ˜ {˜ M1 >n : n 5 Z} ^ {#m>n : m > M1 > n 5 Z> c = 1> = = = > O}

(6.44)

form a pair of dual wavelet frames in O2 (R). By (6.44) and limm$4 hSm i> ji = 0 for all i> j 5 O2 (R), ([(# 1 > = = = > # O )> [(#˜1 > = = = > #˜O )) is a pair of dual wavelet frames in O2 (R).

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The eciency of the frame representation in (6.12) largely lies in two aspects: (1) The frame approximation order. That is, for a given function i 5 O2 (R), how well can i be approximated by SM2 (i ) for a suciently large integer M2 ? (2) The sparseness of the frame representation in (6.12) and the speed of the fast frame transform. Issue (1) has been addressed in Subsection 6.5.2, and the information of the approximating function SM2 i is encoded into the wavelet coecients {hi> ˜M2 >n i}n5Z . Denote c i> ynm := hi> ˜m>n i and znm>c := hi> #˜m>n

m 5 Z> n 5 Z> c = 1> = = = > O=

(6.45)

In order to implement the multilevel frame decomposition in (6.43) eciently, from yM2 , one needs to compute the sequences of wavelet coecients yM1 and zm>c , m = M1 > = = = > M2 1 and c = 1> = = = > O. For this purpose, a fast frame transform is used to compute the coarse-scale low-pass wavelet coecients y m1 and the high-pass wavelet coecients zm1>c , c = 1> = = = > O, from the fine-scale low-pass wavelet coecients ym . For simplicity of discussion, in this W ˆ˜ ˆ section, we assume that the determinant of () () does not vanish (such ˆ and assumption will not aect our main result in Theorem 6.15 since both ˆ ˜ are strongly invertible); otherwise, the fast frame transform discussed in this section needs to be modified accordingly. By (6.14) and (6.17), we deduce that ˆ () ˆ(2) = ˚ d()ˆ ˆ ˆ ˜(2) = ˚ d ˜()ˆ˜()

with with

ˆ ˆ ˆ 1 > ˚ d() := (2)ˆ d()[()] ˆ˜() := (2) ˆ ˆ˜ 1 = ˜ ˚ ˆ˜()[()] d d

(6.46)

Based on (6.43), now the frame decomposition for discrete data is given by m ym1 = W˚ d ˜y

and zm1>c = W˚˜ec y m >

c = 1> = = = > O

(6.47)

and the frame reconstruction transform for discrete data is given by m1 + ym := V˚ dy

O X

V˚ec zm1>c >

(6.48)

c=1

where the operators W˚ d are defined in (6.39), and the new sequences d ˜ and V˚ ˚ c c ˚ e and ˜e are defined to be b b ˆ˜ 1 > ˆ 1 and ˚ ˜ec () := ˜ebc ()[()] ˚ ec () := ebc ()[()]

c = 1> = = = > O=

(6.49)

For the following particular choice of and ˜ in (6.17): W

ˆ˜ = L > ˆ˜ ˆ = () ˆ () () and () u

(6.50)

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the above fast frame transform becomes the one given in [15], that is, the fast frame transform is quite similar to the classical wavelet transform, but a deconvolution appears in the frame reconstruction transform in (6.48). Note W W ˆ ˜ˆ ˆ ˜ˆ () holds, then the conditions in () = () that if the relation () ˜ ˜ being replaced by and , (6.15) and (6.16) are the same with and respectively. Consequently, without loss of any generality of the above described fast frame transform, we may assume by convention that = and ˜ in (6.17) and in the fast frame transform (6.45)—(6.49). ˜ =

6.5.4 Balancing Property of Dual Wavelet Frames by OEP The reader, who is familiar with a multiwavelet transform in applications, may be aware of the approximation ineciency facing most multiwavelet transforms if the multiwavelets do not have high balancing order. In the following, let us address the balancing property of dual multiwavelet frames. In many applications, data y is given as a sequence of scalar numbers, that is, y : Z 7$ C. However, the input data y m in (6.47) is a sequence of 1 × u row vectors, that is, y m : Z 7$ C1×u . Thus, one has to convert the scalar sequence y into a vector sequence y m . A simple way to do this is to take y m := Hy ([39]), where H : c(Z) 7$ (c(Z))1×u is defined to be [Hy]n = [yun > yun+1 > = = = > yun+u1 ]>

n 5 Z> y 5 c(Z)=

(6.51)

Let p1 denote the linear space of all polynomials of degree less than p. For a sequence y, we say that y is a polynomial sequence of degree less than p, denoted by y 5 p1 , if yn = s(n)> n 5 Z for some s 5 p1 . Suppose that a mask d ˆ satisfies the sum rules of order p. In order to have a sparse and ecient wavelet representation, for the fast frame transform in (6.45)—(6.49), it is very important to require that 1. W˚ec ym = 0 for all ym 5 H(p1 ) and c = 1> = = = > O; that is, for a discrete polynomial sequence in p1 , all its high-pass wavelet coecients zm1>c should vanish. m m m 2. V˚ d W˚ d ˜ y = y for all y 5 H(p1 ); that is, any discrete polynomial sequence in p1 can be exactly recovered without using its high-pass wavelet coecients zm1>c > c = 1> = = = > O. m m 3. W˚ d ˜ y 5 H(p1 ) for all y 5 H(p1 ); This condition enables us to keep the properties in (1) and (2) if a multilevel fast frame transform is used since for original input polynomial signals in p1 , the input discrete data for the next level fast frame transform is now W˚ d ˜ H(p1 ). Now we have the following result on the balancing property of dual multiwavelet frames obtained via OEP in Theorem 6.1.

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ˆ˜ be u × u matrices of 2Theorem 6.12. ([28, Theorem 4.4]) Let d ˆ and d periodic trigonometric polynomials. Assume that (6.16) is satisfied for some ˆ˜ ebc > ˜ebc , c = 1> = = = > O, of 2-periodic trigonometric polyˆ , u × u matrices > ˜ in the fast frame transform in (6.45)— nomials. Use = and ˜ = W

ˆ˜ ˆ (6.49) and assume that the determinant of () () does not vanish for all ˜ 5 R. Let ! be an u × 1 vector of compactly supported distributions such ˆ ˆ˜ ˆ˜ ˜ ˆ that !(2) =d ˜()!() and !(0) 6= 0. Denote \ˆ () := [1> hl@u > = = = > hl(u1)@u ]. Then (i) there exists a 2-periodic trigonometric polynomial fˆ with fˆ(0) 6= 0 such that for all c = 1> = = = > O, as $ 0, W

ˆ˜ = R(||p )> and ˜ebc ()!()

ˆ ˆ ˜ ˜ !() () = fˆ()\ˆ () + R(||p ) implies

(6.52)

(ii) the fast frame transform has the balancing order p: W˚ d ˜ y 5 H(p1 )

and

W˚˜ec y = 0

; c = 1> = = = > O> y 5 H(p1 )> (6.53)

and V˚ d W˚ d ˜y = y

; y 5 H(p1 )>

(6.54)

where H(p1 ) := {[s(u·)> s(u · +1)> = = = > s(u · +u 1)] : s 5 p1 } is ˜ec are defined in (6.46) and (6.49). defined in (6.51) and ˚ d> ˚ d ˜, ˚ ec >˚ ˆ˜(0) but all Conversely, if (6.53) is satisfied and 1 is a simple eigenvalue of d m ˆ 2 > m 5 N, are not eigenvalues of d ˜(0), then (i) must hold. Moreover, (6.53) or (i) implies that d ˆ must satisfy the sum rules of order p in (6.21) with ˆ |ˆ() := fˆ()\ˆ ()() for some 2-periodic trigonometric polynomial fˆ with fˆ(0) 6= 0. The following result implies that the balancing property of a dual multiwavelet frame is stronger than the notions of vanishing moments, sum rules, and frame approximation order. Corollary 6.13. ([28, Corollary 4.5]) Under the same notions and the OEP conditions in Theorem 6.12, then (6.52) in Theorem 6.12 implies ˆ 1. d ˆ satisfies the sum rules of order p in (6.21) with |ˆ() := fˆ()\ˆ ()(). 1 O ˜ ˜ 2. {# > = = = > # } has p vanishing moments. 3. the pair ([(# 1 > = = = > # O )> [(#˜1 > = = = > #˜O )) of dual wavelet frames in O2 (R) has the frame approximation order p. The proofs of Theorem 6.12 and Corollary 6.13 in [28] are largely built on the following result, which deepens our understanding of the balancing property of multiwavelet transforms and is of interest in its own right.

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Let | be a finitely supported sequence of 1 × u vectors on Z. As in [18, Section 2] and [22, Page 53], we define a space Pp1>ˆ| by p 5 N> (6.55) Pp1>ˆ| := {s | 5 (p1 )1×u : s 5 p1 }> P where |ˆ() := n5Z |n hln . Note that Pp1>ˆ| is a subspace of the vector polynomial space (p1 )1×u and depends only on |ˆ(m) (0) for m = 0> = = = > p1, by noting s | :=

X

n5Z

s(· n)|n =

4 X m=0

s(m) (·)

(l)m (m) |ˆ (0)> m!

s 5 p1 =

The following result has been established in [28, Propositions 4.2 and 4.3]. Proposition 6.14. Let d and d ˜ be finitely supported sequences of u × u matrices on Z. Let | be a finitely supported sequence of 1 × u vectors on Z. Then for any positive integer p, we have 1. H(p1 ) = Pp1>ˆ| , where the operator H is defined in (6.51), if and only if there is a 2-periodic trigonometric polynomial fˆ such that |ˆ() = fˆ()[1> hl@u > = = = > hl(u1)@u ] + R(||p )> $ 0= (6.56) s | , where ˚ | is a finitely 2. For every s 5 p1 , Wd˜ (s |) = 2s(2·) ˚ W ˆ ˆ supported sequence on Z satisfying ˚ | (2) = |ˆ()d ˜() + R(||p ) as $ 0. fˆ(0) 6= 0>

W

ˆ˜() = R(||p ) as $ 0. 3. Wd˜ (s |) = 0 ; s 5 p1 if and only if |ˆ()d W | (2) + R(||p ) as 4. Wd˜ Pp1>ˆ| Pp1>ˆ| if and only if |ˆ()d ˜ˆ() = fˆ(2)ˆ $ 0 for some 2-periodic trigonometric polynomial fˆ(). (For Wd˜ Pp1>ˆ| = Pp1>ˆ| , we also need fˆ(0) 6= 0.) d( + ) = R(||p ), $ 0. 5. Vd Pp1>ˆ| (p1 )1×u if and only if |ˆ(2)ˆ 6. If Vd Pp1>ˆ| (p1 )1×u , then Vd (s |) = 21 s(·@2) [Vd |] = 21@2 s(·@2) ˚ | for all s 5 p1 , where ˚ |ˆ() := |ˆ(2)ˆ d(). 7. Vd Wd˜ (s |) = s | for all s 5 p1 if and only if W

ˆ ˆ( + ) = |ˆ() + R(||p )> $ 0> = 0> 1= |ˆ()d ˜() d

(6.57)

6.5.5 Main Results on Dual Multiwavelet Frames by OEP Now we have two complementary results on dual multiwavelet frames obtained via OEP in Theorem 6.1 from scalar refinable functions with multiplicity one and from truly refinable function vectors with multiplicity greater than one.

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For dual multiwavelet frames obtained via OEP in Theorem 6.1 from truly refinable function vectors, we have the following result. Theorem 6.15. ([28, Theorem 1.2]) Let ! and !˜ be two u × 1 refinable function vectors of compactly supported functions in O2 (R) such that (6.14) holds ˆ˜ of 2-periodic trigonometric with g = 1 for some u × u matrices d ˆ and d ˆ˜ satisfy the sum rules of orders p and p polynomials. Suppose that d ˆ and d ˜ in (6.21) with 1 × u vectors |ˆ and |ˆ˜ of 2-periodic trigonometric polynoˆ˜ ˆ mials, respectively. Assume that |ˆ(0)!(0) 6= 0 and |ˆ˜(0)!(0) 6= 0. If u A 1, ˆ ˜ˆ eb1 > eb2 > ˜eb1 > ˜eb2 then one can obtain in a constructive way u × u matrices > , of 2-periodic trigonometric polynomials such that

1. All the conditions in Theorem 6.1 are satisfied with O = 2 and g = 1. ˆ˜ are strongly invertible, that is, both ˆ˜ 1 ˆ and ˆ 1 and 2. The matrices are u × u matrices of 2-periodic trigonometric polynomials. 3. ([(# 1 > # 2 )> [(#˜1 > #˜2 )) is a pair of compactly supported dual multiwavelet frames in O2 (R), where # 1 > # 2 > #˜2 > #˜2 are defined in (6.18). ˜ and p vanishing moments, respectively. 4. {# 1 > # 2 } and {#˜1 > #˜2 } have p 5. The pair ([(# 1 > # 2 )> [(#˜1 > #˜2 )) of dual wavelet frames in O2 (R) has the highest possible frame approximation order p. 6. Its fast frame transform is compact and has the highest possible balancing order p.

Based on Theorems 6.12 and 6.15 whose proofs are constructive, now we outline the algorithm for constructing dual wavelet frames with a compact fast frame transform and other desirable properties. Algorithm 6.16. Let ! and !˜ be u × 1 vectors of compactly supported funcˆ˜ ˆ˜ ˆ ˆ ˆ˜()!() = d ˆ()!() and !(2) = d for some tions in O2 (R) such that !(2) ˆ˜ of 2-periodic trigonometric polynomials. Assume that u ×u matrices d ˆ and d ˆ d ˆ and d ˜ satisfy the sum rules of orders p and p ˜ in (6.21) with 1 × u vectors |ˆ and |ˆ ˜ of 2-periodic trigonometric polynomials, respectively. Without loss ˆ of generality, we assume |ˆ(0)!(0) = 1. The construction consists of three steps. ˆ of 2-periodic trigonometric (i) Obtain a strongly invertible u × u matrix ˆ ˆ polynomials satisfying |ˆ() = fˆ()\ ()() + R(||p ), $ 0, where fˆ is a freely chosen 2-periodic trigonometric polynomial with fˆ(0) 6= 0 to ˆ as low as possible. make the degree of ˆ˜ of 2-periodic trigonometric (ii) Obtain a strongly invertible u × u matrix polynomials with low degrees such that W

W

ˆ ˆ˜ ˆ ˆ + R(||p )> ˆ ˆ !() ˜ ˆ (g) d ˜() (g) !(g) = () () W

W

$ 0>

ˆ ˆ˜ ˆ ˆ˜ !() ˜ ˆ = 1 + R(||p+p ˜ ˆ !() ˜ !() () = fˆ()\ˆ () + R(||p )> () )= ()

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127

b b (iii) Construct high-pass filters eb1 > eb2 > ˜e1 > ˜e2 from (6.16) such that (6.16) is satisfied and the following vanishing moment conditions are satisfied

ˆ ˜ ˜ = R(||p ) and ebc ()!() ˆ = R(||p ˜ebc ()!() )> $ 0> c = 1> 2= (6.58)

Define # 1 > # 2 > #˜1 > #˜2 as in (6.18). Then ([(# 1 > # 2 )> [(#˜1 > #˜2 )) is a pair of dual wavelet frames in O2 (R) with a compact fast frame transform and with balancing order p. The frame approximation order is p. Moreover, {# 1 > # 2 } ˜ and p vanishing moments, respectively. and {#˜1 > #˜2 } have p See [28] for examples of balanced dual wavelet frames obtained from spline refinable function vectors via the above algorithm. A natural question one may wonder now about Theorem 6.15 is whether the requirement u A 1 in Theorem 6.15 is essential or not. The OEP in [7, 15] is to improve the vanishing moments of tight and dual wavelet frames derived from refinable functions. There are many results on spline wavelet frames obtained via OEP, see [7, 8, 13, 15, 30, 31, 32, 33]. For example, from any E-spline of order p, a compactly supported spline tight wavelet frame with two generators can be obtained via OEP ([7, 15]) with the highest possible vanishing moments p and frame approximation order p. Moreover, symmetry of a compactly supported tight spline wavelet frame with highest possible vanishing moments can be achieved with three generators ([31, 33]). Let ! be a compactly supported refinable function in O2 (R) with a finitely supported mask d. If ! has stable integer shifts and is symmetric, then there is ([33]) a tight wavelet frame in O2 (R) generated by 3 compactly supported wavelet functions with symmetry and highest possible ˜ be any pair of (symmetric) refinable function vanishing moments. Let (!> !) ˜. Then one can obvectors in O2 (R) with finitely supported masks d and d tain a pair ([(# 1 > # 2 )> [(#˜1 > #˜2 )) of dual wavelet frames in O2 (R) such that all the (symmetric) wavelet functions are compactly supported with highest possible vanishing moments ([13, 30]). Despite all the above interesting results on compactly supported tight and dual wavelet frames with many desirable properties, however, if we require a compact fast frame transform, things will be completely dierent. That is, from the point of view of function setting, things are perfect; however, if we look at the OEP from the point of view of discrete setting and fast frame transform, the diculty stays and we have the following result. Theorem 6.17. ([28, Theorem 1.3]) Let ! and !˜ be two functions in O2 (R) Q4 Q4 ˆ˜ ˆ ˆ˜(2m ) for some finitely ˆ(2m ) and !() such that !() := m=1 d := m=1 d ˆ˜(0) = 1 supported masks d = {dn }n5Z and d ˜ = {˜ dn }n5Z such that d ˆ(0) = d and ¶µ ¶ µ P P dn } n d ˜q } q have modulus one. (6.59) all the roots in C\{0} of n5Z

q5Z

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In particular, (6.59) is satisfied for any pair of compactly supported spline refinable functions (not necessarily having stable shifts) with finitely supported ˆ˜ ˆ , masks d and d ˜. For any 2-periodic rational trigonometric polynomials > b ebc > ˜ec , c = 1> = = = > O, such that (6.15) and (6.16) are satisfied with g = 1, if the associated fast frame transform is compact, letting # c > #˜c be defined in (6.18), then the pair ([(# 1 > = = = > # O )> [(#˜1 > = = = > #˜O )) of dual wavelet frames in O2 (R) obtained via OEP in Theorem 6.1 with u = 1 and g = 1, can have the frame approximation order at most two and each of {# 1 > = = = > # O } and {#˜1 > = = = > #˜O } can have vanishing moments at most one. ˆ˜ = 1 in Theorem 6.17, even if we require ˆ = It is not necessary that that the associated fast frame transform should be compact. In other words, for compactly supported scalar spline refinable functions, the freedom in the ˆ˜ is not enough to help us to achieve both a compact fast ˆ and choice of frame transform and high vanishing moments. Despite the negative result in Theorem 6.17, for a scalar refinable function !, one can still get a pair of dual multiwavelet frames in O2 (R) with a compact fast frame transform and other desirable properties by applying the OEP and Theorem 6.15 to the refinable function vector [!(u·)> !(u · 1)> = = = > !(u · u + 1)]W or even simply [!> 0> = = = > 0]W . Acknowledgments Research supported in part by NSERC Canada under Grant RGP 228051 and by the Alexander von Humboldt Foundation.

References 1. L. W. Baggett, J. E. Courter and K. D.Merrill, The construction of wavelets from generalized conjugate mirror filters in O2 (Rq ), Appl. Comput. Harmon. Anal., 13 (2002), 201—223. 2. L. W. Baggett, P. E. Jorgensen, K. D. Merrill and J. A. Packer, Construction of Parseval wavelets from redundant filter systems, J. Math. Phys., 46 (2005), 083502, 28 pages. 3. J. J. Benedetto and S. Li, The theory of multiresolution analysis frames and applications to filter banks, Appl. Comput. Harmon. Anal. 5 (1998), 389—427. 4. M. Bownik and E. Weber, Ane frames, GMRA’s, and the canonical dual, Studia Math. 159 (2003), 453—479. 5. C. Cabrelli, C. Heil and U. Molter, Accuracy of lattice translates of several multidimensional refinable functions, J. Approx. Theory 95 (1998), 5—52. 6. C. K. Chui and Q. T. Jiang, Balanced multi-wavelets in Rv , Math. Comp. 74 (2000), 1323—1344. 7. C. K. Chui, W. He and J. St¨ ockler, Compactly supported tight and sibling frames with maximum vanishing moments, Appl. Comput. Harmon. Anal. 13 (2002), 224—262. 8. C. K. Chui, W. He and J. St¨ ockler, Nonstationary tight wavelet frames, II: unbounded interval, Appl. Comput. Harmon. Anal. 18 (2005), 25—66. 9. C. K. Chui and X. L. Shi, q× oversampling preserves any tight ane frame for odd q, Proc. Amer. Math. Soc. 121 (1994), 511—517.

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10. W. Dahmen, Multiscale and wavelet methods for operator equations, in Multiscale problems and methods in numerical simulations, 31—96, Lecture Notes in Math. 1825, Springer, Berlin, (2003). 11. I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990), 961—1005. 12. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series, SIAM, Philadelphia, 1992. 13. I. Daubechies and B. Han, Pairs of dual wavelet frames from any two refinable functions, Constr. Approx. 20 (2004), 325—352. 14. I. Daubechies and B. Han, The canonical dual frame of a wavelet frame, Appl. Comput. Harmon. Anal. 12 (2002), 269—285. 15. I. Daubechies, B. Han, A. Ron, and Z. W. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (2003), 1—46. 16. B. Han, On dual wavelet tight frames, Appl. Comput. Harmon. Anal. 4 (1997), 380— 413. 17. B. Han, Analysis and construction of optimal multivariate biorthogonal wavelets with compact support, SIAM Math. Anal. 31 (2000), 274—304. 18. B. Han, Approximation properties and construction of Hermite interpolants and biorthogonal multiwavelets, J. Approx. Theory 110 (2001), 18—53. 19. B. Han, Projectable multidimensional refinable functions and biorthogonal wavelets, Appl. Comput. Harmon. Anal. 13 (2002), 89—102. 20. B. Han, Symmetry property and construction of wavelets with a general dilation matrix. Linear Algebra Appl. 353, (2002), 207—225. 21. B. Han, Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix, J. Comput. Appl. Math. 155 (2003), 43—67. 22. B. Han, Vector cascade transforms and refinable function vectors in Sobolev spaces, J. Approx. Theory 124 (2003), 44-88. 23. B. Han, Symmetric multivariate orthogonal refinable functions, Appl. Comput. Harmon. Anal., 17 (2004), 277—292. 24. B. Han, Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix, Adv. Comput. Math. 24 (2006), 375—403. 25. B. Han, Construction of wavelets and framelets by the projection method, International Journal of Mathematical Sciences, 1 (2007), in Press. 26. B. Han, The projection method in wavelet analysis, in Wavelets and Splines: Athens 2005, G. Chen and M.J. Lai eds., 202—225, Nashboro Press, Brentwood, Tenn., 2006. 27. B. Han, Refinable functions and cascade algorithms in weighted spaces with H¨ older continuous masks, SIAM J. Math. Anal., to appear, (2007). 28. B. Han, Dual multiwavelet frames with high balancing order and compact fast frame transform, preprint, (2007). 29. B. Han, The structure of balanced multivariate biorthogonal multiwavelets, preprint, (2007). 30. B. Han and Q. Mo, Multiwavelet frames from refinable function vectors, Adv. Comput. Math. 18 (2003), 211-245. 31. B. Han and Q. Mo, Tight wavelet frames generated by three symmetric B-spline functions with high vanishing moments, Proc. Amer. Math. Soc. 132 (2004), 77—86. 32. B. Han and Q. Mo, Splitting a matrix of Laurent polynomials with symmetry and its application to symmetric framelet filter banks, SIAM J. Matrix Anal. Appl., 26 (2004), 97—124. 33. B. Han and Q. Mo, Symmetric MRA tight wavelet frames with three generators and high vanishing moments, Appl. Comput. Harmon. Anal. 18 (2005), 67—93. 34. B. Han and Q. Mo, Analysis of some optimal bivariate symmetric refinable Hermite interpolants, Comm. Pure Appl. Anal., 6 (2007), 689—718. 35. B. Han and Z. Shen, Dual wavelet frames and Riesz bases in Sobolev spaces, preprint, (2007).

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36. D. Han and D. R. Larson, Frames, bases and group representations, Mem. Amer. Math. Soc. 147 (2000), no. 697. 37. E. Hern´ andez, D. Labate, G. Weiss and E. Wilson, Oversampling, quasi-ane frames, and wave packets, Appl. Comput. Harmon. Anal. 16 (2004), 111—147. 38. E. Hern´ andez, G. Weiss, A first course on wavelets, CRC Press, 1996. 39. J. Lebrub and M. Vetterli, Balanced multiwavelets: Theory and design, IEEE Trans. Signal Proc. 46 (1998), 1119—1125. 40. A. Ron and Z. Shen, Ane systems in O2 (Rg ): the analysis of the analysis operator, J. Funct. Anal. 148 (1997), 408—447. 41. A. Ron and Z. Shen, Ane systems in O2 (Rg ) II: dual systems, J. Fourier Anal. Appl. 3 (1997), 617—637. 42. I. W. Selesnick, Smooth wavelet tight frames with zero moments, Appl. Comput. Harmon. Anal. 10 (2000), 163—181.

Chapter 7

Characteristic Wavelet Equations and Generalizations of the Spectral Function Veronika Furst Fondly dedicated to Larry Baggett

Abstract Two equations characterize orthonormal (and Parseval) wavelets in O2 (R). We trace the history and development of this characterization and present a dierent argument that uses a generalization of the spectral function of Bownik and Rzeszotnik. By further generalizing this function in the setting of an abstract Hilbert space, we present a compressed proof of the abstract characteristic equation.

7.1 Introduction An orthonormal wavelet is a square-integrable function whose translates and dilates form an orthonormal basis for the Hilbert space O2 (R). That is, given the unitary operators of translation q i ({) = i ({ q), for q 5 Z, and s dilation i ({) = 2i (2{), we call # 5 O2 (R) an orthonormal wavelet if the ane system [(#) = { m q # : m> q 5 Z} = {2m@2 #(2m · q) : m> q 5 Z} is an orthonormal basis for O2 (R). We say # is a Parseval wavelet if [(#) is a other words, if ki k2 = Parseval tight) frame for O2 (R); P in P P P (i.e., normalized m 2 m m m5Z q5Z |h i> q # i| or, equivalently, i = m5Z q5Z h i> q # i q # 2 for all i 5 O (R). We note that the last series converges unconditionally, that is, independently of the enumeration of the frame elements. The focus of this chapter is on the characterization of functions as wavelets via certain equations. In the classical case of O2 (R), these equations occur in the frequency domain, so we need a definition of the Fourier transform: Z i ({)h2l${ g{= ib($) = R

Veronika Furst Department of Mathematics, Fort Lewis College, 1000 Rim Drive, Durango, CO 81301 e-mail: furst [email protected]

131

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Veronika Furst

Orthonormality can be represented through equations in the following way. Proposition 7.1. Let # 5 O2 (R). Then the ane system [(#) is orthonormal if and only if X b + q)|2 = 1 |#($ (7.1) q5Z

and

X

q5Z

b + q) #(2 b m ($ + q)) = 0 #($

for all m 1

(7.2)

hold for almost every $ 5 R.

The orthonormality of the translates of # is equivalent to Equation (7.1) while the orthogonality of the dilates is equivalent to Equation (7.2). See [19] for a proof. By representing the completeness of ane systems as well as orthonormality, the following is a characterization of orthonormal (and Parseval) wavelets. Theorem 7.2. A function # 5 O2 (R), with k#kO2 (R) = 1, is an orthonormal wavelet if and only if the equations X b m $)|2 = 1 |#(2 (7.3) m5Z

and

4 X m=0

b m $) #(2 b m ($ + t)) = 0 #(2

for every t 5 2Z + 1

(7.4)

hold for almost every $ 5 R. In fact, omitting the norm requirement k#kO2 (R) = 1, # is a Parseval wavelet if and only if Equations (7.3) and (7.4) are satisfied almost everywhere. The proof of this theorem appears in [19]. This characterization has been known since the beginning of the study of wavelets and appeared in [25], where it was attributed to Y. Meyer. A hint was provided of the proof in the case of multiresolution analysis (MRA) wavelets and shall be the motivation for our analysis in Sections 7.2 and 7.3. Theorem 7.2 can be used to verify that a function is a Parseval wavelet, as in the following example. ¡ ¢ Corollary 7.3. The Cohen wavelet #F = 13 "[0> 3@2) "[3@2> 3) of [14] is a Parseval wavelet. Proof. Given that the Haar wavelet #K = "[0> 1@2) "[1@2> 1) is an¡ orthonor¢ mal wavelet, we make the following observation: #F ({) = 13 #K {3 , which implies

7 Wavelet Equations and the Spectral Function

#bF ($) =

Z

R

1 3 #K

¡{¢ 3

h2l${ g{ =

Z

R

133

#K ({)h2l(3$){ g{ = #bK (3$)=

Then #F satisfies Equations (7.3) and (7.4) as a consequence of the fact that t u #K does. We now oer a brief account of the developments related to Theorem 7.2 over the past 12 years. Whereas we focus on a few results, we note that a large list of authors have contributed to the extensions of the theorem and can be found as references in the papers discussed here. Although its statement appeared in 1989 in [25] (also [24]), the complete proof of Theorem 7.2 was first published independently by G. Gripenberg and X. Wang ([18], [33]) in 1995. This argument, using the notation of [19], relies on writing, for each i in the dense subspace D = {i 5 O2 (R) : ib 5 O4 (R) and ib is compactly supported in R \ {0}}

of O2 (R), X

m>q5Z

m

2

|hi> q #i|

=

Z

R

+

|ib($)|2 Z

R

where wt ($) =

X m5Z

ib($)

4 X o=0

b m $)|2 g$ |#(2

X X

s5Z t52Z+1

ib($ + 2s t)wt (2s $) g$>

b o $) #(2 b o ($ + t))= #(2

The suciency of Equations (7.3) and (7.4) easily follows, but the proof of their necessity requires delicate approximations. Our focus in Sections 7.2 and 7.3 will therefore be on the necessity of the characteristic equations. The first multidimensional extension of Theorem 7.2 appears in [17]. M. Frazier et al. allow a finite collection of functions (multiwavelet) in O2 (Rg ) with respect to translation by the group Zg and dilation by the matrix 2L, where L is the g × g identity matrix. Furthermore, a larger class of functions is considered by letting the analyzing family = {!1 > !2 > = = = > !O } dier from the synthesizing family = {#1 > #2 > = = = > #O } (called dual frames). In the case when O = 1, Equations (7.3) and (7.4) become X

wt ($) =

m5Z 4 X m=0

b m $) #(2 b m $) = 1> !(2

b m $) #(2 b m ($ + t)) = 0> !(2

(7.5)

(7.6)

134

Veronika Furst

P

P

b m $)|2 and b m 2 where $ 5 Rg and t 5 Zg \ (2Z)g . When m5Z |#(2 m5Z |!(2 $)| are locally integrable in Rg \{0} (i.e., integrable over compact sets), Equations (7.5) and (7.6) are shown to be equivalent to ki k2O2 (Rg ) =

XX

m5Z q5Zg

h i> m q ! ih m q #> i i

for all i 5 D = { i 5 O2 (Rg ) : ib 5 O4 (Rg ) and

is compactly supported in Rg \ {0} }>

a dense subset of O2 (Rg ). The local integrability assumption guarantees that all series converge unconditionally. P Without this assumption, the convergence P in the reconstruction i = m5Z q5Zg h i> m q ! i m q # is not unconditional but still true in a weak sense. Unconditional convergence is established, however, when { m q # : m 5 Z> q 5 Zg } and { m q ! : m 5 Z> q 5 Zg } are Bessel sequences with the same bound. A Bessel P sequence is a collection {hm } of vectors in a Hilbert space H satisfying m |h i> hm i|2 Fki k2 for all i 5 H and some constant F A 0. In [11], A. Calogero further extends the multidimensional analysis, allowing for translation by a general lattice of Rg and dilation by a strictly expanding map P that preserves the lattice . That is, = D · Zg with D 5 Pg (R) and det D A 0, and P is a real g × g matrix thatpsatisfies || A 1 for each eigenvalue and P . Letting P (i )({) = | det P |i (P {) for { 5 Rg and denoting the dual lattice of and the transpose of P the new version of Theorem 7.2 states that by and P P P PO, respectively, m 2 #o i|2 for all i 5 O2 (Rg ) if and only ki kO2 (Rg ) = o=1 m5Z 5 |h i> P if = {#1 > #2 > = = = > #O } satisfies O X¯ ´¯2 ³ X ¯ ¯b m ¯#o P $ ¯ = det D a.e. $ 5 Rg >

(7.7)

o=1 m5Z

4 O X X o=1 m=0

´ ¡ ³ ¢ #bo P m $ #bo P m ($ + ) = 0 a.e. $ 5 Rg > 5 \ P =

(7.8)

The proof follows the approach in [18], [33], and [17] (with added diculty not present in the classical case), in particular, the use of the O2 (Rg )-version of the dense set D. The characterization of minimally supported frequency (MSF) wavelets is presented in [11] as an application. M. Bownik permits the same general lattice and dilation map as Calogero in his generalization of the results of [17], where only dyadic dilations were considered (see [5]). Furthermore, he extends the characterization to quasiane systems, introduced by A. Ron and Z. Shen in [30]. Defined, with respect to integer translation and dilation, to be the collection [ t ( ) =

7 Wavelet Equations and the Spectral Function

{#em>q>o : m 5 Z> q 5 Zg > o = 1> = = = > O} where #em>q>o =

½

m P q # o m | det P |m@2 q P #o

135

if m 0> if m ? 0>

the quasiane system is invariant under translation but not dilation (while the situation is reversed for ane systems). A new proof of Theorem 7.2 and, indeed, an alternative characterization of wavelets is presented in [6]. Instead of using the norm condition and Equations (7.3) and (7.4) to characterize orthonormal wavelets, Bownik employs (the multiwavelet, multidimensional versions of) Equations (7.1), (7.2), and (7.3), thereby separating the requirement of orthonormality from that of completeness. That Equation (7.3), with dilation by 2 replaced by d 5 R> |d| A 1, is equivalent to the completeness of the ane system [(#) whenever this system is orthonormal was conjectured by G. Weiss. The proof in [6] establishes this result for expanding dilation matrices that preserve the lattice ; an alternate proof, in the case of integer dilation matrices, is due to Z. Rzeszotnik ([32]). A larger class of expanding dilation matrices, including all one-dimensional real dilations, is permitted in [7]. For more proofs in the case of arbitrary expansive dilations, see [22], [23], and [31]. On the other hand, Equation (7.4) is singled out in [10] to help conclude whether or not the canonical dual of a quasiane frame is a quasiane system itself. In [12], C. Chui and X. Shi extend Theorem 7.2 to dilations by any real number d A 1, noting that the earlier extensions in [11] and [5], when restricted to the one-dimensional case, apply only to dilations by integers d 2. They generalize the characterization to dual frames. This approach of relaxing the relationship between the translates and dilates (by not requiring that the dilation preserve the translation lattice but rather allowing general dilations by dm for any real d A 1, m 5 Z, and general translations by eq, for any real e A 0, q 5 Z) is further extended to higher dimensions in [13]. The abstract approach of D. Labate in [21] allows the characterization of Gabor systems, as well as ane systems, as Parseval frames for O2 (Rg ). The extension of Theorem 7.2 to finite multiwavelets under dilation by an expanding integer matrix follows as a corollary of such characterization for generalized quasiane systems. E. Hern´ andez et al. ([20]) broaden this approach to include dilation matrices which do not preserve the translation lattice and which are not expanding (thereby obtaining a more general characterization than that in [13]). We note, in [12], [13], [21], and [20], the use of the multidimensional form of the dense subset D O2 (Rg ) or a similar set, which permits functions whose Fourier transforms are supported at the origin. As a technical consideration, we remark on the absence of such a set when O2 (Rg ) is replaced by an abstract Hilbert space and the Fourier transform is replaced by an abstract unitary map (in Section 7.3).

136

Veronika Furst

Our aim in this chapter is to return to P.G. Lemari´e’s original hint in [25] for the proof of the characteristic equations in the MRA case and extend it to non-MRA wavelets that are semiorthogonal. This “GMRA approach” to wavelets was pioneered by L. Baggett, H. Medina, and K. Merrill in [2] (see also [1]). In Section 7.2, our proof of Equation (7.13), the multiwavelet version of Equation (7.3), is a straightforward application of the properties of the spectral function, introduced by Bownik and Rzeszotnik ([8] and [32]); Equation (7.14), the multiwavelet Equation (7.4), requires a generalization of the spectral function. For the sake of comparison with the classical methods, we restrict our attention to dimension 1 and dilation by 2; these techniques can be readily extended to O2 (Rg ), translation by elements of Zg , and dilation by an expanding integer matrix. In Section 7.3, we discuss an analog of Theorem 7.2 in the setting of an abstract Hilbert space and point out (in a sense) the suciency of the semiorthogonality assumption. Although a natural analog of the classical characteristic equations, the abstract equation (7.23) therefore diers in a fundamental way from (7.3) and (7.4). This result, making full use of the structure of GMRAs, can be found in [16], based on the author’s dissertation; our new proof in this chapter uses a generalization of the spectral function for abstract shift-invariant spaces and GMRAs, thereby suppressing much of this structure.

7.2 Connection to Multiresolution in L2 (R) We approach the characteristic equations of Theorem 7.2 in a way that is dierent from those in the references listed previously. The original statement of the theorem (see [25]) is followed by a hint of the proof in the case of those wavelets that correspond to multiresolution analyses, introduced by S. Mallat and Y. Meyer (see [26], [27]). For the remainder of this section, q represents translation by integers and represents dilation by a factor of 2, as first defined in the introduction. Definition 7.4. A collection {Ym : m 5 Z} of closed subspaces of O2 (R) is called a multiresolution analysis, or MRA, if 1.) Ym Ym+1 for all m 5 Z, 2.) Ym+1 = Ym for all m 5 Z, [ 3.) Ym = O2 (R), m5Z

4.)

\

m5Z

Ym = {0},

5.) There exists a function ! 5 Y0 such that the set {q ! : q 5 Z} of translates of ! is an orthonormal basis for Y0 . The function ! is called a scaling function.

7 Wavelet Equations and the Spectral Function

137

The Mallat—Meyer construction of orthonormal wavelets from MRAs begins with the refinement equation, relating the scaling function ! to its negative dilate, using a periodic function k, called the low-pass filter: ¡ ¢ ¡ ¢ b (7.9) !($) = s12 k $2 !b $2 = A corresponding high-pass filter j defines an orthonormal wavelet # by ¡ ¢ ¡ ¢ b #($) = s12 j $2 !b $2 ; (7.10)

moreover, Ym = span{ l q # : l ? m> q 5 Z}. The Mallat—Meyer construction oers a converse, in which a suciently nice, smooth, periodic function k yields a scaling function and thereby an MRA. Of the various filter equations expressing the relationship between k and j, one proves most beneficial for our purposes: |k($)|2 + |j($)|2 = 2. By Equations (7.9) and (7.10), 2 b = |#(2$)|

2 1 b 2 |!($)|

¡ ¢ 2 2 b b 2 |k($)|2 = |!($)| |!(2$)| = (7.11)

As a consequence of this “telescoping relation,” ³ ´ X b m $)|2 = lim |!(2 b m $)|2 > b m1 $)|2 |!(2 |#(2 m5Z

m$4

and Equation (7.3) follows from establishing b m $)|2 = 1 lim |!(2

m$4

and

b m $)|2 = 0 lim |!(2

m$4

almost everywhere. Equation (7.4) can be proved similarly. The downside is that this argument can only work for MRA wavelets. In [28] and [29], M. Paluszy´ nski et al. extend the notion of MRA wavelets by studying those Parseval (instead of orthonormal) wavelets # that happen to correspond to functions j and ! as in Equation (7.10). That is, instead of considering the scaling function of an MRA, the authors define a pseudoscaling function ! that corresponds only to generalized filters k and j and satisfies the refinement equation (7.9). When # can be constructed from these functions via Equation (7.10), # is known as an MRA Parseval wavelet (MRA TFW, or MRA tight frame wavelet, in the words of the authors). In proving, for example, that the function # thus defined is indeed a Parseval wavelet, the authors need to exhibit Equations (7.3) and (7.4). Instead of complicated calculations, they apply Lemari´e’s trick of Equation (7.11), albeit in the case when # is not an orthonormal wavelet corresponding to an MRA but a Parseval wavelet corresponding to a frame MRA (that is, the translates of the pseudo-scaling function ! form a frame for Y0 , not an orthonormal basis).

138

Veronika Furst

Of special interest to us in this chapter are semiorthogonal Parseval wave0 lets, which satisfy h m q #> m q0 # i = 0 whenever m 6= m 0 . In other words, Zm B Zm 0 where Zm = span{ m q # : q 5 Z}. The authors of [28] use Lemari´e’s trick again to show that # is a semiorthogonal MRA Parseval L1 wavelet if and only if the translates of ! form a Parseval frame for Y0 = m=4 Zm and a nice enough generalized filter can be found. The trick of [25] appears in reverse in [29] during the proof of the following characterization of semiorthogonal MRA Parseval wavelets: a Parseval wavelet # is a semiorthogonal MRA Parseval wavelet if and only if its dimension P4 P b m ($ + q))|2 only takes on values in the set function G# ($) = m=1 q5Z |#(2 {0> 1}. In the proof of the suciency of this condition on the dimension function, the authors must construct a pseudo-scaling function and generalized filters, so that # is given by Equation (7.10). Together Equations (7.4) and (7.11) yield the necessary filter equations, proving that k and j are indeed generalized filters. While the arguments in [28] and [29] rely on Theorem 7.2, our goal is to use Lemari´e’s trick to prove Equations (7.3) and (7.4). However, we wish to allow all semiorthogonal Parseval wavelets satisfying one additional restriction, not only those that correspond to pseudo-scaling functions (and frame MRAs). To this end, we begin with a definition that first appeared in [1] and was formalized in [2]. Definition 7.5. A generalized multiresolution analysis, or GMRA, is a collection {Ym : m 5 Z} of closed subspaces of O2 (R) that satisfy conditions (1)—(4) in Definition 7.4 of an MRA and 50 .) Y0 is shift-invariant; i.e., q (Y0 ) Y0 for all q 5 Z. Instead of restricting ourselves to a single function, we allow multiwavelets consisting of countably many functions (the definitions encountered so far extend naturally). Given a countable index set L and a collection = {#o : o 5 L}, we define the resolution spaces Ym ( ) = span{ l q #o : l ? m> q 5 Z> o 5 L} for each m 5 Z. If is a semiorthogonal Parseval wavelet, then {Ym ( ) : m 5 Z} is a GMRA ([1], [9]). For a general Parseval wavelet , conditions (1), (2), (3), and T (5’) of the definition of a GMRA are satisfied, but the question of whether m5Z Ym ( ) = {0} remains open; however, it is shown in [9] that the answer is negative if frame bounds of 1 and 1 + , for any A 0, are allowed. For now, the aspect of GMRAs we use is the shiftinvariance of the core subspace Y0 . The following result of [15] (see also [2], [3]) is a consequence of Stone’s Theorem for unitary representations of Z and of spectral multiplicity theory for commuting unitary operators. Theorem 7.6. Let {Ym : m 5 Z} be a GMRA in O2 (R). Then there exists a unique finite Borel measure class [] on T, unique (a.e. ) Borel subsets V1 V2 · · · of T and a (not necessarily unique) unitary operator M: Y0 $ L 2 l5I O (Vl > ) such that for all i 5 Y0 and for each l 5 I,

7 Wavelet Equations and the Spectral Function

139

[M(q i )]l ($) = h2lq$ [M(i )]l ($) for almost every $ 5 T. Note that the index set I of the sets PVl may be countably infinite. We define the multiplicity function p by p = l5I "Vl ; then Vl = {$ 5 T : p($) l}.

Remark 7.7. Without additional information, we know from [15] that V1 = T almost everywhere. However, it is shown in [2] that is absolutely continuous with respect to Haar measure on T. Redefining V1 to be the support of the Radon—Nikodym derivative = g@g, we find that (T \ V1 ) = 0; that is, we still have V1 = T a.e.(). Furthermore, restricted to V1 , the function 1@ serves as the Radon—Nikodym derivative of |V1 with respect to . Hence, |V1 . Since it is only determined up to equivalence, we take to be |V1 . By [34], this choice of is necessary and sucient for Proposition 7.8 below to hold. We will often consider statements that are true “almost everywhere.” By this, we shall mean “for almost every $ 5 T” or “for almost every $ 5 R,” as appropriate. Our use of Haar measure is permitted whenever a result is a consequence of standard Fourier analysis techniques applied to functions in O2 (T) or O2 (R). The distinction between and becomes important only when the operator M, and not the Fourier transform F, is applied. However, in all such instances, the resulting functions will be supported on V1 , on which and agree. Therefore, all integrals will be written with respect to Haar measure , and we will use the abbreviation g$ = g($). L Set !m = M 1 ("m ), where "m is the element of l5I O2 (Vl ) whose mth component is "Vm and whose other components are all 0. The following result is proved in [15]. Proposition 7.8. The set {q !m : q 5 Z> m 5 I} is a Parseval frame for Y0 . The functions !m are called generalized scaling functions for Y0 . As their name indicates, these functions play a role similar to that of the scaling function for an MRA but are more general than the pseudo-scaling function of [28] and [29]. We note here that while {q !m : q 5 Z> m 5 I} is not necessarily an orthonormal basis for Y0 , we still have 0

h q !m > q0 !m 0 i = h Mq !m > Mq0 !m 0 i = h h2lq· "m > h2lq · "m 0 i = 0 whenever m 6= m 0 . Parseval frame generators, such as {!m }, whose translates form a Parseval frame for their span, play an important role in the theory of shift-invariant spaces. In particular, it is shown in [32] that if T is a countable collection in a shift-invariant subspace Y O2 (R) such that {q * : q 5 Z> * 5 T } is a Parseval frame for Y , then the spectral function

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Veronika Furst

Y ($) =

X

*5T

2 |*($)| b

is well-defined and independent of the choice of T . Since Theorem 7.6 and hence Proposition 7.8 apply to any shift-invariant space Y O2 (R), we see that a Parseval frame generator T always exists. Alternatively, the spectral function can be defined via the dimension function (given by the dimension of the range function) of a shift-invariant space; see [8] for a detailed treatment. In what follows, we will need two key properties of ; for a proof of the next lemma, see [32]. Lemma 7.9. The spectral function satisfies the following: P L (i) If Y = q5N¡Yq¢where each Yq is shift-invariant, then Y = q5N Yq . (ii) Y ($) = Y $2 .

The methods in the proof of this lemma will provide the foundation of our argument in proving Theorems 7.12 and 7.16. The many other properties of the spectral function can be found in [8]. The connection between Equation (7.3) and the spectral function is clear (see the proof of Theorem 7.12), but we need the following modification in order to arrive at Equation (7.4). Lemma 7.10. Let T be a countable collection contained in a shift-invariant 2 subspace Y PO (R) such2 that {q * : q 5 Z> * 5 T } is a Parseval frame for b ? 4 for a.e. $ 5 R. For m> p 5 Z, the function Y . Suppose *5T |*($)| m>p Y ($) =

X

*5T

*(2 b m $) *(2 b m ($ + p))

is well-defined (a.e.) and independent of the choice of T . Proof. We mimic the proof of Theorem 1.8 of [32]. Consider the standard Gabor orthonormal basis {ho>n : o> n 5 Z} of O2 (R) defined by 2lo$ "T ($ n)> hd o>n ($) = h

where T = [0> 1]. Let SY denote the orthogonal projection of O2 (R) onto Y . For every p> o> n 5 Z, XX h h\ d h SY h0>n+p > SY ho>n i = 0>n+p > d q * i h d q *> h o>n i =

XZ *>q

=

X

*5T

=

*5T q5Z

T+(n+p)

*5T

2lo·

h "T+n *h b

XZ

2lq$ *($)h b g$

T+n

Z

T+n

2lq$ 2lo$ *($)h b h g$

> "T+n *(· b + p) i

*($) b *($ b + p)h2lo$ g$=

7 Wavelet Equations and the Spectral Function

141

We remark that the original sum converges unconditionally, and for all n 5 Z, XZ 2 |*($)| b g$ = kSY h0>n k2O2 (R) ? 4= *5T

T+n

Two applications of the Cauchy—Schwarz inequality justify our use of Fubini’s theorem to conclude that Z X *($) b *($ b + p)h2lo$ g$ (7.12) h SY h0>n+p > SY ho>n i = T+n *5T

P 2 for every p> o> n 5 Z. Since *5T |*($)| b ? 4 for almost every $ 5 R, the P sum *5T *($) b *($ b + p) is a well-defined function on R for each p 5 Z. The left-hand side of Equation (7.12) P does not involve T while the rightb *($ b + p) via its Fourier hand side fully describes the function *5T *($) coecients on T + n. This justifies our claim that for each m> p 5 Z, m>p Y is independent of the order of summation and independent of the choice of Parseval frame generator T . t u Although we allow a wavelet to consist of a countable collection of functions, we do require a certain finiteness assumption, as evident in the previous theorem. Recall that a GMRA has an associated multiplicity function p. We say that the GMRA is integrable if p 5 O1 (T> |V1 ). The following is an easy consequence of this assumption. P 2 b Lemma 7.11. If p 5 O1 (T> |V1 ), then l5I |!l ($)| ? 4 almost everywhere. Proof. Z X R l5I

|!bl ($)|2 g$ =

X l5I

k"l k2

=

XZ l5I

T

"Vl ($) g$ =

Z

p($) g$=

T

t u

We are ready for the main result of this section. More accurately, we are ready for the main proof of this section since the result is known. We stress again, however, that this proof diers from those discussed in the Introduction in the sense that it is true to Lemari´e’s original hint, namely, Equation (7.11). The MRA and scaling function of the original proof are replaced by a GMRA and generalized scaling functions, thereby allowing a larger class of wavelets. However, since we want to use not only the resolution spaces Ym of a GMRA but also the wavelet spaces Zm , we restrict the class of Parseval wavelets to semiorthogonal ones, thereby guaranteeing the pairwise orthogonality of the wavelet spaces.

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Theorem 7.12. Let L be a countable index set, and let = {#o : o 5 L} be a semiorthogonal Parseval wavelet for O2 (R). If the collection {Ym ( ) : m 5 Z} where Ym ( ) = span{ l q #o : l ? m> q 5 Z> o 5 L} is an integrable GMRA, then the equations XX |#bo (2m $)|2 = 1 (7.13) m5Z o5L

and 4 X X m=0 o5L

#bo (2m $) #bo (2m ($ + t)) = 0

for every t 5 2Z + 1

(7.14)

hold for almost every $ 5 R.

Proof. As stated earlier, it is well-known that the collection {Ym = Ym ( ) : m 5 Z} is a GMRA (see [1], [9]). Consequently, we need only assume that its multiplicity function p belongs to O1 (T). By Proposition 7.8, there exists a set of generalized scaling functions {!l : l 5 I} such that {q !l : q 5 Z> l 5 I} is a Parseval frame for Y0 . Moreover, by the definition of a semiorthogonal Parseval wavelet, {q #o : q 5 Z> o 5 L} is a Parseval frame for Z0 = span{q #o : q 5 Z> o 5 L}, and Z0 = Y1 ª Y0 . Immediately, Lemma 7.9 implies X X |!bl ($)|2 + |#bo ($)|2 = Y0 ($) + Z0 ($) l5I

o5L

= Y0 ($) =

X ¯¯ ¡ ¢¯¯2 ¯!bl $2 ¯ = l5I

By Lemma 7.11, we have a telescoping relation analogous to Equation (7.11): X X X |#bo (2$)|2 = |!bl ($)|2 |!bl (2$)|2 > (7.15) o5L

l5I

l5I

and it follows that, for almost every $ 5 R, Ã ! X X XX m 2 m1 2 m 2 |#bo (2 $)| = lim |!bl (2 $)| |!bl (2 $)| = m5Z o5L

m$4

l5I

It remains to be shown that X |!bl (2m $)|2 = 1 and lim m$4

l

l5I

lim

m$4

X l

|!bl (2m $)|2 = 0 for a.e. $=

nP o4 bl (2m $)|2 | ! is a nonincreasing sequence, so a By Equation (7.15), l m=0 limit

7 Wavelet Equations and the Spectral Function

O1 ($) = lim

m$4

X l5I

143

|!bl (2m $)|2

exists. By Fatou’s lemma and the proof of Lemma 7.11, Z X Z O1 ($) g$ lim |!bl (2m $)|2 g$ = lim 2m kpkO1 (T) = 0> 0 R

m$4

R l5I

yielding O1 ($) = 0 almost everywhere, as desired. P To show that lim l |!bl (2m $)|2 = 1 for a.e. $, fix a constant E A 0 and o4 nP m$4 bl (2m $)|2 | ! is nondecreasing, so we let 0 ? { E. The sequence l m=1

can define

O2 ($) = lim

m$4

X l5I

|!bl (2m $)|2

almost everywhere. Next, we note that { m q !l : q 5 Z> l 5 I} is a Parseval frame for Ym . Using the same idea as in the proof of Theorem 1.7 in Chapter 2 of [19], let i 5 O2 (R) be such that ib = "[0> {] , and let Sm be the orthogonal projection of O2 (R) onto Ym . For m large enough so that [0> 2m {] [0> 1], we have XX |h Sm i> m q !l i|2 kSm i k2O2 (R) = l5I q5Z

¯2 X X ¯¯Z ¯ m $ b m@2 2l(q)2 m b ¯ = h !l (2 $) g$ ¯¯ ¯ i ($)2 R

l5I q5Z

= 2m

X X ¯¯Z ¯ ¯ l5I q5Z

=

XZ

0

l5I

Since Z

0

{

S

m5Z

{

0

1

"[0>

m

2

¯2 ¯ 2lq b ¯ () ()h g ! l {] ¯

|!bl (2m $)|2 g$=

Ym = O2 (R), the Monotone Convergence theorem implies

O2 ($) g$ = lim

m$4

Z

0

{

X l5I

|!bl (2m $)|2 g$ =

lim kSm i k2O2 (R) = {=

m$4

Since { 5 [0> E] is arbitrary, dierentiating yields O2 ({) = 1 for a.e. { 5 [0> E], from which we conclude that O2 ($) = 1 for a.e. $ 5 R, thus establishing Equation (7.13). To prove Equation (7.14), we begin by noting that for each m> p 5 Z, = m>p + m>p m>p Y1 Y0 Z0

(7.16)

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Veronika Furst

since the translates of the elements of T ^ S form a Parseval frame for Y1 = Y0 Z0 whenever T and S are Parseval frame generators for Y0 and Z0 , respectively. Given t 5 2Z + 1, Equation (7.16) gives us 1>t Y1 ($) =

X l5I

!bl (2$) !bl (2($ + t)) +

X o5L

#bo (2$) #bo (2($ + t))=

On the other hand, the set {!l : l 5 I} ^ {1 !l : l 5 I} also generates, via translates, a Parseval frame for Y1 = Y0 (see [32]). By Lemma 7.10, we have ¶ Xµ cl (2($ + t)) + cl (2$) ! [ [ 1>t ($) = ! (2$) ! (2($ + t)) ! 1 l 1 l Y1 l5I

=

X l5I

!bl ($) !bl ($ + t) = 0>t Y0 ($)=

By Lemma 7.11, we once again arrive at a telescoping relation: X o5L

#bo (2$) #bo (2($ + t)) =

X l5I

!bl ($) !bl ($ + t)

X l5I

!bl (2$) !bl (2($ + t))=

(7.17)

Repeating the above argument with m = 0 instead of m = 1 yields 0>t 0>t 0>t Y1 = Y0 + Z0 =

X l5I

and 0>t 0>t Y1 = Y0 =

!bl ($) !bl ($ + t) +

X o5L

#bo ($) #bo ($ + t)

¶ Xµ cl ($ + t) + cl ($) ! [ [ ! ($) ! ($ + t) =0 ! 1 l 1 l l5I

since t is odd. Then 4 X X m=0 o5L

P b P b b b o #o ($) #o ($ + t) = l !l ($) !l ($ + t), and we have

#bo (2m $) #bo (2m ($ + t)) = lim

m$4

X l5I

!bl (2m $) !bl (2m ($ + t))=

Equation (7.14) now follows from O1 ($) = limm$4

P b m 2 l |!l (2 $)| = 0.

t u

In the statement of Theorem 7.2, satisfaction of the characteristic equations (7.3) and (7.4) is sucient for # to be a Parseval wavelet. Therefore, in Theorem 7.12, Equations (7.13) and (7.14) are clearly not sucient for to be a semiorthogonal Parseval wavelet. The Cohen wavelet of Corollary 7.3 is a Parseval wavelet and therefore satisfies either pair of characteristic equations but is not semiorthogonal. Although no converse of Theorem 7.12 is possible, the next section illustrates how the situation changes when O2 (R) is replaced by an abstract Hilbert space.

7 Wavelet Equations and the Spectral Function

145

7.3 Multiresolution in an Abstract Hilbert Space In this section, we present the main results of [16]; our new compressed argument hides some of the structure evident in that paper. Let H be a separable (complex) Hilbert space, and let be a countable Abelian group of unitary operators on H, equipped with counting measure. Let be a unitary operator on H such that () = 1 is an isomorphism of onto a subgroup of finite index Q A 1 in . Then induces a map on the dual group b, 2 defined by ($) = $ for each $ 5 b. In the case s when H = O (R), as in the previous section, = {q : q 5 Z}> i ({) = 2i (2{)> (q ) = 2q , so Q = 2, and b = T maps into T by $(q ) = $ q . Given 5 , let b bb denote the element of defined by b($) = $(). We do not restate the definitions of semiorthogonality, Parseval wavelet, or GMRA. Their abstract analogs require only that we replace O2 (R) with H, {q : q 5 Z} with , and the concrete dilation with the abstract operator . However, we now restate the abstract version of Theorem 7.6 (the references remain the same), applied to any “shift”-invariant subspace, not only the core subspace of a GMRA. Theorem 7.13. Given a subspace Y H such that Y Y for all 5 , there exists a unique Borel measure class [] on b, unique (a.e. ) Borel b subsets V 1 V2 = = = of , and a (not necessarily unique) unitary operator L 2 M: Y $ l5I O (Vl > ) such that, for each i 5 Y and 5 , M(i ) = bM(i )

almost everywhere. The index set I may be countably infinite. Now and for the remainder of this chapter, “almost everywhere” shall refer to “ almost every $ 5 b,” where is normalized Haar measure on b, and we will use the abbreviation g$ = g($). By assuming ?? |V1 , we can take = |V1 . Recall from Remark 7.7 that this happens automatically in O2 (R). We define generalized scaling functions m 5 Y by the property that M(m ) = "m , where ["m ]l = lm "Vm = Then {m : 5 > m 5 I} forms a Parseval frame for Y ([15]). In the case when L the invariant subspace comes from a GMRA, we can say more. If M: Y0 $ l5I O2 (Vl > ), where Y0 is the core subspace of a GMRA, 2 e e Z0 $ L and M: e), where Z0 = Y1 ª Y0 , then the multiplicity Pn5K O (Vn > " and the complementary multiplicity function p e = function p = V l l P en are related by the consistency equation n "V e ($)) = p( ($)) + p(

X

5ker

p($)

(7.18)

146

Veronika Furst

almost everywhere ([2]). We say that a GMRA is integrable if p 5 O1 (b> |V1 ) and note that the integrability of p implies that of p. e Continuing to cite [15], suppose {!m : m 5 I} are the generalized scaling functions in Y0 and {n : n 5 K} are the complementary generalized scaling functions in Z0 . Since 1 !m 5 Y0 , there exist functions klm 5 O2 (Vl ), called generalized low-pass filters, such that, for each m 5 I, M klm = (7.19) M( 1 !m ) = l5I

Similarly, since 1 n 5 Y0 , there exist generalized high-pass filters jln 5 O2 (Vl ) such that, for each n 5 K, M M( 1 n ) = jln = (7.20) l5I

These functions satisfy various filter equations (see [15], [3], [4]), of which one is crucial for us: b |V ), then K = [klm ] Theorem 7.14. If p 5 O1 (> 1 l5I>m5I and J = [jln ]l5I>n5K 0 satisfy, for all > 5 ker and almost every $ 5 b, the filter equation X

klm ($)kl0 m ($ 0 ) +

m5I

X

jln ($)jl0 n ($ 0 ) = Q l>l0 > 0 "Vl ($)=

(7.21)

n5K

As our last preparatory result, we refer to the following abstract analog of Lemma 7.11 (see [16]). Lemma 7.15. If p 5 O1 (b> |V1 ), then

4 XX X 1 | [M m !s ]l ($)|2 ? 4 m Q m=1

(7.22)

s5I l5I

for almost every $ 5 b. A similar results holds if !s is replaced with n .

The main result of this section is the abstract analog of Theorem 7.12:

Theorem 7.16. Let L be a countable index set, and let = {#o : o 5 L} be a semiorthogonal Parseval wavelet for a separable Hilbert space H. For each m 5 Z, define Ym ( ) = span{ l #o : l ? m> 5 > o 5 L}. Then {Ym ( ) : m 5 Z} is a GMRA, and a unitary operator M corresponds to the core subspace Y0 . If this GMRA is integrable, then for every 5 ker , 4 X X 1 [M m #o ]l ($)[M m #o ]l0 ($) = l>l0 >hb "Vl ($) m Q m=1

(7.23)

o5L

holds for almost every $ 5 b, where hb represents the identity element of b.

7 Wavelet Equations and the Spectral Function

147

The two characteristic equations in O2 (R) have only one analog in H. We no longer have the Fourier transform in an abstract Hilbert space, so its role must now be performed by the unitary operator M; since M is defined only on Y0 , only negative dilates of #o appear in Equation (7.23). The complete proof of Theorem 7.16 that appears in [16] is a true generalization of Lemari´e’s trick: not only is a telescoping relation reached, but it is also done so using the generalized filters, modeling the intermediate step in Equation (7.11). We now provide a new argument using a further generalization of the spectral function. Lemma 7.17. Let {Ym : m 5 Z}Lbe an integrable GMRA in H with corresponding unitary map M: Y0 $ l5I O2 (Vl > |V1 ). Suppose that Y is a subspace of H such that m Y Y0 for all m 1. If T is a countable collection of vectors in Y such that {* : 5 > * 5 T } is a Parseval frame for Y , then for each m 1 and l> l0 5 I, the function X 0 Ym>l>l ($> ) = [M m *]l ($)[M m *]l0 ($) *5T

is well-defined for almost every $ 5 b and each 5 ker and independent of the choice of T .

Such a result has been proved in the case of Y = Z0 (see Lemma 3.2 in [16]). A straightforward modification of the proof shows that for any appropriate T , X X [M m *]l ($)[M m *]l0 ($) = [M m ]l ($)[M m ]l0 ($) *5T

5S

where { : 5 S} is the set of generalized scaling functions for the invariant space Y . As a technicality in this modification, we note that by the additivity of multiplicity functions on direct sums of -invariant subspaces, guarantees the the integrability of the multiplicity function of Y0P P integrability of the multiplicity function of Y ; consequently, * k*k2H = kk2H ? 4. Lemma 7.17 is the abstract analog of Lemma 7.10, only severely restricted (and with an entirely dierent proof). A direct analog would be nonsense: each -invariant subspace corresponds to a dierent unitary operator M. Proof of Theorem 7.16. Suppose that is a semiorthogonal Parseval wavelet. Then {Ym ( ) : m 5 Z} is a GMRA ([1], [9]), which we assume to be integrable. Let {!l : l 5 I} Y0 be the set of generalized scaling functions for Y0 ; then {!l : 5 > l 5 I} is a Parseval frame for Y0 . Fix l> l0 5 I and 5 ker . The -invariant space Y1 = Y0 clearly satisfies the requirement that m Y1 Y0 for all m 1. Let Z0 = span{#o : 5 > o 5 L}. By the semiorthogonality of , {#o : 5 > o 5 L} is a Parseval frame for Z0 , Y1 = Y0 Z0 , and {!l : l 5 I} ^ {#o : o 5 L} is a Parseval frame generator for Y1 . By Lemma 7.17, for each m 1,

148

Veronika Furst 0

0

0

m+1>l>l Ym+1>l>l ($> ) = Ym+1>l>l ($> ) + Z ($> ) 1 0 X0 = [M (m+1) !s ]l ($)[M (m+1) !s ]l0 ($) s5I

+

X

[M (m+1) #o ]l ($)[M (m+1) #o ]l0 ($)

o5L

for almost every $5b. On the other hand, choosing coset representatives 1 > = = = > Q of @ , the collection {q !s : q = 1> = = = > Q> s 5 I} generates a Parseval frame for Y1 = Y0 (see [32]). By the invariance property of Lemma 7.17, we have 0 ($> ) Ym+1>l>l 1

=

Q X X

[M (m+1) q !s ]l ($)[M (m+1) q !s ]l0 ($)

q=1 s5I

=

Q X X

m ( )($) [M m ! ] ($) \ m ( )($) [M m ! ] ($) \ q s l q s l0

q=1 s5I

= Q

X

[M m !s ]l ($)[M m !s ]l0 ($)

s5I

since 5 ker . Therefore, for each m 1, X o5L

1 Q m+1

[M (m+1) #o ]l ($)[M (m+1) #o ]l0 ($) =

X 1 [M m !s ]l ($)[M m !s ]l0 ($) Qm s5I X 1 [M (m+1) !s ]l ($)[M (m+1) !s ]l0 ($) Q m+1 s5I

for almost every $ 5 b. This telescoping relation is the analog of Equation (7.11) in the abstract non-MRA setting. Although the remainder of the proof can be found in [16], we provide it here for the sake of completeness. The series 4 X X 1 [M m !s ]l ($)[M m !s ]l0 ($) m Q m=1 s

converges for almost every $ 5 b by Lemma 7.15, from which we conclude that X 1 [M m !s ]l ($)[M m !s ]l0 ($) = 0= lim m m$4 Q s

By Lemma 7.17, applied to Y = Z0 and its complementary generalized scaling functions {n : n 5 K}, and Equations (7.19), (7.20), and (7.21),

7 Wavelet Equations and the Spectral Function

149

4 X X 1 [M m #o ]l ($)[M m #o ]l0 ($) m Q m=1 o5L X 1 X 1 [M 1 n ]l ($)[M 1 n ]l0 ($) + [M 1 !s ]l ($)[M 1 !s ]l0 ($) = Q Q n5K s5I 3 4 X 1 CX = jln ($)jl0 n ($) + kls ($)kl0 s ($) D = l>l0 >hb "Vl ($) Q n5K

s5I

for almost every $ 5 b, as desired.

t u

We remark that the unitary operator M corresponding to a -invariant space Y0 is not necessarily unique and that the definition of requires us to choose a particular M. Nonetheless, if = {#o : o 5 L} is a semiorthogonal Parseval wavelet such that the GMRA {Ym ( ) : m 5 Z} is integrable, then Equation (7.23) implies that the sum 4 X 4 X X 1 1 m>l>l0 m m [M # ] ($)[M # ] ($) = ($> ) 0 o l o l m Q Q m Z0 m=1 m=1

(7.24)

o5L

is independent of the choice of M (see [16]). We end by contrasting the abstract Hilbert space H to O2 (R). We remarked after the proof of Theorem 7.12 that no converse to that theorem is possible; namely, the two characteristic equations (7.13) and (7.14) do not imply semiorthogonality. This is not the case in an abstract Hilbert space. In particular, a Parseval wavelet that yields an integrable p and satisfies Equation (7.23) must be semiorthogonal. To be more precise, we state a result of [16]. Theorem 7.18. Let = {#o : o5L} be a countable collection of vectors in a separable Hilbert space H. For each m5Z, let Ym ( ) = span{ l #o : l ? m> 5> o5L}. S Suppose is a semiorthogonal collection such that m5Z Ym ( ) is dense in H. Then {Ym ( )} is a GMRA. If this GMRA is integrable and for every 5 ker , Equation (7.23) holds almost everywhere, then is a Parseval wavelet. Conversely, suppose is a Parseval wavelet and assume that the multiplicity function of the collection {Ym ( )} is integrable. If for every 5 ker , Equation (7.23) holds almost everywhere, then is semiorthogonal. The combination of Theorems 7.16 and 7.18 can be stated, in the context of integrable GMRAs, as follows: given a countable collection H, of the three statements (a) is a Parseval wavelet, (b) is semiorthogonal,

150

Veronika Furst

(c) satisfies Equation (7.23), any two statements imply the third. Acknowledgments I thank Palle Jorgensen, Kathy Merrill, and Judith Packer for inviting me to contribute to this volume in honor of my advisor, Lawrence Baggett. I appreciate the many valuable comments of Judith Packer and Kathy Merrill, the latter’s careful reading of my recent work, and a helpful e-mail from Ziemowit Rzeszotnik. Most of all, I thank Lawrence Baggett for all his inspiration, help, and guidance.

References 1. Baggett, L., Carey, A., Moran, W., Ohring, P.: General existence theorems for orthonormal wavelets, an abstract approach. Publ. RIMS, Kyoto Univ. 31, 95—111 (1995). 2. Baggett, L., Medina, H., Merrill, K.: Generalized multi-resolution analyses and a construction procedure for all wavelet sets in Rq . J. Fourier Anal. Appl. 5, 563—573 (1999). 3. Baggett L., Courter, J., Merrill, K.: The construction of wavelets from generalized conjugate mirror filters in O2 (Rq ). Appl. Comput. Harmon. Anal. 13, 201—223 (2002). 4. Baggett, L., Jorgensen, P., Merrill, K., Packer, J.: Construction of Parseval wavelets from redundant filter systems. J. Math. Phys. 46, 083502 (2005). 5. Bownik, M.: A characterization of ane dual frames in O2 (Rq ). Appl. Comput. Harmon. Anal. 8, 203—221 (2000). 6. : On characterizations of multiwavelets in O2 (Rq ). Proc. Amer. Math. Soc. 129, 3265—3274 (2001). : Quasi-ane systems and the Calder´ on condition. Contemp. Math. 320, 29— 7. 43 (2003). 8. Bownik, M., Rzeszotnik, Z.: The spectral function of shift-invariant spaces. Mich. Math. J. 51, 387—414 (2003). 9. : On the existence of multiresolution analysis for framelets. Math. Ann. 332, 705—720 (2005). 10. Bownik, M., Weber, E.: Ane frames, GMRA’s, and the canonical dual. Studia Math. 159, 453—479 (2003). 11. Calogero, A., A characterization of wavelets on general lattices. J. Geom. Anal. 10, 579—622 (2000). 12. Chui, C.K., Shi, X.: Orthonormal wavelets and tight frames with arbitrary real dilations. Appl. Comput. Harmon. Anal. 9, 243—264 (2000). 13. Chui, C.K., Czaja, W., Maggioni, M., Weiss, G.: Characterization of general tight wavelet frames with matrix dilations and tightness preserving oversampling. J. Fourier Anal. Appl. 8, 173—200 (2002). 14. Cohen, A.: Wavelets and multiscale signal processing (Ryan, R., trans.). Chapman and Hall, London (1995). 15. Courter, J.: Construction of dilation-d wavelets. In: Baggett, L., Larson, D. (eds.) The functional and harmonic analysis of wavelets and frames (San Antonio, TX, 1999). Contemp. Math. 247, pp. 183—206. Amer. Math. Soc., Providence (1999). 16. Furst, V.: A characterization of semiorthogonal Parseval wavelets in abstract Hilbert spaces. Preprint (2007). 17. Frazier, M., Garrig´ os, G., Wang, K., Weiss, G.: A characterization of functions that generate wavelet and related expansion. J. Fourier Anal. Appl. 3, supplement 1, 883— 906 (1997).

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18. Gripenberg, G.: A necessary and sucient condition for the existence of a father wavelet. Studia Math. 114, 207—226 (1995). 19. Hern´ andez, E., Weiss, G.: A first course on wavelets. CRC Press, Boca Raton (1996). 20. Hern´ andez, E., Labate, D. Weiss, G.: A unified characterization of reproducing systems generated by a finite family, II. J. Geom. Anal. 12, 615—662 (2002). 21. Labate, D.: A unified characterization of reproducing systems generated by a finite family. J. Geom. Anal. 12, 469—491 (2002). 22. Laugesen, R.: Completeness of orthonormal wavelet systems for arbitrary real dilations. Appl. Comput. Harmon. Anal. 11, 455—473 (2001). : Translational averaging for completeness, characterization and oversampling 23. of wavelets. Collect. Math. 53, 211—249 (2002). 24. Lemari´ e-Rieusset, P.G.: Ondelettes a ` localisation exponentielle. J. Math. Pures Appl. (9) 67, 227—236 (1988). : Analyse multi-´ echelles et ondelettes a ` support compact. In: Lemari´ e, P.G. 25. (ed.) Les ondelettes en 1989 (Orsay, 1989), pp. 26—38. Lecture Notes in Mathematics 1438, Springer-Verlag, Berlin (1990). 26. Mallat, S.: Multiresolution approximations and wavelet orthonormal bases of O2 (R). Trans. Amer. Math. Soc. 315, 69—87 (1989). 27. Meyer, Y.: Wavelets and operators. Cambridge Studies in Advanced Mathematics 37, Cambridge University Press, Cambridge (1992). ˇ c, H., Weiss, G., Xiao, S.: Generalized low pass filters and MRA 28. Paluszy´ nski, M., Siki´ frame wavelets. J. Geom. Anal. 11, 311—342 (2001). : Tight frame wavelets, their dimensions functions, MRA tight frame wavelets 29. and connectivity properties. Adv. Comput. Math. 18, 297—327 (2003). 30. Ron, A., Shen, Z.: Ane systems in O2 (Rg ): the analysis of the analysis operator. J. Funct. Anal. 148, 408—447 (1997). : Generalized shift-invariant systems. Constr. Approx. 22, 1—45 (2005). 31. 32. Rzeszotnik, Z.: Calder´ on’s condition and wavelets. Collect. Math. 52, 181—191 (2001). 33. Wang, X.: The study of wavelets from the properties of their Fourier transforms. Ph. D. thesis, Washington University in St. Louis (1995). 34. Weber, E.: Frames and single wavelets for unitary groups. Canad. J. Math. 54, 634— 647 (2002).

Chapter 8

Baggett’s Problem for Frame Wavelets Marcin Bownik Dedicated to Larry Baggett for his insightful contributions to the theory of wavelets

Abstract Baggett’s problem asks whether every Parseval wavelet # is associated with a generalized multiresolution analysis (GMRA). Equivalently, one can ask whether the intersection of all dilates of the space Y (#) of negative dilates of # must be necessarily trivial. We present the current state of knowledge and ramifications of this open problem for the wavelet theory. We also construct an example of (nontight) frame wavelet # with many desirable properties (such as smoothness, good decay, having a dual frame wavelet) such that its corresponding space of negative dilates is equal to the entire space Y (#) = O2 (R). This improves the original example of this kind by Rzeszotnik and the author [Math. Ann. 332 (2005), 705—720]. Key words: wavelet, Parseval wavelet, frame wavelet, framelet, GMRA, space of negative dilates

8.1 Introduction For a function # 5 O2 (R), we define its ane (or wavelet) system by m

W(#) = {#m>n ({) = 2 2 #(2m { n) : m> n 5 Z} If the system is an orthonormal basis of O2 (R), then we call # a wavelet. In the more general case when the system forms a frame for O2 (R), we call # a frame wavelet,or simply a framelet. If W(#) is a tight frame (with constant 1), i.e.,

Marcin Bownik Department of Mathematics, University of Oregon, Eugene, OR 97403-1222 e-mail: [email protected]

153

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||i ||2 =

XX m5Z n5Z

|hi> #m>n i|2

for all i 5 O2 (R)>

then # is a tight framelet and also called a Parseval wavelet. One of the fundamental problems in the theory of wavelets is a problem posed by Baggett in 1999. Baggett’s problem asks whether every Parseval wavelet # must necessarily come from a generalized multiresolution analysis (GMRA). The precise meaning of this statement is explained later. Nonetheless, this problem can be reformulated in terms of the space of negative dilates of # defined as (8.1) Y (#) = span{#m>n : m ? 0> n 5 Z}= Question 8.1 (Baggett, 1999). Let # be a Parseval wavelet with the space of negative dilates Y = Y (#). Is it true that \ Gm (Y ) = {0} ? m5Z

Despite its simplicity, Question 8.1 is a dicult open problem and only partial results are known. For example, Rzeszotnik and the author proved in [15] that if the dimension function (also called multiplicity function) of Y (#) is not identically 4, then the answer to Question 8.1 is armative. Question 8.1 is not only interesting for its own sake, but it also has several implications for other aspects of the wavelet theory. Rzeszotnik and the author [14] showed that a positive answer to Question 8.1 would imply that all compactly supported Parseval wavelets come from a MRA, thus generalizing the well-known result of Lemari´e-Rieusset [1, 31] for compactly supported (orthonormal) wavelets. Furthermore, the answer to Question 8.1 would help in understanding the structure of the set of Parseval wavelets that was reˇ c, Speegle, and Weiss [37]. cently studied by Siki´ However, there is some evidence that the answer to Question 8.1 might be negative. This is because there exists a (nontight) frame wavelet # with a very large space of negative dilates. The first example of such # was given by Rzeszotnik and the author in [14]. In fact, # has a dual frame wavelet, and the space of negative dilates of # is the largest possible Y (#) = O2 (R). Here, we improve this result by showing that one can find such # with good smoothness and decay properties, e.g., # in the Schwartz class S(R).

8.2 Preliminaries Despite the fact that all of our results are motivated by the classical case of dyadic dilations in R, we will adopt a more general setting of an expansive integer-valued matrix, i.e., an q × q matrix whose eigenvalues have modulus greater than 1. That is, we shall assume that we are given an q × q expansive matrix D with integer entries, which plays the role of the usual dyadic

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dilation. The dilation operator G is given by G#({) = | det D|1@2 #(D{), and the translation operator Wn is given by Wn i ({) = i ({ n), n 5 Zq . We say that a finite family = {# 1 > = = = > # O } O2 (Rq ) is a wavelet if its associated ane system #m>n = Gm Wn #>

m 5 Z> n 5 Zq > # 5

is an orthonormal basis of O2 (Rq ). In the more general case, when the ane system is a frame or tight frame (with constant 1), we say that is a frame wavelet or a Parseval wavelet, resp. Moreover, a frame wavelet is called semiorthogonal if 0

Gm Z B Gm Z

for all m 6= m 0 5 Z=

where Z = Z ( ) = span{Wn # : n 5 Zq > # 5 }=

(8.2)

q

The support of a function i defined on R is denoted by supp i = {{ 5 Rq : i ({) 6= 0}= Note that we are not taking the closure, since most of our functions are elements of O2 (Rq ) and hence they are defined a.e. Given a Lebesgue measurable set N Rq , define the space ˇ 2 (N) = {i 5 O2 (Rq ) : supp iˆ N}= O Here, the Fourier transform is defined by Z ˆ i ({)h2lh{>i g{= Fi () = i () = Rq

8.2.1 GMRAs Definition 8.2. A sequence {Gm (Y )}m5Z of closed subspaces of O2 (Rq ) is called a generalized multiresolution analysis (GMRA) if (M1) Wn Y = Y for all n 5 Zq , (M2) Y G(Y ), S (M3) m5Z Gm (Y ) = O2 (Rq ), T (M4) m5Z Gm (Y ) = {0}. In addition, if (M5) holds,

(M5) < * 5 Y such that {Wn *}n5Zq is an orthonormal basis of Y , then {Gm (Y )}m5Z is a multiresolution analysis (MRA).

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A GMRA {Gm (Y )}m5Z is customarily written as {Ym }m5Z , where Ym = G (Y ). The space Y is called the core space of the GMRA. Condition (M1) means that Y is shift-invariant (SI) and allows us to use the theory of shiftinvariant spaces for understanding the connections between the GMRA structure and wavelets or framelets. This is a subject of an extensive study by several authors, e.g., [3, 4, 5, 7, 11, 13, 17, 29, 30]. For a family O2 (Rq ), we define its space of negative dilates by m

Y = Y ( ) = span{#m>n : m ? 0> n 5 Zq > # 5 }=

(8.3)

We say that a frame wavelet is associated with a GMRA, or shortly comes from a GMRA, if its space Y = Y ( ) satisfies (M1)—(M4). In addition, if Y satisfies (M5), then Y is associated with an MRA. It turns out that every semiorthogonal frame wavelet comes from a GMRA. That is, the space Y = Y ( ) satisfies the conditions (M1)—(M4) and, therefore, Y is a core space of a GMRA. This is an easy consequence of the fact that the spaces Y and Z given by (8.2) and (8.3) satisfy M m5Z

Gm (Z ) = O2 (Rq )>

Y =

M

Gm (Z ) =

m6 1

µM m> 0

¶B Gm (Z ) =

(8.4)

Conversely, if we want to see when a GMRA gives rise to a wavelet, or a semiorthogonal frame wavelet, then some knowledge of shift-invariant spaces is useful.

8.2.2 The Spectral Function of Shift-Invariant Spaces Every shift-invariant space Y O2 (Rq ) has a set of generators , that is, a countable family of functions whose integer shifts form a tight frame (with constant 1) for Y , see [10, Theorem 3.3]. Although this family is not unique, the function X 2 |*()| ˆ Y () = *5

does not depend (except on a set of null measure) on the choice of the family of generators. We call Y the spectral function of Y . This notion was introduced by Rzeszotnik and the author in [13]. The basic property of is that it is additive on countable orthogonal sums of SI spaces and that O2 (Rq ) = 1. The spectral function also behaves nicely under dilations since G(Y ) () = Y ((DW )1 ). Moreover, if Y is generated by a single function *, then ( P 2 |*()| ˆ ( n5Zq |*( ˆ + n)|2 )1 for 5 supp *> ˆ Y () = 0 otherwise.

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We also mention that there are several other equivalent ways of defining the spectral function among which we note the following formula Y () = lim ||SYˆ (1(%@2>+%@2)q )||2 @%q %$0

for a.e. 5 Rq >

where SYˆ denotes the orthogonal projection of F(Y ) = Yˆ onto O2 (Rq ). The spectral function also allows us to define the dimension function of Y X Y ( + n)= dimY () = n5Zq

The dimension function (also called the multiplicity function) takes values in N ^ {0> 4}. It is additive on countable orthogonal sums as the spectral function. Moreover, the minimal number of functions needed to generate Y is equal to the O4 norm of dimY . In particular, Y can be generated by a single function if and only if dimY 6 1. Moreover, condition (M5) is equivalent to the equation dimY 1. We refer the reader to [10, 13] for the proofs of all these facts.

8.2.3 Semiorthogonal Parseval Wavelets and GMRAs The dimension function can be applied to connect GMRAs to semiorthogonal Parseval wavelets. If Y is a core space of a GMRA, then the space Z = G(Y ) ª Y is shift-invariant and has a (possibly infinite) set of generators . From (M2), (M3), and (M4) it follows that M Gm (Z )> O2 (Rq ) = m5Z

so we conclude that is a Parseval wavelet possibly of infinite order. That is, may have an infinite number of generators, and the ane system generated by the elements of forms a tight frame for O2 (Rq ). Moreover, is clearly semiorthogonal. Conversely, if is a semiorthogonal Parseval wavelet (possibly of infinite order), then the space Y of its negative dilates satisfies conditions (M1)—(M4) due to (8.4). Therefore, there is a perfect duality between GMRA structures and semiorthogonal Parseval wavelets (with possibly infinite number of generators). Since we are interested in finitely generated frame wavelets, the following result provides the required connection. Theorem 8.3. Suppose that is a semiorthogonal Parseval wavelet with O generators and Y is the space of negative dilates of . Then, {Gm (Y )}m5Z is a GMRA such that

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dimY () ? 4 and

X

for a.e. >

dimY ((D )1 ( + g)) dimY () 6 O

(8.5) for a.e. >

(8.6)

g5D

where D consists of representatives of distinct cosets of Zq @(D Zq ). Conversely, if {Gm (Y )}m5Z is a GMRA satisfying (8.5) and (8.6), then there exists a a semiorthogonal Parseval wavelet (with at most O generators) associated with this GMRA. Theorem 8.3 is a variant of the following well-known result of Baggett et al. [4]. For simplicity we state Theorem 8.4 in a shorter form. Its full form appears analogous to Theorem 8.3. Theorem 8.4 (Baggett, Medina, Merrill, 1999). A GMRA gives rise to a wavelet with O generators if and only if the dimension function of its core space Y satisfies (8.5) and X dimY ((D )1 ( + g)) dimY () = O for a.e. = (8.7) g5D

Equation (8.7) is often referred to as the consistency equation of Baggett. In order to establish Theorem 8.3, we recall the following fact shown in [13]. Lemma 8.5. If is a semiorthogonal Parseval wavelet and Y is the space of negative dilates of , then Y () =

4 XX

#5 m=1

m ˆ |#((D ) )|2 =

In particular, dimY () = G () where G () :=

4 X X X

#5

n5Zq

m=1

for a.e. > m ˆ |#((D ) ( + n))|2 =

(8.8)

The function G is often referred to as the wavelet dimension function [1, 2, 16, 27, 35]. Proof (Theorem 8.3). Suppose that is a semiorthogonal Parseval wavelet with O generators and the spaces Z and Y are given by (8.2) and (8.3). We already know that {Gm (Y )}m5Z is a GMRA. By Lemma 8.5, Z

[0>1]q

dimY ()g = =

Z

Rq

X

#5

Y ()g =

4 Z XX

#5 m=1

R

m ˆ |#((D ) )|2

||#|| @(| det D| 1) 6 O@(| det D| 1) ? 4= 2

(8.9)

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Hence, (8.5) holds. Since Z Y = G(Y ), we have Z () + Y () = G(Y ) () = Y ((D )1 )= This implies that dimZ () + dimY () =

X

dimY ((D )1 ( + g))

for a.e. >

(8.10)

g5D

where D consists of representatives of distinct cosets of Zq @(D Zq ). Since dimZ () 6 O, (8.6) holds. Conversely, let {Gm (Y )}m5Z be a GMRA satisfying (8.5) and (8.6). Let Z = G(Y ) ª Y . The consistency equation (8.10) and (8.6) yields dimZ () 6 O

for a.e. =

By [10, Theorem 3.3] this implies that Z has a set of 6 O generators. Since M Gm (Z )> Y = m6 1

we infer that is a semiorthogonal Parseval wavelet associated with the GMRA {Gm (Y )}m5Z .

8.3 Baggett’s Problem for Parseval Wavelets Baggett posed the following open problem during his talk at Washington University in 1999. Question 8.6 (Baggett, 1999). Is every Parseval wavelet associated with a GMRA? For the sake of historical accuracy, one should add that Baggett actually attempted to answer armatively Question 8.6 during his momentous lecture. This has sparked the interest of two listeners, Rzeszotnik and the author, who pointed out a missing argument in Baggett’s approach. Despite several attempts in succeeding years, Question 8.6 remains unanswered as of now. Nonetheless, in his talk Baggett proved that Questions 8.1 and 8.6 are equivalent. Indeed, the following observation is due to Baggett. Proposition 8.7 (Baggett, 1999). If is a Parseval wavelet, then its space of negative dilates Y is shift-invariant. Proof. It is enough to prove that the orthogonal complement Y B of Y is shift-invariant. It is clear that this complement is given by

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Y B = { i 5 O2 (Rq ) : ki k22 =

4 X XX

#5 m=0

n5Zq

|hi> #m>n i|2 }

by the tight frame property. Thus, we can see immediately that the space t u Y B is shift-invariant. We remark that the above result also holds if we assume that the framelet has a canonical dual framelet with the same number of generators, or equivalently, that has period one in the terminology of Daubechies and Han [23]. However, Proposition 8.7 in general is false for nontight framelets and even for framelets that have a dual framelet. These facts were shown by Weber and the author in [17]. Proposition 8.7 proves that the space of negative dilates of a Parseval wavelet satisfies condition (M1). The other two conditions, (M2) and (M3), are clearly satisfied leaving only (M4). This crucial obstacle leads naturally to Question 8.1. Consequently, Questions 8.1 and 8.6 are equivalent. In general, one might want to know what conditions on a shift-invariant space Y guarantee that \ Gm (Y ) = {0}= (8.11) m5Z

A nontrivial result of this type was shown by Rzeszotnik in [36]. Proposition 8.8 (Rzeszotnik, 2001). Let Y be a shift-invariant space. If Y 5 O1 (Rq ), then condition (8.11) holds. In the case when Y is a space of negative dilates, we have a stronger result due to Rzeszotnik and the author [15]. Theorem 8.9 (Bownik, Rzeszotnik, 2006). Let O2 (Rq ) be a Parseval wavelet and Y be its space of negative dilates. If |{ 5 Rq : dimY () ? 4}| A 0>

(8.12)

then (8.11) holds and generates a GMRA. While the complete proof of Theorem 8.9 can be found in [15], we present its outline containing the key idea of semiorthogonalization appearing later in the proof of Theorem 8.12. This procedure constructs a semiorthogonal wavelet that is associated to the same GMRA as a given Parseval wavelet. In practice, it may not even be known whether a Parseval wavelet , as in Theorem 8.9, is associated with a GMRA. Nevertheless, one can use the idea of semiorthogonalization to eventually deduce this property. Proof. Let Z = G(Y ) ª Y . Observe that Z is a shift-invariant space generated by {# SY #}#5 , where SY is the orthogonal projection on Y . Since is finite, Z has a finite number of generators. That is, we have dimZ 6 O for some O 5 N. The equation G(Y ) = Y Z implies that

8 Baggett’s Problem for Frame Wavelets

X

p(E 1 + g) = p() + dimZ () 6 p() + O>

161

(8.13)

g5D

where p = dimY and E = D . To complete the proof we need the following result from [15]. Lemma 8.10. Suppose that p : Rq $ [0> 4) is Zq -periodic, measurable function such that X p( + g) 6 p(E) + O for a.e. 5 Tq > (8.14) g5G

for some O > 0. Then, Z

Tq

p()g 6 O@(| det D| 1)=

(8.15)

To apply Lemma 8.10, we need to show that p is finite a.e. This can be done using a simple ergodic argument. Since the matrix E = D : Rq $ Rq preserves the lattice Zq , it induces a ˜ is ergodic by ˜ : Tq $ Tq . Moreover, E measure-preserving endomorphism E [38, Corollary 1.10.1] because E is expansive. Define the set H = { 5 Tq : p() ? 4}= ˜ is measure-preserving, ˜ 1 H H. Since E The condition (8.13) implies that E 1 ˜ we must have E H = H (modulo null sets). Finally, by the ergodicity of ˜ we have either |H| = 0 or |H| = 1. Combining this with our hypothesis E, |H| A 0 proves that p() ? 4 for a.e. 5 Rq . Since all the assumptions of Lemma 8.10 are satisfied for our p, we get that p 5 O1 (Tq ). Equivalently, we have Y 5 O1 (Rq ). By Proposition 8.8, (8.11) holds and generates a GMRA. We end this section by mentioning an interesting variant of Baggett’s problem for singly generated Parseval wavelets [37]. ˇ c, Speegle, and Weiss, 2007). Let Y be the space of negQuestion 8.11 (Siki´ ative dilates of a Parseval wavelet #. Is it true that # 65 Y=

(8.16)

Naturally, an armative answer to Question 8.1 implies a positive answer to Question 8.11. However, the converse implication is not known. Nonetheless, the following equivalent statements about a Parseval wavelet # can be easily shown [37]: (i) # 5 Y , (ii) Y = GY , (iii) Y = O2 (R).

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Once we relax the assumption that # is a Parseval wavelet, then Questions 8.1 and 8.11 are distinct. In Theorem 8.27, we shall exhibit a frame wavelet # such that # 65 Y , but (8.11) fails.

8.4 Ramifications of Baggett’s Problem A positive answer to Baggett’s problem influences many other problems involving Parseval wavelets. The reason behind it is a semiorthogonalization procedure that was introduced by Rzeszotnik and the author in [14]. Theorem 8.12. Suppose that is a Parseval wavelet with O generators and its space of negative dilates Y satisfies (8.11). Then, there exists a semiorthogonal Parseval wavelet with 6 O generators such that its space of negative dilates is also Y . In other words, both and are associated with the same GMRA {Gm (Y )}m5Z . Proof. Let Y be the space of negative dilates of . By the hypothesis (8.11), the sequence {Gm (Y )}m5Z is a GMRA. Let Z = G(Y )ªY . Observe that Z is generated by O functions, namely #SY #, # 5 , where SY is the orthogonal projection onto Y . Therefore, we can find a set of 6 O generators for Z . As in the proof of Theorem 8.3, we have M Gm (Z )= Y = m6 1

Hence, we can infer that that is a semiorthogonal Parseval wavelet and Y is the space of negative dilates of . Therefore, is associated to the same GMRA as . Remark 8.13. A more explicit semiorthogonalization procedure for the subˇ c et al. [37]. class of MRA Parseval wavelets was introduced recently by Siki´ Suppose that # 5 O2 (R) is a dyadic Parseval wavelet associated with an MRA. Let p be its generalized low-pass filter [32, 33, 37]. Then, the authors of [37] proved that one can modify the filter p in some minimal way to obtain a new filter corresponding to a semiorthogonal Parseval wavelet ! that is associated with the same MRA as #. As a corollary of Theorems 8.3 and 8.12 we deduce that Parseval wavelets give rise to the same class of GMRAs as semiorthogonal Parseval wavelets. A priori, this is only true for Parseval wavelets associated with a GMRA, which may (or may not) encompass all Parseval wavelets depending on the answer to Question 8.6. Corollary 8.14. Suppose that is a Parseval wavelet with O generators. Then, either {Gm (Y )}m5Z is a GMRA satisfying (8.5) and (8.6), or

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dimY 4= Proof. If dimY is not identically 4, then {Gm (Y )}m5Z is a GMRA by Theorem 8.9. Hence, Theorems 8.3 and 8.12 imply that (8.5) and (8.6) hold. Next, we deduce that an armative answer to Baggett’s problem implies that a compactly supported Parseval wavelet comes from an MRA [14]. Theorem 8.15 (Bownik, Rzeszotnik, 2005). Let be a Parseval wavelet with O = | det D| 1 generators such that its space of negative dilates Y satisfies condition (8.11). Then, is associated with an MRA if and only if G () =

4 X XX

#5 n5Z m=1

m ˆ |#((D ) ( + n))|2 A 0

a.e.

(8.17)

Remark 8.16. We recall that the restriction on the number of generators O = | det D| 1 in Theorem 8.15 is a necessary condition for (orthogonal) wavelet to be associated with an MRA due to Lemma 8.5. In the case of Parseval wavelets, it is possible to have MRA constructions resulting with bigger number of generators, see [20, 21, 24, 26, 34]. However, Theorem 8.15 is false if we relax the assumption O = | det D| 1. Remark 8.17. We must emphasize that for general Parseval wavelets, G is not equal to dimY . This is unlike the case of semiorthogonal wavelets, where Lemma 8.5 yields (8.18) G dimY = Conversely, by the results of Paluszy´ nski et al. [33], the identity (8.18) forces a Parseval wavelet to be semiorthogonal, see also [37, Theorem 3.15]. For the sake of accuracy, we should add that this result was shown only for dyadic, singly generated, one-dimensional Parseval wavelets. Despite that (8.18) may fail, we have that for any Parseval wavelet supp G = supp dimY >

(8.19)

see [14]. Indeed, by Proposition 8.7, Y is a shift-invariant space generated by the functions {Gm # : # 5 > m = 1> 2> = = =}= This, combined with an equivalent definition of the dimension function of shift-invariant spaces in terms of its range function, see [8, 10], yields m ˆ dimY () = dim span{(#((D ) ( + n))n5Zq : # 5 > m = 1> 2> = = =}>

which shows (8.19). Proof (Theorem 8.15). First, suppose that is associated with an MRA, i.e., its space of negative dilates satisfies dimY 1. By (8.18), we have that supp G = Rq and thus (8.17) holds.

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Conversely, assume (8.17). We need to show that (M5) is satisfied, or equivalently that dimY 1. Let be the semiorthogonal Parseval wavelet obtained from by Theorem 8.12. By Lemma 8.5 and the estimate (8.9) with taking place of , we have Z X dimY () g = k*k ˆ 2 @(| det D| 1) 6 O@(| det D| 1) = 1= [0>1]q

*5

On the other hand, (8.17) and (8.18) imply that dimY () A 0 for a.e. . Since dimY is integer-valued, we have that dimY 1, which concludes the proof of Theorem 8.15. t u As a corollary of Theorem 8.15, we have the following extension of a result of Lemari´e-Rieusset [31] to Parseval wavelets. Corollary 8.18 (Bownik, Rzeszotnik, 2005). Suppose that a Parseval wavelet satisfies the assumptions of Theorem 8.15 and at least one generator of is compactly supported. Then, is associated with an MRA. Combining Corollary 8.18 with Theorem 8.9, we have the following corollary. Corollary 8.19. Suppose that a Parseval wavelet has O = | det D| 1 generators and at least one of them is compactly supported. If the space Y of negative dilates of satisfies (8.12), then comes from an MRA.

8.5 Frame Wavelets with Large Spaces of Negative Dilates In this section, we prove that the assumption in Question 8.1 on # being a Parseval wavelet is necessary. This result is due to Rzeszotnik and the author [14] who constructed an example of a dyadic framelet # 5 O2 (R), such that its space of negative dilates Y is the largest possible, i.e., Y = O2 (R). Furthermore, such a framelet can have frame bounds arbitrarily close to 1 and it has a dual framelet. Here, we shall improve the example in [14] by showing that such a framelet can also have good smoothness and decay properties. Theorem 8.20. For any A 0, there exists a frame wavelet # 5 O2 (R) such that: (i) #ˆ is F 4 and all its derivatives have exponential decay, (ii) the frame bounds of W(#) are 1 and 1 + , (iii) the space of negative dilates of # is equal to O2 (R), (iv) # has a dual frame wavelet.

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While the proof of Theorem 8.20 follows the general construction method of [14], there are also some significant changes due to the additional smoothness requirement on #. In the proof of Theorem 8.20, we will use the following two standard results. Lemma 8.21 gives a sucient condition for an ane system to be a Bessel sequence. Its proof can be found in [28, Theorem 13.0.1]. Lemma 8.22 is a basic perturbation result for frames and can be found in [19, Corollary 15.1.5]. Lemma 8.21. Suppose that # 5 O2 (R) is such that #ˆ 5 O4 (R) and ˆ #() =R(|| ) as $ 0> ˆ as || $ 4> #() =R(||1@2 )

(8.20) (8.21)

for some A 0. Then the ane system W(#) is a Bessel sequence. Lemma 8.22. Suppose that H is a Hilbert space, {im } H is a frame with constants F1 and F2 , X F1 ||i ||2 6 |hi> im i|2 6 F2 ||i ||2 for all i 5 H> m

and {jm } H is a Bessel sequence with constant F0 , X |hi> jm i|2 6 F0 ||i ||2 for all i 5 H= m

If F0 ? F1 , then {im + jm } is a frame with constants ((F1 )1@2 (F0 )1@2 )2 and ((F2 )1@2 + (F0 )1@2 )2 . We will also need the following fact about the scale averaging of periodic functions. Lemma 8.23 can be considered as a special case of a result due to Bui and Laugesen [18, Lemma 9], which also holds for functions in Osorf (Rq ) and fairly general dilation matrices. This result is very close in spirit to the classical results of Banach—Saks and Szlenk asserting that weak convergence in Os implies norm convergence of arithmetic means. Since we impose weaker assumptions on than in [18], we present the proof of Lemma 8.23 for completeness. Lemma 8.23. Suppose 5 O2 (T), where T = R@Z. In other words, is a R1 1-periodic function in O2orf (R). Let m ({) = (2m {), and f = 0 . Then, for any strictly increasing sequence (om )m5N N, M 1X om = f M$4 M m=1

lim

in O2orf (R)=

(8.22)

R1 Proof. Without loss of generality, we can assume that f = 0 = 0. Otherwise, it suces to apply (8.22) for a function f. For the purpose of Lemma

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R1 8.23, let || || = ( 0 | |2 )1@2 be the norm in O2 (T) with the corresponding scalar product h·> ·i. We claim that the sequence ( m ) converges to f weakly in O2 (T). Indeed, let i be 1-periodic and continuous. Take any % A 0 and choose m 5 N such that |{ || 6 2m =, |i ({) i (|)| 6 %. Since Z

(n+1)@2m

(2m {)g{ = 0

n@2m

we have ¯Z ¯ ¯ ¯

1 0

m ¯ ¯ 2X Z ¯ ¯ 1 ¯ ¯ m i ¯ = ¯

n=0

(n+1)@2m

¯ Z ¯ ¯ (2 {)(i ({) i (n@2 ))g{¯ 6 % m

n@2m

1

m

0

| |=

A standard approximation argument using Luzin’s theorem and || m || = || || shows the claim. In particular, we have that gm := |h > m i| $ 0

as m $ 4=

(8.23)

For any m 6 n 5 N, the change of variables and 1-periodicity of yields |h m > n i| = |h > nm i| = gnm = Thus, we have the estimate ¯X ¯ X M M M X M1 X ¯ M X ¯ h om > on i¯¯ 6 g|om on | 6 2M gm = || o1 + = = = + oM ||2 = ¯¯ m=1 n=1

m=1 n=1

m=0

Here, we used g|om on | 6 g|mn| , where gm = sup{gn : n > m} is a decreasing sequence dominating (gm ). Hence, by (8.23) ° ° M1 X ° o1 + = = = + oM °2 ° 6 2 ° g $ 0 ° ° M M m=0 m

as M $ 4=

This shows (8.22) and completes the proof of Lemma 8.23. Proof (Theorem 8.20). Define the sets ]1 > ]2 by [ ]1 = (n + (1@4> 1@4))> n5Z

]2 = R \ ]1 =

ˇ 2 (]1 ) and # 2 5 O ˇ 2 (]2 ). As usual, Suppose that # 0 = # 1 + # 2 , where # 1 5 O define o : n 5 Z} for o = 0> 1> 2= Zmo = span{#m>n

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167

Lemma 8.24. Zm0 = Zm1 Zm2

for m 5 Z=

(8.24)

Proof. It suces to show (8.24) for m = 0. Take any i 5 Z01 and j 5 Z02 . By the results in [8, 10] we have Z0o = {i 5 O2 : iˆ() = p()#ˆo ()>

p is measurable and 1-periodic}= (8.25) Since supp iˆ ]1 , supp jˆ ]2 we have i B j. Thus, Z01 B Z02 . Finally, it suces to prove Z01 Z02 Z00 , as the converse inclusion is trivial. Take any i 5 Z01 Z02 . By (8.25), there are 1-periodic measurable functions p1 and p2 such that iˆ() = p1 ()#ˆ1 () + p2 ()#ˆ2 () = p1 ()1]1 ()#ˆ0 () + p2 ()1]2 ()#ˆ0 ()= (8.26) Since the sets ]1 and ]2 are invariant under integer shifts, p = p1 1]1 + p2 1]2 is 1-periodic. Hence, by (8.25) and (8.26) i 5 Z00 , which shows Z00 = Z01 Z02 . It now remains to choose # 1 and # 2 appropriately. The idea is that negative dilates of # 1 will generate functions whose Fourier transform is supported near the origin, whereas the negative dilates of # 2 will exhaust all functions that are supported away from the origin (in the Fourier domain). Let # 1 be a Parseval wavelet such that #ˆ1 is F 4 and supp #ˆ1 = (1@4> 1@16) ^ (1@16> 1@4)= Such a frame wavelet can be constructed by a standard method, for example see [12]. Indeed, it suces to take the convolution of 1(3@16>1@8)^(1@8>3@16) with a nonnegative smooth bump function supported on (1@16> 1@16) and normalize the result to obtain the Calder´on condition X |#ˆ1 (2m )|2 = 1 for 5 R \ {0}= m5Z

ˇ 2 (]1 ) and by (8.25), Z01 = O ˇ 2 ((1@4> 1@16)^(1@16> 1@4)). Note that # 1 5 O Hence, ˇ 2 ((2m2 > 2m4 ) ^ (2m4 > 2m2 )) Zm1 = O

for any m 5 Z>

and therefore, the space of negative dilates of # 1 is 1

Y = span

[

Zm1

ˇ2 =O

m?0

µ [ 1

m2

(2

m=4

ˇ2

= O (1@8> 1@8)=

m4

> 2

m4

) ^ (2

m2

>2

)

¶

(8.27)

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The function # 2 should be regarded as a perturbation term of # 0 = # 1 + # 2 . We are now ready to describe the construction procedure of # 2 . Let {*p : p 5 N} be some enumeration of the “truncated” Gabor system {1(n>n+1) h2lm : m 5 Z> n 5 Z> n 6= 1> 0}= Clearly, {*p : p 5 N} is an orthonormal basis of O2 ((4> 1) ^ (1> 4)). For any p 5 N, let np 5 Z denote the left endpoint of the support of *p , i.e., supp *p = (np > np + 1). Let be a 1-periodic function such that Z 1 supp ]2 > = 1= (8.28) 5 F 4> 0

Let (ps )s5N be a sequence of natural numbers such that each natural number occurs infinitely many times. We construct by induction a sequence of functions {!s : s 5 N} and a sequence of natural numbers (os )s5N . Let !1 = Go1 (*p1 ) and o1 = 1. Suppose we have constructed functions !1 > = = = > !s and integers o1 > = = = > os up to some s 5 N. Define os+1 to be the smallest integer such that supp !1 ^ = = = ^ supp !s (2os+1 > 2os+1 )>

(8.29)

!s+1 = Gos+1 (*ps+1 ) =

(8.30)

and It is easy to see that the sequence (os )s5N is increasing and the supports of !s ’s ˇ 2 (]2 ) are included in pairwise disjoint open intervals. Finally, define # 2 5 O by X X c2 () = fs !s () = fs Gos (*ps ) > (8.31) # s5N

s5N

for some suciently fast decaying sequence (fs )s5N of positive numbers. More precisely, we can choose fs ’s such that 0 ? fs+1 ? fs @(s + 1) for all s 5 N c2 have exponential decay. This will guarantee that and all derivatives of # # 0 = # 1 + # 2 satisfies property (i) of Theorem 8.20. In particular, by Lemma 8.21, the ane system generated by # 2 is a Bessel sequence. Our next goal is to show the following key fact. Lemma 8.25. Suppose that # 2 given by (8.31) is constructed as above. Let Y 2 be the space of negative dilates of # 2 and S be the orthogonal projection ˇ 2 ((4> 1) ^ (1> 4)), i.e., onto O [ (S i )() = iˆ()1(4>1)^(1>4)

for i 5 O2 (R)=

ˇ 2 ((4> 1) ^ (1> 4)). Then, S (Y 2 ) is dense in O Proof. Since

8 Baggett’s Problem for Frame Wavelets

169

2 : s 5 N} Y 2 > Y˜ 2 := span{#o s >0

ˇ 2 ((4> 1) ^ (1> 4)). Hence, we it suces to show that S (Y˜ 2 ) is dense in O need to show that each basis element *p , p 5 N, of O2 ((4> 1) ^ (1> 4)) belongs to the closure of F(S (Y˜ 2 )). Given u 5 N, ou c 2 \ 2 # ou >0 = G (# ) =

X

fs Gou (!s )=

s5N

By (8.29) and (8.30), supp Gou (!s ) (1> 1) for s ? u, and we have X X 2 (S (#o ))ˆ= fs Gou (!s ) = fs Gou os (*ps ) ou u >0 s> u

s> u

¸ X fs = fu ou *pu + Gou os (*ps ) = f sAu u

Since fu+1 @fu ? 1@(u + 1), ¯¯ X ¯¯ X ¯¯ ¯¯ fs ou os 1 ¯¯ ¯¯ ||Gou os (*ps )|| ? 2@u> G (* ) ps ¯¯ 6 ¯¯ f (u + 1)(u + 2) = = = s u sAu sAu

we conclude that ou (*pu + u ) belongs to F(S (Y˜ 2 )) for some u 5 O2 with ||u || ? 2@u. For a fixed p 5 N, let U = {u 5 N : pu = p}. By our construction, U = {u1 > u2 > = = =} is infinite. By Lemma 8.23 ou1 + = = = + ouM $ 1 as M $ 4 M

in O2 (np > np + 1)=

Hence, as M $ 4 ou1 (*pu1 + u1 ) + = = = + ouM (*puM + uM ) M ou1 + = = = + ouM ou1 u1 + = = = + ouM uM + $ *p = *p M M

in O2 (R)>

since || o ||4 = || ||4 ? 4 and u1 + = = = + uM $0 M

in O2 (R) as M $ 4=

Therefore, *p belongs to the closure of F(S (Y˜ 2 )). Since p 5 N is arbitrary and {*p : p 5 N} is an orthonormal basis of O2 ((4> 1) ^ (1> 4)), this completes the proof of Lemma 8.25. Lemma 8.26. Suppose that Y 2 is the same as in Lemma 8.25. Let Sm be the ˇ 2 ((4> 2m ) ^ (2m > 4)), i.e., orthogonal projection onto O

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Marcin Bownik

\ ˆ (S m i )() = i ()1(4>2m )^(2m >4)

for i 5 O2 (R)=

ˇ 2 ((4> 2m ) ^ (2m > 4)) for any m 5 Z. Then, Sm (Y 2 ) is dense in O Proof. Since the case m > 0 follows immediately from Lemma 8.25, we may assume that m ? 0. A straightforward calculation shows that Sm = Gm S Gm . ˇ 2 ((4> 1)^(1> 4)), ˇ 2 ((4> 2m )^(2m > 4)). Since Gm i 5 O Take any i 5 O by Lemma 8.25 there exists a sequence {in : n 5 N} Y 2 such that S0 in $ Gm i as n $ 4. Hence, Sm Gm in $ i as n $ 4. Since Gm in 5 Y 2 for m 6 0, this shows Lemma 8.26. We are now ready to conclude the proof of Theorem 8.20. Let Y 0 be the space of negative dilates of # 0 . By (8.24), ¶ µ[ µ[ ¶ 0 0 1 2 Y = span Zm = span (Zm ^ Zm ) = span(Y 1 ^ Y 2 )= m?0

m?0

Therefore, by (8.27) and by Lemma 8.26 ˇ 2 ((4> 1@8) ^ (1@8> 4))> S3 (Y 2 ) = O we have that Y 0 is dense in O2 (R). Since Y 0 is closed, it must be equal to O2 (R). It remains to show that one can also find a framelet with this property. Recall that # 0 = # 1 + # 2 , where # 1 is a Parseval wavelet and # 2 generates a Bessel ane system. Therefore, by Lemma 8.22, there exists % A 0 such that # 0 = # 1 + %# 2 is a framelet with frame bounds 1 @3 and 1 + @3. Moreover, since %# 2 is also of the form (8.31), the space of negative dilates of # 0 is also O2 (R). Therefore, # = (1@3)1@2 # 0 is a framelet with constants 1 and 1+ whose space of negative dilates is O2 (R). In fact, a more delicate argument shows that the lower frame bound of # 0 is > 1 and the last normalization step is not necessary. Finally, to show that # has a dual frame wavelet, we employ the well-known characterizing equations [9, 25, 27]. We recall that functions !> # 5 O2 (R) whose respective ane systems are Bessel sequences form a pair of dual framelets if and only if X ˆ m )#(2 ˆ m ) = 1 a.e. > !(2 m5Z

4 X

ˆ m )#(2 ˆ m ( + t)) = 0 !(2

a.e. and for odd t=

m=0

Thus, using supp #ˆl ]l , l = 1> 2, one can show that ! = (1 @3)1@2 # 1 is a dual framelet to # = (1 @3)1@2 (# 1 + %# 2 ). This completes the proof of Theorem 8.20.

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We finish this section by showing that the armative answer to Question 8.11 does not imply a positive answer to Question 8.1 for general frame wavelets #. Theorem 8.27. For any A 0, there exists a frame wavelet # 5 O2 (R) such that: (i) #ˆ is F 4 and all its derivatives have exponential decay, (ii) the frame bounds of W(#) are 1 and 1 + , T (iii) the space Y of negative dilates of # satisfies m5Z Gm (Y ) 6= {0}> (iv) # 65 Y , (v) # has a dual frame wavelet. Proof. Let #1 and #2 be the same as in the proof of Theorem 8.20. Then, a frame wavelet constructed by Theorem 8.20 is of the form # 0 = f1 # 1 + f2 # 2 for some constants f1 , f2 . ˆ (0>4) . Define a function # = f1 # 1 + f2 #+ , where #+ is given by #ˆ+ = #1 We claim that # satisfies all properties of Theorem 8.27. Indeed, (i) is trivial. The property (ii) follows from the same perturbation argument as in Theorem 8.20. Likewise, the same argument as in Theorem 8.20 shows that the ˇ 2 (0> 4) Y . This is mainly due to the space of negative dilates Y satisfies O decomposition O2 (R) = K+ (R)K (R)>

2 2 ˇ 2 (4> 0)> K+ ˇ 2 (0> 4)> where K (R) = O (R) = O

2 2 (R) and K+ (R) are invariant under the and the fact that Hardy spaces K action of G and Wn . On the other hand, it is clear that Y 6= O2 (R) and hence (iv) holds. Finally, (v) is shown exactly in the same way as in Theorem 8.20 with ! = (f1 )1 #1 being a dual framelet to #.

References 1. P. Auscher, Solution of two problems on wavelets, J. Geom. Anal. 5 (1995), 181—236. 2. L. Baggett, An abstract interpretation of the wavelet dimension function using group representations, J. Funct. Anal. 173 (2000), 1—20. 3. L. Baggett, J. Courter, K. Merrill, The construction of wavelets from generalized conjugate mirror filters in O2 (Rq ), Appl. Comput. Harmon. Anal. 13 (2002), 201—223. 4. L. Baggett, H. Medina, K. Merrill, Generalized multi-resolution analyses and a construction procedure for all wavelet sets in Rq , J. Fourier Anal. Appl. 5 (1999), 563—573. 5. L. Baggett, K. Merrill, Abstract harmonic analysis and wavelets in Rq , The functional and harmonic analysis of wavelets and frames (San Antonio, TX, 1999), 17—27, Contemp. Math. 247, Amer. Math. Soc., Providence, RI, 1999. 6. L. Baggett, P. Jorgensen, K. Merrill, J. Packer, Construction of Parseval wavelets from redundant filter systems, J. Math. Phys. 46, 083502 (2005). 7. J. Benedetto, O. Treiber, Wavelet frames: multiresolution analysis and extension principles, Wavelet transforms and time-frequency signal analysis, 3—36, Appl. Numer. Harmon. Anal., Birkh¨ auser Boston, Boston, MA, 2001.

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8. C. de Boor, R. DeVore, A. Ron, The structure of finitely generated shift-invariant spaces in O2 (Rg ), J. Funct. Anal. 119 (1994), 37—78. 9. M. Bownik, A characterization of ane dual frames in O2 (Rq ), Appl. Comput. Harmon. Anal. 8 (2000), 203—221. 10. M. Bownik, The structure of shift-invariant subspaces of O2 (Rq ), J. Funct. Anal. 177 (2000), 282—309. 11. M. Bownik, G. Garrig´ os, Biorthogonal wavelets, MRA’s and shift-invariant spaces, Studia Math. 160 (2004), 231—248. 12. M. Bownik, J. Lemvig, The canonical and alternate duals of a wavelet frame, Appl. Comput. Harmon. Anal. 23 (2007), 263—272. 13. M. Bownik, Z. Rzeszotnik, The spectral function of shift-invariant spaces, Michigan Math. J. 51 (2003), 387—414. 14. M. Bownik, Z. Rzeszotnik, On the existence of multiresolution analysis for framelets, Math. Ann. 332 (2005), 705—720. 15. M. Bownik, Z. Rzeszotnik, Construction and reconstruction of tight framelets and wavelets via matrix mask functions, preprint (2006). 16. M. Bownik, Z. Rzeszotnik, D. Speegle, A characterization of dimension functions of wavelets, Appl. Comput. Harmon. Anal. 10 (2001), 71—92. 17. M. Bownik, E. Weber, Ane frames, GMRA’s, and the canonical dual, Studia Math. 159 (2003), 453—479. 18. H.-Q. Bui, R. Laugesen, Ane systems that span Lebesgue spaces, J. Fourier Anal. and Appl. 11 (2005), 533—556. 19. O. Christensen, An introduction to frames and Riesz bases, Birkh¨ auser, Boston, 2003. 20. C. Chui, W. He, Compactly supported tight frames associated with refinable functions, Appl. Comput. Harmon. Anal. 8 (2000), 293—319. 21. C. Chui, W. He, J. St¨ ockler, Compactly supported tight and sibling frames with maximum vanishing moments, Appl. Comput. Harmon. Anal. 13 (2002), 224—262. 22. C. Chui, W. He, J. St¨ ockler, Q. Sun Compactly supported tight ane frames with integer dilations and maximum vanishing moments, Adv. Comput. Math. 18 (2003), 159—187. 23. I. Daubechies, B. Han, The canonical dual frame of a wavelet frame, Appl. Comput. Harmon. Anal. 12 (2002), 269—285. 24. I. Daubechies, B. Han, A. Ron, Z. Shen, Framelets: MRA-based constructions of wavelet frames, Appl. Comput. Harmon. Anal. 14 (2003), 1—46. 25. M. Frazier, G. Garrig´ os, K. Wang, G. Weiss A characterization of functions that generate wavelet and related expansion, J. Fourier Anal. Appl. 3 (1997), 883—906. 26. K. Gr¨ ochenig, A. Ron Tight compactly supported wavelet frames of arbitrarily high smoothness, Proc. Amer. Math. Soc. 126 (1998), 1101—1107. 27. E. Hern´ andez, G. Weiss A first course on wavelets, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1996. 28. M. Holschneider, Wavelets: An analysis tool, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995. 29. H.-O. Kim, R.-Y. Kim, J.-K. Lim, Characterizations of biorthogonal wavelets which are associated with biorthogonal multiresolution analyses, Appl. Comput. Harmon. Anal. 11 (2001), 263—272. 30. D. Larson, W. Tang, E. Weber, Multiwavelets associated with countable groups of unitary operators in Hilbert spaces, Int. J. Pure Appl. Math. 6 (2003), 123—144. 31. P. Lemari´ e-Rieusset, Existence de “fonction-p` ere” pour les ondelettes a ` support compact, C. R. Acad. Sci. Paris S´ er. I Math. 314 (1992), 17—19. ˇ c, G. Weiss, S. Xiao, Generalized low pass filters and MRA 32. M. Paluszy´ nski, H. Siki´ frame wavelets, J. Geom. Anal. 11 (2001), 311—342. ˇ c, G. Weiss, S. Xiao, Tight frame wavelets, their dimension 33. M. Paluszy´ nski, H. Siki´ functions, MRA tight frame wavelets and connectivity properties, Adv. Comput. Math. 18 (2003), 297—327.

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34. A. Ron, Z. Shen, Compactly supported tight ane spline frames in O2 (Rg ), Math. Comp. 67 (1998), 191—207. 35. A. Ron, Z. Shen, The wavelet dimension function is the trace function of a shiftinvariant system, Proc. Amer. Math. Soc. 131 (2003), 1385—1398. 36. Z. Rzeszotnik, Calder´ on’s condition and wavelets, Collect. Math. 52 (2001), 181—191. ˇ c, D. Speegle, G. Weiss, Structure of the set of dyadic PFW’s, Contemp. Math., 37. H. Siki´ Amer. Math. Soc., Providence, RI, (to appear). 38. P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982.

Chapter 9

Simple Wavelet Sets for Scalar Dilations in R2 Kathy D. Merrill Dedicated to Larry Baggett for his contributions of multiplicity and consistency

Abstract Wavelet sets for dilation by any scalar g A 1 in O2 (R2 ) are constructed that are finite unions of convex polygons. Such simple wavelet sets for dilation by 2 were widely conjectured to be impossible. The examples are built using the generalized scaling set technique of Baggett et al. [3]. Generalizations to other expansive dilations in O2 (R2 ) are discussed.

9.1 Introduction The term wavelet set was coined by Dai and Larson [13] in the late 1990s to describe a set Z such that "Z , the characteristic function of Z , is the Fourier transform of an orthonormal wavelet on O2 (Rq ). Here, by orthonormal wavelet they meant a single function # whose successive dilates by a scalar of all translates by the integer lattice form an orthonormal basis for O2 (R). This definition was later generalized to higher dimensions and to allow for other dilation and translation sets. In this chapter, we will restrict our attention to translations by the integer lattice in Rq and dilations by an expansive real-valued (sometimes restricted to be integer-valued) q × q matrix D, where by expansive we mean that all the eigenvalues have absolute value greater than 1. At about the same time as the Dai/Larson paper, Fang and Wang [18] first used the term MSF wavelet (minimally supported frequency wavelet) to describe wavelets whose Fourier transforms are supported on sets of the smallest possible measure, and they noted that such a wavelet ˆ a characteristic function. Since it is also true # would necessarily have |#| that any multiple of an MSF wavelet by a function of absolute value 1 is

Kathy D. Merrill Department of Mathematics, Colorado College, Colorado Springs, CO 80903 e-mail: [email protected]

177

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Kathy D. Merrill

still an MSF wavelet, MSF wavelets can be characterized as precisely those wavelets associated with wavelet sets. Even before this formal beginning of their study, wavelet sets and MSF wavelets were important to wavelet theory as a source of examples. One of the simplest and earliest wavelets to be studied was the Shannon or Littlewood— Paley wavelet for dilation by 2 in O2 (R), an MSF wavelet with wavelet set Z = [1> 12 )^[ 12 > 1) (see [16]). This wavelet derives its names from its important connections to the Shannon Sampling Theorem [32] and to Littlewood— Paley theory [28]. It can be thought of as a complementary example to another early and simple example for dilation by 2 in O2 (R): the Haar wavelet [21], # = "[0> 1 ) "[ 1 >1) . While the Haar wavelet is well-localized but not 2 2 smooth, the Shannon is smooth but not well-localized. These two examples each characterize one of the two properties desired of wavelets for the purpose of applications. In the mid-1980s, Meyer [29], Battle [5], and Lemari´e [26] found wavelets for dilation by 2 in O2 (R) that have both good smoothness and good localization properties, culminating in Daubechies’ [17] construction of a single wavelet that was both smooth and compactly supported. In contrast, MSF wavelets in general are not directly useful for applications, in that their discontinuous Fourier transforms mean that they cannot be well-localized. However, the importance of MSF wavelets as a source of examples and counterexamples has continued throughout wavelet history. A famous example due to Journ´e [16] first showed that not all wavelets have an associated structure called a multiresolution analysis (MRA). This structure, which grades O2 (Rq ) into a nested sequence of closed subspaces controlled by level of dilation, requires that the base space has an orthonormal basis consisting of translates of a single function called a scaling function. The Journ´e wavelet fails this last requirement. The fact that this counterexample is an MSF wavelet is a symptom of a larger fact: Auscher proved in [1] that every wavelet whose Fourier transform satisfies a weak smoothness and decay condition must be associated with an MRA. The discovery of a non-MRA wavelet gave an important push to the development of more general structures such as frame multiresolution analyses (FMRAs) (see, e.g., [8], [31]) and generalized multiresolution analyses (GMRAs) [3]. Counterbalancing the desire for smooth, well-localized wavelets and MRA wavelets is a desire for single wavelets. If a set of n functions {#1 > · · · > #n } has the property that their dilates of translates form an orthonormal basis for O2 (Rq ), we say that {#1 > · · · > #n } is a k-wavelet. For example, the simplest wavelets to build for dilation by 2 in O2 (Rq ) are (2q 1)-wavelets, formed using tensor products of wavelets and scaling functions from one-dimensional space. Moreover, it was shown in 1995 (see [1], [2], or [19]), that an MRA wavelet for dilation by an expansive integer-valued matrix D in O2 (Rq ) must be a (| det D| 1)-wavelet. For applications set in high-dimensional spaces, though, it is desirable to keep the number of wavelets under control, and information about how many wavelets are required is of theoretical interest. Until the late 1990s, it was an open question whether single wavelets (nec-

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essarily non-MRA except for the determinant-2 case) existed in dimension higher than 1. This question was settled by Dai, Larson, and Speegle [14], who showed that wavelet sets, and thus single wavelets, exist for an arbitrary expansive matrix in any dimension. Here again, it was MSF wavelets that provided an essential example. In addition to their usefulness as examples or counterexamples, MSF wavelets have been used as building blocks for more well-localized wavelets. Hern´andez, Wang, and Weiss ([22], [23]) smoothed the filters associated with MSF wavelets for dilation by 2 in O2 (R) to produce wavelets whose Fourier transforms were arbitrarily smooth. In the same spirit, Bownik and Speegle [11] used smoothing of MSF wavelet filters to show that (| det D|1)wavelets with compactly supported smooth Fourier transforms exist for every expansive integral matrix in O2 (R2 ). By the theorem of Auscher mentioned above, the Fourier transforms of non-MRA wavelets cannot be smoothed while retaining their non-MRA and orthonormal characteristics. However, the non-MRA wavelet based on the Journ´e wavelet set has been smoothed to give a non-MRA Parseval wavelet (dilates of translates form a normalized tight frame rather than an orthonormal basis) whose Fourier transform is both smooth and rapidly decaying [4]. Using a dierent approach than that of smoothing filters, Dai and Larson [13] developed a procedure they call interpolation to find well-behaved wavelets that lie between two MSF wavelets in an operator-theoretic sense. Finally, Lim, Packer, and Taylor [27] used wavelet sets as building blocks in a dierent sense, as a space over which to decompose wavelet representations as direct integrals of irreducibles. Thus, for wavelet theory, wavelet sets are a desirable commodity. Examples are relatively easy to construct in one dimension, and many appear in the literature, mostly for dilation by 2, but also (see, e.g., [3], [13]) for arbitrary dilations. Higher-dimensional examples are more dicult to find. The first two-dimensional examples for dilation by 2 appeared in the late 1990s in [15], [33] and [36]. Shortly after these examples appeared, general construction techniques were introduced by Baggett, Medina, and Merrill [3] and Benedetto and Leon [6] that can be used for arbitrary expansive matrix dilations in any dimension. All the construction techniques are iterative, and until now, all the wavelet set examples for scalar dilations in dimension 2 and higher showed the fingerprint of these iterative procedures and thus have had a fractal-like structure. (See, e.g., Figure 9.1 below for previous examples of dilation-by-2 wavelet sets in O2 (R2 ).) In fact, the only previously known simple wavelet sets in dimension greater than one are those for dilation by matrices of determinant ±2 in dimension 2. (See Section 9.4.) Many researchers believed that dimension 2 or higher dyadic wavelet sets, in particular, must necessarily have a complicated geometric structure. For example, in [9], Benedetto and Sumetkijakan showed that a wavelet set for dilation by 2 in Rq > q 2, cannot be the union of q or fewer convex sets, and they conjectured that it could not be the union of a finite number of convex sets. Soardi and Wieland in [33] made the weaker conjecture that a

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Fig. 9.1 Hole in the middle wavelet set of [33] and Windmill wavelet set of [3] and [6].

dilation-by-2 wavelet set in dimension greater than 1 could not be the finite union of polygons. The primary purpose of this chapter is to construct counterexamples to these conjectures for dilation by any real scalar g A 1 in O2 (R2 ). In Section 9.2, we develop the construction technique (based on [3]) that we used to build our examples. In Section 9.3, we first build one-dimensional examples needed for the two-dimensional constructions and then build the two-dimensional examples themselves. Finally, Section 9.4 contains some possible directions for generalizations.

9.2 The Construction Technique A well-known characterization, expressed in the following theorem, describes wavelet sets as precisely those sets that tile Rq both by translation and dilation. Theorem 9.1. A measurable set Z Rq is a wavelet set for dilation by an invertible real-valued matrix D if and only if X "Z ({ + n) = 1 a.e. { 5 Rq > (9.1) n5Zq

X

"Z (Dm {) = 1

a.e. { 5 Rq =

(9.2)

m5Z

Proof. See, e.g., [13]. This theorem gives an easy method to verify a set’s claim to be a wavelet set, but no easy method to discover them. In this section, we outline the

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construction technique that was used to discover the wavelet sets described in this chapter. This method was first developed in [3] for integer-valued matrix dilations, as a technique for building all wavelet sets. The original technique uses the theory of a generalized multiresolution analysis (GMRA), a collection of nested subspaces {Ym } that can be built from any wavelet # by letting Ym = vsdq{#(Dn · o) : n ? m> o 5 Zq }. However, as noted in [10], the essential ideas behind the wavelet set construction of [3] can be described independently of GMRA theory, using only the characterization of wavelet sets given by Theorem 9.1. In order to make the presentation given here selfcontained, we will take the latter approach, and point out the connection to GMRAs only briefly in passing. This approach has the additional advantage of allowing the dilation matrix D to be non-integer-valued. Although weµwill be ¶ g0 mostly interested in the case where q = 2 and the dilation matrix is , 0g we present the technique for a general expansive matrix in any dimension, as it introduces no additional complications. The construction depends upon first building a generalized scaling set. Motivated by the idea of a GMRA, or in applications by the idea of controlling the level of detail considered for an image described by a wavelet, it is useful to look at the subspaces obtained from all translates of #, but only those dilates less than a fixed cuto. The idea of a generalized scaling set comes from doing this for a wavelet # with #ˆ = "Z . Note that in this case, the Fourier transforms of just the translates {#(· o) : o 5 Zq } form an orthonormal basis for O2 (Z ), and thus the Fourier transforms of all the negative dilates of translates {#(Dn · o) : o 5 Zq > n ? 0} form an orthonormal basis for O2 (H), where H = ^m?0 Dm Z . Definition 9.2. A set H Rq is called a generalized scaling set for dilation by D if H = ^m?0 Dm Z for some wavelet set Z , or equivalently, if H D H and D H \ H is a wavelet set.

The equivalence of these two formulations is easily established (see, e.g., [10]), and means that the construction of a scaling set will lead immediately to a wavelet set. If H Rq is a generalized scaling set for dilation by D, we can use the fact that Z tiles Rq by both translations and dilations to see that H must satisfy the following consistency equation, which is a special case of the consistency equation satisfied by the multiplicity function of GMRA theory [3]. X X ¡ ¢ "H D1 ({ + m) = "D H ({ + m) m5Zq

m5Zq

=

X

m5Zq

=

"^

n Z n0 D

XX

m5Zq n0

({ + m)

"Dn Z ({ + m)

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=

XX

"Dn Z ({ + m)

n0 m5Zq

=

X

"Z ({ + m) +

m5Zq

= 1+

X

X

"Dn Z ({ + m)

n?0 m5Zq

"^

n Z n?0 D

m5Zq

= 1+

XX

({ + m)

"H ({ + m) a.e.

m5Zq

The following theorem uses the consistency equation to give sucient conditions for a set to be a generalized scaling set. Theorem 9.3. Suppose that D is an invertible real q × q matrix and that the measurable set H Rq is invariant under D1 ; contains a neighborhood of the origin; and that "H satisfies the consistency equation X X ¡ ¢ 1+ "H ({ + m) = "H D1 ({ + m) a.e. (9.3) m5Zq

m5Zq

Then Z = D H \ H is a wavelet set for dilation by D. Proof. The consistency equation (9.3) can be rewritten: X ¡ ¢ X 1= "H D1 ({ + m) "H ({ + m) m5Zq

=

X

"D H ({ + m)

m5Zq

=

X

X

m5Zq

"H ({ + m)

m5Zq

"D H\H ({ + m)

m5Zq

=

X

"Z ({ + m) a.e.

m5Zq

This shows that Z tiles Rq by translations. Since D is one-to-one and H is invariant under D1 , we have that the dilates of Z are disjoint. Because H contains a neighborhood of the origin, we can conclude that the dilates of Z will (up to measure 0) cover Rq . The result then follows by Theorem 9.1. Theorem 9.3 is the basis of our construction technique. We use it to build a set H = t4 l=1 Hl that is guaranteed to be a generalized scaling set as follows. We first build a set H1 that contains a neighborhood of the identity and satisfies X ¡ ¢ "H1 D1 ({ + m) a.e. (9.4) 1= m5Zq

We then recursively build disjoint sets Hl such that

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183

D1 Hl ^nl+1 Hn and

X

"Hl ({ + m) =

m5Zq

X

m5Zq

¡ ¢ "Hl+1 D1 ({ + m) a.e.

(9.5) (9.6)

The disjoint union of these sets, H = t4 l=1 Hl , will then satisfy all three hypotheses of Theorem 9.3 and thus will be a scaling set for dilation by D. This technique is described in [3] in terms of two measurable one-to-one maps W , taking [ 12 > 12 )q to H1 , and W 0 , taking Hl to Hl+1 . Note that both maps take a point { to a translate of D1 { by the lattice D1 Zq . The requirement (9.5) demands that W 0 ({) = D1 { whenever such an assignment does not violate the condition that the Hl be disjoint.

9.3 The Examples Our ultimate goal in this section is to use the technique outlined in Section 9.2 to build dilation-g wavelet sets in O2 (R2 ) that are finite unions of convex polygons. It will be useful to first carry out a similar construction in O2 (R1 ), building dilation-g wavelet sets that are finite unions of intervals. The key strategy in one dimension is to pick an interval H1 in such a way that some later interval Hl will be adjacent to it. This can be done in many ways; we concentrate here on constructions that most easily generalize to O2 (R2 ). Note that Theorem 9.1 and the construction technique of Section 9.2 only determine wavelet sets up to a set of measure 0. Thus, endpoints of intervals in the one-dimensional constructions or edges of polygons in the two-dimensional constructions can be dealt with arbitrarily. We take as a convention using intervals closed on the left and open on the right, and appropriate generalizations to R2 . 1 1 1 > f+ 2g ) with 2g ? First we note that any interval of the form H1 = [f 2g 1 f ? 2g will contain a neighborhood of 0 and satisfy Equation (9.4). To form H2 , we will then split H1 in half and use a translate of its dilate by go for the left half, and a translate by ng for the right half, where o> n 5 Z will be deter¢ £ ¢ £ mined later. That is, we let H2 = gf 2g12 go > gf go ^ gf + ng > gf + 2g12 + ng , which can be seen to satisfy Equation (9.6) and also (9.5). For H3 , we will take the dilate H3 = g1 H2 , which also satisfies Equation (9.6) and will be required by (9.5). Now we go back and choose f> o> n such that the two subintervals of H3 are adjacent to the the two subintervals of H1 . That is, we require µ ¶ 1 1 f n + =f+ (9.7) g g g 2g and

1 g

µ

o f g g

¶

=f

1 2g

(9.8)

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Kathy D. Merrill

A simultaneous solution to these two equations requires that n + o = g, and thus that g is an integer. In that case, one solution is to take o = g 1, n = 1, and f = 2g2g 2 2 . With this choice, we then build the rest of the sets Hl , for l 4 to continue this adjacency. That is, by defining W 0 to be the same on all odd Hl as it is on H1 , we use the adjacency of H3 to H1 to force H4 to be adjacent to H2 , and so on. Similarly, we define W 0 to be the same on all the even Hl , namely W 0 ({) = {@g. (This last definition is actually forced by (9.5).) The resulting scaling set for dilation by any integer g A 2 consists of three intervals: one formed from H1 , which is centered at f and has length g f n g2 1 ; one formed from the right half of H2 , so with left endpoint at g + g 1 and length 2g2 2 ; and one formed from the left half of H2 , which gives the previous interval shifted to the left by 1. That is, ¶ ¶ ¶ 2g 1 1 1 g 2g 1 g > 2 1 ^ > 2 ^ > > (9.9) H= g + 1 2g 2 g+1 g 1 2g2 2 g2 1 so that we have a wavelet set for dilation by an arbitrary integer g given by ¶ ¶ 1 g2 2g2 g 2g 1 > g ^ 1> Z = g + 1 2g2 2 2g2 2 g+1 ¶ 2 ¶ 1 2g 1 g2 2g g ^ 2 > ^ > = g 1 2g2 2 2g2 2 g2 1 We can alter this construction slightly as follows. Using the same values for the whole dilate¢ of H1 to the right by f and n, we form H2 by translating £f n 1 1 1 f 1 1 g = g , thus taking H2 = g 2g2 + g > g + 2g2 + g . We then form H3 by splitting the dilate of H2 in half and translating the left half by g1 . This makes H3 the same set as in the first example, so that its two halves match up with the two ends of H1 . Continuing the construction by giving W 0 the same definition on the rest of the odd Hl as on H1 , and the same definition on the rest of the even Hl as on H2 , we get a two interval scaling set: ¶ ¶ 1 1 g 1 > 2 ^ > 2 = (9.10) H= g+1 g 1 g+1 g 1 Unlike the previous example, this construction does not use translation by o g , and thus does not require that g be an integer. However, in order to avoid overlap in our intervals (which would violate the requirement that W 0 be oneto-one), we need g 2. The resulting wavelet set, for any real dilation g 2, is given by ¶ ¶ ¶ 1 1 g2 g 1 g > ^ 2 > ^ > 2 = (9.11) Z = g+1 g+1 g 1 g+1 g+1 g 1 The family of wavelet sets (9.11) appears as an example in [13].

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185

Other wavelet sets can be built by this same technique if we ask for dierent adjacencies. For example, for dilation by 2, if we ask that the second dilate of the halves of H2 be adjacent to H1 instead of the first dilate being adjacent, 1 2 2 1 16 we get the Journ´e wavelet set Z = [ 16 7 > 2)^[ 2 > 7 )^[ 7 > 2 )^[2> 7 ). The same construction can be used to produce analogs of Journ´e for any dilation g such that g2 5 Z. If we alter the Journ´e construction by translating the whole dilate of H1 to the right to form H2 , as we did before to construct scaling set (9.10), but still requiring the halves of the second dilate of H2 match up with H1 , we can build a wavelet set for any real dilation g A 32 . For 16 g = 2, the resulting wavelet set is Z = [ 47 > 27 ) ^ [ 27 > 37 ) ^ [ 12 7 > 7 ). We can generalize both example (9.10) and this one-sided Journ´e-like construction by asking that the m wk dilate of the halves of H2 be adjacent to H! , where m 1. (Here, m = 1 corresponds to scaling set (9.10) and m = 2 corresponds to the one-sided Journ´e.) There are many solutions for f and n in the analogous equations to (9.7); one consistent with our previous examples is n = m>

f=

2m gm = 2(gm+1 1)

(9.12)

The resulting scaling set H has m + 1 pieces, with the largest (formed from mgm m H1 ) given by [ gm+1 1 > gm+1 1 ), the second given by the dilate of the first translated by gm , and the rest given by successive dilates of the second. This 1 generalized example can be built for any dilation g max(m m > m+1 m ). (The first restriction is required to ensure that the origin is in H1 and the second to avoid overlap.) Thus by taking m large enough, we can get a wavelet set consisting of a finite number of intervals for any real dilation g A 1. Other one-dimensional examples can be found in [3]. We will now use these one-dimensional constructions as a starting point to build the two-dimensional examples. As before, we will make our first example by splitting H1 at the center and forcing the images of this center in H3 to match up with the edges of H1 . We can make the positions match up by using the same values for f> o> n as in the first one-dimensional construction, here used in both horizontal and vertical directions. In two dimensions, however, we have the additional complication of needing the size and shape to match as well. We will accomplish this by making H1 a truncated diamond with axis along the line | = {. We think of the truncated diamond as the union of two trapezoids, with the longer of two parallel sides coinciding at the center of H1 , and the shorter at the edges with length g12 times the length of the center line. (See Figure 9.2(a) below.) The fact that slopes are preserved by integer dilations is essential to making this construction work. For the sake of clarity, we describe the details of this construction first for the specific case g = 2. Recall that H1 must satisfy Equation (9.4), and so must consist of translates by Z2 @2 of the points in the dilated square [ 14 > 14 )2 . Because H1 had a center value of f = 0 in the one-dimensional example for dilation by 2,

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Kathy D. Merrill

we do not need to move the dilated square before forming the correct shape in this case. To make the truncated diamond, we remove triangles from the corners of the dilated square in the second and fourth quadrants and translate these triangles to positions adjacent to the corners of the square in the first and third quadrants. We choose the size of the triangle to make the outer edges of the truncated diamond have length 14 times the length of the centerline. Specifically, we translate the triangle in the fourth quadrant with 2 2 > 10 ), and (0> 14 ) up by (0> 12 ) and the one with vertices vertices ( 14 > 14 ), ( 10 1 1 2 2 at ( 4 > 4 ), ( 10 > 10 ), and ( 14 > 0) to the left by ( 12 > 0). Symmetrically defined translations of triangles in the second quadrant complete the construction of the truncated diamond H1 . (See Figure 9.2(a).) Then, to form H2 , we first dilate H1 to 12 H1 and then translate the dilated upper right trapezoid by ( 12 > 12 ) and the dilated lower left trapezoid by ( 12 > 12 ). (See Figure 9.2(b).) It is now easy to see that by taking H3 to be 12 H2 , we can make H3 be just an extension of the truncated diamond H1 . For the construction of Hl , l A 3, we define W 0 following the first one-dimensional example. That is, for { 5 Hl with l odd, we let W 0 (({> |)) = ( {2 + 12 > |2 + 12 ) for | { and W 0 (({> |)) = ( {2 12 > |2 12 ) for | ? {. For even l, W 0 takes ({> |) 5 Hl to ( {2 > |2 ), as is required by the rules of the construction technique. The resulting scaling set H and its wavelet set Z = 2H \ H are shown below in Figure 9.3. Variations on this construction as described for the one-dimensional case will work here as well. If we leave all the odd Hl as in the previous construction but put both halves of the even Hl0 v in the first quadrant, we get a scaling set analogous to the one-dimensional (9.10), which results in the wavelet set shown in Figure 9.4a. If instead we form H2 by translating the upper right trapezoid of H1 to the lower left, and the lower left trapezoid to the upper

(=5> =5) (=2> =3)

H1

H1 H1 (=2> 3=2)

H2

(a) Fig. 9.2 Building H1 from [3 14 > 14 )2 and H2 from H1 .

(b)

H2

9 Simple Wavelet Sets for Scalar Dilations in R2

187

(2@3> 2@3) 0=5

(4@3> 4@3) 1

(=6> =4)

(1=2> =8) (=6> =4)

(1@3> 1@3)

30=5

31

0=5

1

(=2> 3=2)

(=4> 3=4)

30=5

31

Fig. 9.3 A generalized scaling set and the corresponding wavelet set for dilation by 2.

right, we get the “connected” wavelet set shown in Figure 9.4b. Notice that these alternative generalized scaling sets could also be formed directly from the first by translating pieces of H by lattice elements in such a way that the containment H 2H is maintained. We give two final variations that cannot easily be seen as resulting from translates of the original H. If we form H2 by translating the two dilated trapezoids of H1 out by ±(1> 1), then dilate them twice to get adjacency to H1 , the resulting two-dimensional wavelet set is analogous to Journ´e. (See Figure 9.5a.) This provides an example of a wavelet set in R2 whose multiplicity function (dimension function) takes on values greater than 1. Finally, if we build H1 from the square [ 14 > 14 )2 by translating rounded triangles, and then complete H as for Figure 9.4a, we get the rounded wavelet set pictured

(4@3> 4@3)

1 (2@3> 2@3)

1 (1=2> =8)

(1=2> =8)

(=6> =4) 31

1

31

1 (=4> 3=4)

(=4> 3=4) (32@3> 32@3) 31

31

(a)

(b)

Fig. 9.4 A two-piece wavelet set and a connected wavelet set for dilation by 2.

188

Kathy D. Merrill (16@7> 16@7) 2

1

32 31 (3=5> 3=5)

1

(4@9> 5@9) 1

2

31

1

(4@9> 34@9) 31

(32> 32)

32

(a)

31

(b)

Fig. 9.5 A wavelet set analogous to Journ´ e and a rounded wavelet set for dilation by 2.

in Figure 9.5b. Unlike the previous two-dimensional examples in this section, Example 9.5b is not the finite union of convex sets, but it does have the property that its boundary is smooth except at a finite number of points. Many other variations are possible, including the obvious rotations of the given examples by multiples of 2 . Now we describe briefly how to construct analogous two-dimensional examples for any scalar dilation g. Just as in the one-dimensional example, 1 1 2 1 1 > f + 2g ) with 2g ? f ? 2g we note that any square of the form [f 2g will contain a neighborhood of the origin and satisfy Equation (9.4). So, to generalize Example 9.3 or Example 9.4, we first locate the center of square 2g as in the one-dimensional example to be (f> f) = ( 2g2g 2 2 > 2g2 2 ). To form H1 , we then modify the square as we did for dilation by 2, to form a truncated diamond whose cuto edges are g12 as long as the parallel centerline. Again following the one-dimensional example, we let H2 be the set that results from translating the dilate of the upper right trapezoid by ( g1 > g1 ) and the lower left g1 1 by ( g1 g > g ). The set H3 = g H2 then exactly matches up with the outer edge of H1 . We finish the construction as before by letting W 0 have the same definition on all odd and even Hl as on H1 and H2 , respectively. The wavelet set for dilation by the integer g 2 can then be formed by Z = gH \ H. The other forms discussed for dilation by 2 above can also be built for dilation by an arbitrary integer g 2. For example, Figure 9.6 shows the one-sided wavelet set for dilation by 3. Just as in the one-dimensional constructions, the one-sided forms 9.4a and 9.5b can also be built for any scalar dilation g 2; the Journ´e-like 9.5a can be built for any scalar dilation g such that g2 5 Z. Wavelet sets for any other scalar dilation g 1 can be obtained using the one-sided generalized Journ´e

9 Simple Wavelet Sets for Scalar Dilations in R2

189

(63@80>87@80)

(9@8>9@8)

1 (3@4>3@4)

(29@80>21@80) 31

1

(21@80>351@80) (33@4>33@4) 31

Fig. 9.6 A two-piece wavelet set for dilation by 3.

with (m + 1)-piece scaling sets. We use the values of n and f given in Equation (9.12), and build the original truncated diamond H1 so that the lengths of the 1 as long as the center line. As in the one-dimensional truncated edges are gm+1 1 case, this will yield a wavelet set for any g max(m m > m+1 m ), so that by taking m large enough, we can build a wavelet set that is the finite union of polygons for any real scalar dilation g A 1. Theorem 9.3 guarantees that all of the examples constructed in this section are wavelet sets. It is interesting to note that therefore they all necessarily satisfy Theorem 9.1. Figure 9.7 shows this property for the dilation-by-3 wavelet set of Figure 9.6.

Fig. 9.7 Tiling R2 by translation and dilation with dilation-3 wavelet set.

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Kathy D. Merrill

9.4 Generalizations The two-dimensional examples built in the previous section depend on picking the maps W and W 0 so that one of the sets Hl , l 2 matches up nicely with the outer edges of H1 . This match-up depends on the fact that scalar dilations preserve slopes. Thus, the techniques used in Section 9.3 cannot easily be generalized to other expansive matrix dilations. The one type of nonscalar dilation that can be seen by previous work to have a simple wavelet set is any expansive integer matrix with determinant of absolute value 2. This class of matrices had already been marked as a special case since they are the only dilations in dimension 2 that can give MRA wavelets. Such matrices can be grouped into six classes, and then representatives of each class can be shown to have simple wavelet sets as follows. Two matrices D and E are integrally similar if there is an integer matrix F of determinant ±1 such that D = FEF 1 . It is easy to see that if D has a wavelet set that is the union of a finite number of polygons, then any integrally similar matrix E does as well. All determinant-±2 are µ ¶ matrices µ ¶ 0 ±2 1 1 integrally similar to one of the following 6 matrices: , ± , 1 0 1 1 µ ¶ 02 ± (see [11], [24], [25]). Simple wavelet sets can be found for the first 1 1 pair of matrices (Figure 9.8(a)) using tensor products of one-dimensional wavelets and scaling functions (see, e.g., [35]). Calogero [12] and Gu and Han [20] produced a simple wavelet set for the quinconx matrix µ independently ¶ 1 1 (Figure 9.8(b)) that is symmetric with respect to the origin, and 1 1 µ ¶ 1 1 thus a wavelet set for as well. Finally, Bownik and Speegle [11], 1 1 µ ¶ 02 found a simple wavelet set for ± (Figure 9.8(c)), thus showing that 1 1 simple wavelet sets do exist for all matrices with determinant of absolute value 2. We know that the construction technique described in Section 9.2 can be used to build these determinant-±2 wavelet sets, since that technique was shown in [3] to produce all wavelet sets. However, our construction of the determinant-±2 wavelet sets depends on a lucky guess to find the maps W and W 0 , rather than a general procedure for choosing these maps, such as the one described in Section 9.3. Thus, the determinant-±2 wavelet sets are currently hard to generalize using our procedure. Since many dierent pairs of maps W and W 0 can be used to build the same wavelet set, it is possible that a more general approach to the determinant-±2 sets will be found. For now, it remains an open question whether simple wavelet sets (either the finite union of polygons or even the finite union of convex sets) exist for arbitrary expansive integer-valued matrix dilations in R2 . One negative result along

9 Simple Wavelet Sets for Scalar Dilations in R2 1

1

31

30=5

191

0=5

1

31

1

1 31

31

1

31

31 (a)

(b)

(c)

Fig. 9.8 Wavelet sets for determinant-2 matrices.

these lines is known for dilation by a nonexpansive, non-integer-valued ¶2 ×2 µ 2 0 does matrix. Darrin Speegle showed in 2003 [34] that the matrix s 21 have wavelet sets, but that any wavelet set must have empty interior. For dimensions higher than 2, the question is almost entirely open. For example, it is unknown whether a wavelet set for any scalar dilation in R3 can be a finite union of polyhedra or even a finite union of convex sets.

References 1. P. Auscher, Solution of two problems on wavelets, J. Geom. Anal. 5 (1995), 181—236. 2. L. Baggett, A. Carey, W. Moran, P. Ohring, General existence theorems for orthonormal wavelets, an abstract approach, Publ. Res. Inst. Math. Sci. Kyoto Univ. 31 (1995), 95—111. 3. L. Baggett, H. Medina, K. Merrill, Generalized multi-resolution analyses and a construction procedure for all wavelet sets in Rq , J. Fourier Anal. Appl. 5 (1999), 563—573. 4. L. Baggett, P. Jorgensen, K. Merrill, J. Packer, A non-MRA F u frame wavelet with rapid decay, Acta Appl. Math. 89 (2005), 251—270. 5. G. Battle, A block spin construction of ondelettes, Part 1: Lemari´ e functions, Comm. Math. Phys. 110 (1987), 601—615. 6. J. J. Benedetto, M. T. Leon, The construction of multiple dyadic minimally supported frequency wavelets on Rg , Contemp. Math. 247 (1999), 43—74. 7. J. J. Benedetto, M. T. Leon, The construction of single wavelets in d-dimensions, J. Geom. Anal. 11 (2001), 1—15. 8. J. J. Benedetto, S. Li, The theory of multiresolution analysis frames and applications to filter banks, Appl. Comput. Harmon. Anal. 5 (1998), 389—427. 9. J. J. Benedetto, S. Sumetkijakan Tight frames and geometric properties of wavelet sets, Adv. Comput. Math. 24 (2006), 35—56. 10. M. Bownik, Z. Rzeszotnik, D. Speegle, A characterization of dimension functions of wavelets, Appl. Comput. Harmon. Anal. 10 (2001), 71—92. 11. M. Bownik, D. Speegle, Meyer type wavelet bases in R2 , J. Approx. Theory 116 (2002), 49—75.

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12. A. Calogero, A characterization of wavelets on general lattices, J. Geom. Anal. 10 (2000) 597—622, 13. X. Dai, D. R. Larson, Wandering vectors for unitary systems and orthogonal wavelets, Mem. Amer. Math. Soc. 134, No. 640 (1998), 14. X. Dai, D. R. Larson, and D. M. Speegle, Wavelet sets in Rq , J. Fourier Anal. Appl. 3 (1997), 451—456. 15. X. Dai, D. R. Larson, and D. M. Speegle, Wavelet sets in Rq II, Contemp. Math. 216 (1998), 15—40. 16. I. Daubechies, Ten Lectures on Wavelets, American Mathematical Society, Providence, RI, 1992. 17. I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), 909—996. 18. X. Fang, X. Wang, Construction of minimally supported frequency wavelets, J. Fourier Anal. Appl. 2 (1996), 315—327. 19. G. Gripenberg, A necessary and sucient condition for the existence of a father wavelet, Studia Math., 114(1995), 207—226. 20. Q. Gu, D. Han, On multiresolution analysis (MRA) wavelets in Rq , J. Fourier Anal. Appl. 6 (2000), 437—447. 21. A. Haar, Zur theorie der orthogonalen funktionene systems, Math. Ann. 69 (1910), 331—271. 22. E. Hern´ andez, X. Wang, G. Weiss, Smoothing minimally supported frequency wavelets I, J. Fourier Anal. Appl. 2 1996), 329—340. 23. E. Hern´ andez, X. Wang, G. Weiss, Smoothing minimally supported frequency wavelets II, J. Fourier Anal. Appl. 3 (1997), 23—41. 24. I. Kirat, K. S. Lau, Classification of integral expanding matrices and self-ane tiles, Discrete Comput. Geom. 28 (2002), 49—73. 25. J. Lagarias, Y. Wang, Haar-type orthonormal wavelet bases in R2 , J. Fourier Anal. Appl. 2 (1995), 1—14. 26. P. G. Lemari´ e-Rieusset, Ondelettes a ` localisation exponentielle, J. Math. Pures Appl. (9) 67 (1988), 227—236. 27. L.-H. Lim, J. Packer, K. Taylor, Direct integral decomposition of the wavelet representation, Proc. Amer. Math. Soc. 129 (2001), 3057—3067. 28. J. E. Littlewood, R. E. A. C. Paley, Theorems on Fourier series and power series, J. London Math. Soc. 6 (1931), 230—233. 29. Y. Meyer, Principe d’incertitude, bases hilbertiennes et alg´ ebres d’op´ erateurs, S´ eminaire Bourbaki 662 (1986). 30. G. Olafsson, D. Speegle, Wavelets, wavelet sets and linear actions on Rq , in Wavelets, Frames and Operator Theory (C. Heil, P. Jorgensen, D. Larson, Editors), Contemp. Math. vol. 345, American Mathematical Society, Providence, RI, 2004, 253—279. 31. M. Papadakis, Generalized frame multiresolution analysis of abstract Hilbert space and its applications, in Wavelet Applications in Signal and Image Processing VIII (A. Aldroubi, A. Laine, M. Unser, Editors), SPIE Proceedings, vol. 4119, SPIE, Bellingham, WA, 2000, 165—175. 32. C. E. Shannon, Communications in the presence of noise, Proc. Inst. Radio Eng. 37 (1949), 10—21. 33. P. M. Soardi, D. Weiland, Single wavelets in n-dimensions, J. Fourier Anal. Appl. 4 (1998), 299—315. 34. D. Speegle, On the existence of wavelets for non-expansive dilation matrices, Collect. Math. 54(2003), 163—179. 35. P. Wojtaszczyk, A Mathematical Introduction to Wavelets, Cambridge Univ. Press, Cambridge, UK, 1997. 36. V. Zakharov, Nonseparable multidimensional Littlewood—Paley like wavelet bases, Centre de Physique Th´ eorique, CNRS Luminy 9 (1996).

Chapter 10

Interpolation Maps and Congruence Domains for Wavelet Sets Xiaofei Zhang and David R. Larson Dedicated to Larry Baggett for his great friendship, his love of mathematics, and his continued support of young mathematicians

Abstract It is proved that if an interpolation map between two wavelet sets preserves the union of the sets, then the pair must be an interpolation pair. We also construct an example of a pair of wavelet sets for which the congruence domains of the associated interpolation map and its inverse are equal, and yet the pair is not an interpolation pair. The first result solves armatively a problem that the second author had posed several years ago, and the second result solves an intriguing problem of D. Han. The key to this counterexample is a special technical lemma on constructing wavelet sets. Several other applications of this result are also given. In addition, some problems are posed. We also take the opportunity to give some general exposition on wavelet sets and operator-theoretic interpolation of wavelets. Key words: Wavelet set, interpolation pair, interpolation family, congruence domain; 2000 Mathematics Subject Classification 42C15, 42C40, 47A13

10.1 Introduction An orthonormal wavelet is a single function # in L2 (Rq ) whose translates by all members of a full-rank lattice followed by dilates by all integral powers of a real expansive matrix on Rq generates an orthonormal basis for L2 (Rq ). By the term wavelet set, we mean a measurable subset H Rq with the property that the inverse Fourier transform of its normalized characteristic function, F 1 ( 1 q2 "H ), is an orthonormal wavelet. In [DL] an operator-theoretic tech(2)

nique for working with certain problems concerning wavelets was introduced that was called operator-theoretic interpolation. If # and are orthonormal Xiaofei Zhang, David R. Larson Department of Mathematics, Texas A&M University, College Station, TX 77843 e-mail: [email protected], [email protected]

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wavelets in the same space, and #q>o and q>o are the corresponding wavelet bases, then the unitary operator determined by the mapping #q>o to q>o was called the interpolation unitary between # and . These interpolation operators associated with ordered pairs of wavelets play an essential role in the theory. They are associated with the von Neumann subalgebras of the so-called local commutant space, whose unitary groups provide natural parameterizations of certain families of wavelets. In the special case where the interpolation operator is involutive (i.e., has square I), the pair of wavelets is called an interpolation pair of wavelets. One surprising feature of the theory is that interpolation pairs occur not infrequently. In general, interpolation unitaries can be hard to work with. However, in the special case where # and are s-elementary wavelets (also called MSFwavelets with phase 0, or wavelet-set wavelets), the interpolation unitary takes the form of a composition operator with measure-preserving symbol. Every pair of wavelet sets gives rise to a measure-preserving transformation on the underlying measure space in a natural way, called the interpolation map determined by the pair. The interpolation theory for such wavelets and their associated wavelet sets is, in many concrete cases, computable by hands-on experimental paper-and-pencil computations. This permits experimentation in the form of testing of hypotheses in potential theorems for more general types of wavelets. The simplest case is where a pair of wavelet sets has the property that the measure-preserving transformation is an involution (i.e., = lg). In this case, the composition unitary has square L, so the pair of wavelets is indeed an interpolation pair. The pair of wavelets sets is, by analogy, called an interpolation pair of wavelet sets. More generally, an interpolation family of wavelets (and analogously, of wavelet sets) is a finite (or even infinite) family of wavelets for which the associated family of interpolation unitaries (interpolation maps) forms a group. It is appropriate to give a bit of background and history that serves to indicate why interpolation pairs of wavelet sets are relevant to the theory of wavelets. More exposition on this, including specific details and statements of theorems involved, can be found in the semiexpository articles [La2], [La3], [La4], and [La5]. In [DL], for any interpolation pair of wavelet sets (E, F), the authors constructed a 2 × 2 complex matrix-valued function (called the Coecient Criterion; see [DL, Proposition 5.4], and also [La2, section 5.22]) that specifies precisely when a function i on R with frequency support contained in H ^ I is an orthonormal wavelet. If vxss{iˆ} is contained in H ^ I , then this criterion shows that i is an orthonormal wavelet i this matrixvalued function is a unitary matrix (a.e.), and it is a Riesz wavelet i it is an invertible matrix (a.e.). Moreover, it shows that a Parseval frame wavelet (resp. Riesz frame wavelet) with frequency support contained in H ^ I is necessarily an orthonormal wavelet (resp. Riesz wavelet). It follows that the set of orthonormal wavelets with frequency support contained in the union of H ^ I , where (H> I ) is an interpolation pair of wavelet sets, is pathwise

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connected in L2 (R). The set of Fourier transforms of this set is also connected in the L4 norm on the frequency space. These results were the main motivating factor in posing the first open problem discussed in [DL], namely the question of whether the set of all orthonormal wavelets in L2 (R) is norm-pathwise connected. This was the same problem that was posed completely independently by G. Weiss and his research group in [HWW1, HWW2] for dierent reasons. Their reasons included the interesting discovery that certain wavelet sets (rather, the associated MSF wavelets) could be “smoothed” in a continuous fashion to obtain wavelets that were continuous in the frequency domain. It turned out that our operator-interpolation approach, in certain key cases, was equivalent to the “smoothing” approach of G. Weiss, and the cases involved included the derivation of Y. Meyer’s classic family of wavelets that are compactly supported and continuous in the frequency domain. Exploring common interest in the relationships between smoothing of a wavelet set on the one hand, and operator-theoretic interpolation between a pair of wavelet sets on the other hand, and the general “connectedness” problem that was motivated by both approaches independently (as described above) led to the formation of the WUTAM Consortium (short for Washington University and Texas A&M University) and the joint work [Wut] of the consortium, in which the connectedness problem was shown to have a positive answer for the case of MRA wavelets. The basic idea behind operator interpolation is elementary. If { and | are elements of a vector space Y , we say that a vector } is linearly interpolated from { and | if } is a convex combination of {> |. More generally, it is convenient to allow arbitrary linear combinations. So the set of vectors interpolated by { and | is the linear span of { and |. More generally, we can say that } is linearly interpolated from a collection F of vectors if } is a linear combination of vectors from the family. And more generally yet, if the vector space Y is a left module over some operator algebra D, we can consider linear combinations from F with coecients that are operators from D, called modular linear combinations. If } is a modular linear combination of {> |, then we say that } is derived from { and | by operator-theoretic interpolation (or operator interpolation for short). In the case of wavelets, the operator algebra D is the von Neumann algebra of all bounded linear operators acting on L2 (Rq ) that commute with the dilation and translation unitary operators for the wavelet system. When we conjugate this with the Fourier transform (which is unitary), so we are working in L2 (Rq ) as the ˆ then D ˆ is an frequency space, and if we denote this conjugated algebra by D, 2 q algebra of multiplication operators on L (R ). In particular, it is a commutative algebra. If a wavelet is a modular linear combination of wavelets # and with coecients that are operators in D, then we say is derived by operator interpolation between # and . Not all modular linear combinations of # and are orthonormal wavelets. They are all Bessel wavelets to be sure, but a certain unitarity condition needs to be satisfied to be an orthonormal

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wavelet. Let Y# be the interpolation unitary from # to . If D and E are operators in G and = D# + E, the necessary and sucient condition for to be an orthonormal wavlet is that the operator X := D + EY# needs to be unitary. (More generally, for a frame wavelet the criterion is that X must be surjective, and for a Riesz wavelet X must be invertible.) The reason that interpolation pairs of orthonormal wavelets are special is that if (#> ) is an interpolation pair, and if D and E are operators in G such that D D + E E = L, then under certain circumstances D# + E can also be an orthonormal wavelet. In particular, if 5 [0> 2] is arbitrary, then := cos # + l sin is an orthonormal wavelet. Indeed, in this case the operator X above is just cos L + l sin Y# , and since (Y# )2 = L, it follows that X X = L, so X is unitary, as required by the criterion. Letting vary continuously it follows, in particular, that # and are pathwise connected via a path of orthogonal wavelets. For wavelet-set wavelets, i.e., MSF wavelets with phase 0, more is true, and the operator-algebraic geometry involved is fairly rich. For any pair of wavelets sets H and I , with associated MSF wavelets #H and #I , the interpolation unitary Y##HI normalizes the von Neumann algebra D in the sense that Y##HI D(Y##HI ) = D. This was proved in Chapter 5 of [DL] and was a key result of that memoir. (We note that it is an open question (see [La2, Problem 4]) as to whether arbitrary interpolation operators (i.e., for nonwavelet-set wavelets) normalize D.) The reason that interpolation pairs of wavelet sets are even more special than general interpolations of wavelets is the following: First, (H> I ) is an interpolation pair of wavelet sets if and only if the pair of wavelets (#H > #I ) is an interpolation pair of wavelets. And second, since in this case the interpolation unitary normalizes D, and since (Y#H #I )2 = L, it follows that the set of operators {D + EY##HI | D> E 5 D} is closed under multiplication and is in fact a von Neumann algebra. (For a more general interpolation pair of wavelets whose interpolation operator normalizes D, the same thing is true.) Since the unitary group of a von Neumann algebra is pathwise connected in the operator norm, the interpolated family of wavelets is also connected. Much more is true. Since the elements of D are multiplication operators in a certain family (the dilation-periodic operators), if we write D = Pi and E = Pj we obtain (D + EY##HI )#H = i #H + j#I . Thus the outcome is an actual formula as well as a criterion (the Coecient Criterion mentioned above) for constructing all orthonormal wavelets whose frequency support is contained in the union H ^ I of a given interpolation pair (H> I ) of wavelet sets. By choosing i and j appropriately, so they are continuous and vanish on the boundary of the union H ^ I , and agree on H _ I , and satisfy the unitarity condition referred to above, one can obtain wavelets in this fashion that are smooth in the frequency domain. Not all interpolation pairs (H> I ) can be so smoothed. But some can. As alluded to above, Y. Meyer’s famous class of orthonormal wavelets that are continuous and compactly supported in the frequency domain can be derived in this way from a special interpolation pair of wavelet sets: namely the pair

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4 2 4 4 2 4 8 H = [ 8 3 > 3 ) ^ [ 3 > 3 ) and I = [ 3 > 3 ) ^ [ 3 > 3 ). Even for cases in which smoothing cannot work (for instance, H ^ I may have too many boundary points), the operator algebra involved can be interesting. The purpose of this chapter is to provide solutions to two related problems concerning dyadic orthonormal wavelet sets in the line. One problem asked whether a certain containment relation for an interpolation map implies that the associated pair of wavelet sets is an interpolation pair. This was posed by the second author in a VIGRE seminar course at Texas A&M several years ago. We answer this question armatively, and we also observe that the analogous result does not hold for a general interpolation family of wavelet sets. The second problem was posed by D. Han, who asked whether the equality of the congruence domains of an interpolation map and its inverse implies that the associated pair of wavelet sets is an interpolation pair. We were able to give a counterexample to this problem. Our work on this interesting problem motivated a useful lemma on constructing wavelet sets, which is apparently dierent from those methods that have appeared in the literature to date, and which is used in this counterexample as well as in the construction of several wavelet sets concerning some related questions, and also some wavelet sets in the plane. Much of the work presented in this chapter is material from the doctoral dissertation of the first author [Zh], which has not appeared elsewhere. Much work has been accomplished on the topic of wavelet sets since the mid-1990s. We have outlined some of the background and history in the opening paragraph of Section 10.5, for the interested reader. Our main results in this chapter are for the special case n = 1 with the dilation scale factor 2 and integer translates (the dyadic case). In Section 10.5, we apply one of our techniques to the construction of certain dyadic wavelet sets in the plane. Based on our results, some further directions are suggested for higher dimensions.

10.2 Two Problems A dyadic orthonormal wavelet is a function # 5 L2 (R) (Lebesgue measure) q with the property that the set { 2 2 #(2q ·o) | q> o 5 Z } forms an orthonormal basis for L2 (R). More generally, if D is any real invertible q × q matrix, then a single function # 5 O2 (Rq ) is an orthonormal wavelet for D if q

{|ghwD| 2 #(Dq · o) | q 5 Z> o 5 Z(q) } is an orthonormal basis of O2 (Rq ). If D is expansive (equivalently, all eigenvalues of D are required to have absolute value strictly greater than 1), then it was shown in [DLS1] that orthonormal wavelets for D always exist. By the support of a measurable function, we mean the set of points in its domain at which it does not vanish. By the support of an element i of

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L2 (R), we mean the support of any measurable representative of i , which is well-defined in the measure algebra of equivalence classes of sets modulo null sets. By the frequency support of a function, we mean the support of its Fourier transform. Let F denote the q-dimensional Fourier transform on L2 (Rq ) defined by Z 1 hvw i (w)gp (Fi )(v) := q (2) 2 Rq for all i 5 O2 (Rq ). Here, v w denotes the real inner product. A measurable set H Rq is a wavelet set for D if 1 F 1 ( p "H ) (H)

is an orthonormal wavelet for D. A sequence S of measurable sets {Hq } is called a measurable partition of H if H = H ( q Hq ) is a null set and Hq _Hp has measure zero if q 6= p, where denotes the symmetric dierence of sets. Measurable subsets H and I of R are called 2-translation congruent to each other, denoted by H 2 I , if there exists a measurable partition {Hq } of H, such that {Hq + 2q} is a measurable paritition of I . Similarly, H and I are called 2-dilation congruent to each other, denoted by H 2 I , if there is a measurable partition {Hq } of H, such that {2q Hq } is a measurable partition of I . A measurable set H is called a 2-translation generator of a measurable partition of R if {H + 2q}q5Z forms a measurable partition of R. Similarly, a measurable set I is called a 2-dilation generator of a measurable partition of R if {2q I }q5Z forms a measurable partition of R. Lemma 4.3 in [DL] gives the following characterization of wavelet sets, which was also obtained independently in [FW] using dierent techniques. Let H R be a measurable set. Then H is a wavelet set if and only if H is both a 2-translation generator of a measurable partition of R and a 2-dilation generator of a measurable partition of R. Again from [DL], suppose that H> I are wavelet sets, and let : H $ I be the bijective map (modulo null sets) implementing the 2-translation congruence. Then can be extended to a bijective (modulo null sets) measurable map on R by defining (0) = 0 and (v) = 2q (2q v) for each I and called the interpolation map v 5 2q H> q 5 Z. This map is denoted by H for the ordered pair of wavelet sets (H> I ). An ordered pair of wavelet sets (H> I ) is called an interpolation pair if I I I 2 := (H ) = lgR . In general, an interpolation family of wavelet sets H H I | I 5 F } is a group under is a family F of wavelet sets such that { H composition of maps for some H 5 F. I Question A. Let H> I be wavelet sets. Does H (H ^ I ) H ^ I imply that I 2 (H ) = lgR ?

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We answer Question A armatively. We then give an elementary example that shows that this result need not hold for a triple of wavelet sets; however, there is a natural modification that does make sense for q-tuples, and we pose it as an open question (Question C). Given a pair of wavelet sets, it is not obvious at all upon initial inspection whether they actually form an interpolation pair. And constructing interpolation pairs can be hard. A basic problem from [DL] that still remains open is the question: Given an arbitrary dyadic wavelet set H in R1 , is there necessarily a second distinct wavelet set I such that (H> I ) is an interpolation pair? Partly to address this problem, and partly for intrinsic interest, Han introduced and studied properties of congruence domains in [Han1] and [Han2]. Given a pair of wavelet sets (H> I ), the domain of 2-congruence of I I I , denoted by DH , is the set of all points v 5 R such that H (v) v is an H integral multiple of 2. There is a close relation between this and the interpolation map of a pair of wavelet sets. Han asked the following question. I I 2 = DIH imply that (H ) = Question B. Let H> I be wavelet sets. Does DH lgR ?

If (H> I ) is an interpolation pair, then it is easily verified that the domains of 2-congruence of (H> I ) and (I> H) are the same. So the above question just asks if the converse is true. In many cases it is true. But it is not universally true. We answer Question B negatively by constructing a counterexample. The key is a special lemma on constructing wavelet sets, which is also used to build examples for several other questions.

10.3 Solution to Question A The following theorem provides an answer to Question A. Theorem 10.1. Let (H> I ) be wavelet sets. The following two statements are equivalent: (i) (H> I ) is an interpolation pair, I (H ^ I ) H ^ I . (ii) H I 2 ) = Proof. (i) , (ii). Suppose that (H> I ) is an interpolation pair. Then (H I 1 H I H I lgR . Since (H ) = I , it follows that H = I . Observe that H (H) = I , IH (I \H) = H\I , and so I I I (H ^ I ) = H (H) ^ H (I \H) = I ^ IH (I \H) = I ^ H\I = H ^ I= H I I I (H ^ I ) H ^ I . Since H (H) = I and H is (ii) , (i). Suppose that H I bijective (modulo null sets), we must have H (I \H) H\I . We will prove this by way of contradiction, and a diagram is included at the end of the I . proof to help in navigating between H and I , with arrows representing H

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Assume that (H> I ) is not an interpolation pair. Let J0 = { v 5 H | I 2 ) (v) 6= v }. Then J0 H\I is Lebesgue measurable and has positive (H measure. Since { J0 _ (I 2q) }q5Z forms a measurable partition of J0 , it follows that J0 _ (I 2q1 ) has positive measure for some q1 5 Z. Denote I (J1 ) = J1 + 2q1 I \H. Since H\I is 2-dilation this set by J1 . Then H congruent to I \H, following the similar discussion, there exists a measurable subset J2 of J1 with positive measure, such that 2n1 (J2 + 2q1 ) H\I for some integer n1 . Then, there exists a measurable subset J3 of J2 with I n1 (2 (J3 + 2q1 )) = 2n1 (J3 + 2q1 ) + 2q2 positive measure such that H for some integer q2 . Thus, I 2 I I n1 (H ) (J3 ) = H (J3 + 2q1 ) = 2n1 · H (2 (J3 + 2q1 ))

= 2n1 · (2n1 (J3 + 2q1 ) + 2q2 ) = J3 + 2q1 + 2n1 · 2q2 H\I=

Since both J3 > J3 + 2q1 + 2n1 · 2q2 H\I and they are distinct by @ Z, which implies that n1 6 1. assumption, we must have q1 + 2n1 · q2 5 Similarly, there exists a measurable subset J4 of J3 with positive measure, I (J4 + 2q1 + 2n1 · 2q2 ) = J4 + 2q1 + 2n1 · 2q2 + 2q3 for such that H some integer q3 . Then, I 2 n1 I n1 ) (2 (J4 + 2q1 )) = H (2 (J4 + 2q1 ) + 2q2 ) (H I (J4 + 2q1 + 2n1 · 2q2 ) = 2n1 · H

= 2n1 · (J4 + 2q1 + 2n1 · 2q2 + 2q3 )

= 2n1 (J4 + 2q1 ) + 2q2 + 2n1 · 2q3 H\I= Since both 2n1 (J4 + 2q1 ) and 2n1 (J4 + 2q1 ) + 2q2 + 2n1 · 2q3 are contained in H\I and q2 + 2n1 · q3 must be an integer, we must have I maps both J4 and J4 + 2q1 + 2n1 · 2q2 q2 + 2n1 · q3 = 0. Then, since H n1 I 2 ) (v) = to J4 +2q1 , J4 and J4 +2q1 +2 ·2q2 must be the same, then (H I 2 v> ;v 5 J4 J0 , contradicting the assumption. Thus, (H ) = lgR almost everywhere. H\I : I \H : H\I :

I \H :

2n1 (J2 + 2q1 )

J0 ? J1 + 2q1 ? J3 + 2q1 + 2n1 · 2q2 ? J4 + 2q1 + 2 · 2q2 + 2q3 n1

? 2n1 (J3 + 2q1 ) + 2q2 n1

2

? (J4 + 2q1 ) + 2q2 +2n1 · 2q3 ? ···

t u

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Let F be a general interpolation family of wavelet sets, and fix H1 5 F. H2 H3 H4 H = H for some Then for arbitrary H2 > H3 5 F, we have H 1 1 1 H2 H2 H3 H4 H4 5 F. Thus H1 (H3 ) = H1 H1 (H1 ) = H1 (H1 ) = H4 . It follows that S H S H ( H5F H) H5F H for each H 5 F. However, the converse may be 1 false as the following example shows. Example 10.2. Let (H> I ) be an interpolation pair of wavelet sets and suppose J H ^ I is a wavelet set contained in the union that is distinct from H and I . Observe that H\I is both 2-translation congruent and 2-dilation congruJ = ent to I \H, thus J H^I must contain H_I to be a wavelet set. Since H J I J 2 J 2 lgR on H _ J and H = H on H\J, (H ) = lgR . Similarly, (I ) = lgR . I J I J H = IJ 6= lgR , which implies that {lgR > H > H } is not a group Then, H under composition of maps. Hence {H> I> J} is not an interpolation family. However, notice that {H\J> J\I> H _ I> J\H> I \J} forms a measurable partition of H ^ I . H\J is both 2-translation and 2-dilation congruent to J\H and I \J is both 2-translation and 2-dilation congruent to J\I , which I I (H ^ I ^ J) = H (H ^ I ) = H ^ I , and is contained in H _ J. We have H J J J J (H ^ I ^ J) = H (H ^ I ) = H (H) ^ H (I \H) H

J J (J\H) ^ H (I \J) = J ^ H\J ^ I \J = H ^ I= = J ^ H

t u

Observe that in the above example, if let K = H\J ^ (H _ I ) ^ I \J, K I J J I = H H = H H . then K H ^ I is also a wavelet set and H H I J K Then {H> I> J> K} is an interpolation family since {H = lgR > H > H > H } is actually isomorphic to the Klein four group. This motivates the following question, which seems dicult and which we pose as an open problem: Question C. Let F be a finite collection of wavelet sets. Fix H1 5 F. Suppose S H S ( H) is contained in H for each H 5 F. Is F a subset of a H H5F H5F 1 finite interpolation family of wavelet sets? That is, can F always be extended to a finite interpolation family?

10.4 A Counterexample to Question B Before giving a counterexample to Question B, we present a characterization of interpolation pairs of wavelet sets and equality of 2-congruence domains of associated interpolation maps. This is actually the motivation that led us to find a counterexample. Given a pair of wavelet sets (H> I ), there exists a measurable partition {Hq }q5Z of H such that {Hq + 2q}q5Z is a measurable partition of I . Similarly, there exists another measurable partition {H n }n5Z of H such that {2n H n }n5Z is another measurable partition of I . Denote Hq _ H n by Hq>n , then {Hq>n }q>n5Z is a measurable partition of H, and both {Hq>n + 2q}q>n5Z and {2n Hq>n }q>n5Z are measurable partitions of I . Observe that if Hq>n

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has positive measure, then q = 0 whenever n = 0, and Hq>n _ (2o Hp>o 2q)> q> n> p> o 5 Z\0 forms a partition of H\I . Theorem 10.3. Let H> I be wavelet sets, and let {Hq>n }q>n5Z be as defined above. Then: (i) (H> I ) is an interpolation pair if and only if Hq>n _ (2o Hp>o 2q) with positive measure for some nonzero integers q> n> p> o implies that q+2o ·p = 0. Furthermore, for such a set, we also have n = o. I = DIH if and only if Hq>n _(2o Hp>o 2q) with positive measure for (ii) DH some nonzero integers q> n> p> o implies that [q] + o = [p]. Here, [q] denotes the smallest integer n such that 2n · q 5 Z. I 2 ) = lg. Proof. (i) Suppose that (H> I ) is an interpolation pair. Then (H Since I 2 I ) (Hq>n _ (2o Hp>o 2q)) = H ((Hq>n + 2q) _ 2o Hp>o ) (H I o = 2o · H (2 (Hq>n + 2q) _ Hp>o ) o = 2 · ( (2o (Hq>n + 2q) _ Hp>o ) + 2p) = (Hq>n + 2q + 2o · 2p) _ 2o (Hp>o + 2p)>

it follows that if Hq>n _ (2o Hp>o 2q) has positive measure, then 2q + 2o · 2p = 0, i.e., q + 2o · p = 0. Conversely, suppose that Hq>n _ (2o Hp>o 2q)> q> n> p> o 6= 0 having positive measure implies that q + 2o · p = 0. Observe that {Hq>n _ (2o Hp>o 2q)}q>n>p>o6=0 forms a measurable partition of H\I . Then, a similar arguI 2 ) = lgR . ment shows that (H Furthermore, if Hq>n _ (2o Hp>o 2q) having positive measure implies that q + 2o · p = 0, then (Hp>o + 2p) _ 2o Hq>n = 2o · (Hq>n _ (2o Hp>o 2q) ) also has positive measure. Since Hp>o + 2p I and 2o Hq>n I only if n = o, we must have n = o. (ii) Observe that I = DH

[ ˙

(

[ ˙

[ ˙ m 2m Hq>n ) ^˙ 2 (H _ I ) m

q>n6=0 m> [q]

[ ˙

=

(

[ ˙

q>n>p6=0 m> [q]

[ ˙ m 2m (Hq>n _ 2n (Hp + 2p))) ^˙ 2 (H _ I )> m

and DIH = =

[ ˙

(

[ ˙

q>n6=0 m> [q]

[ ˙

(

[ ˙ m 2m (Hq>n + 2q)) ^˙ 2 (H _ I )

[ ˙

m

[ ˙ m 2m ((Hq>n + 2q) _ 2o Hp>o )) ^˙ 2 (H _ I )

q>n>p>o6=0 m> [q]

m

10 Interpolation Maps and Congruence Domains for Wavelet Sets

[ ˙

=

(

[ ˙

m

q>n>p>o6=0 m> [q]

[ ˙

=

(

[ ˙

q>n>p>o6=0 m> [p]

=

[ ˙

(

[ ˙

q>n>p6=0 m> [p]

203

[ ˙ m 2m+o (Hp>o _ 2o (Hq>n + 2q))) ^˙ 2 (H _ I )

[ ˙ m 2m+n (Hq>n _ 2n (Hp>o + 2p))) ^˙ 2 (H _ I ) m

[ ˙ m 2m+n (Hq>n _ 2n (Hp + 2p))) ^˙ 2 (H _ I )> m

where ^˙ denotes disjoint union. The next to last equality comes from changing I = DIH implies that [p] + n = [q], if Hq>n _ 2n (Hp + 2p) indices. Thus, DH has positive measure, or equivalently, if Hp>o _ (2n Hq>n 2p) has positive measure for some o 6= 0, since 2n (Hq>n _ 2n (Hp + 2p)) 2p = (2n Hq>n S 2p) _ Hp = (2n Hq>n 2p) _ ( ˙ o Hp>o ) . The converse direction can be shown by reversing the above discussion. t u The basic idea we use is to construct two certain measurable subsets of R, which are both 2-translation and 2-dilation congruent to each other, yet the interpolation map restricted to the union of 2-dilates of which does not have the property that its square equals the identity map, and then to construct the remaining pieces of wavelet sets for these two sets so that the congruency domains match up. This type of approach was used by D. Speegle in [S1] and Q. Gu in [Gu1], and a necessary and sucient condition has also been given for a measurable set being contained in some wavelet set in [IP]. However, the methods and the constructions we use in this chapter are completely dierent from those in [S1], [Gu1], and [IP]. Theorem 10.4. The answer to Question B is no. Before proving Theorem 10.4, we require a technical lemma. This will also be used in constructing several other examples in the next section. Lemma 10.5. Let H> I Rq be Lebesgue measurable sets with finite positive measure. If there exist q1 > q2 > n1 > n2 5 Z(q) , such that modulo null sets, 2n1 I H + 2q1

and

H + 2q2 2n2 I>

and (H + 2q2 ) _ 2n1 I is a null set, then there exists a Lebesgue measurable set J (H + 2q2 ) ^ 2n1 I such that J is both 2-translation congruent to H and 2-dilation congruent to I . In fact J can be taken as: J=

4 [

l=0

V l (2n1 I \2n1 n2 (H + 2q2 ))

^ (H + 2q2 )\(

4 [

l=1

2n2 n1 · V l (2n1 I \2n1 n2 (H + 2q2 )))>

where V({) = 2n1 n2 ({ + 2(q2 q1 ))> ;{ 5 Rq .

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Xiaofei Zhang and David R. Larson

Proof. By hypothesis, 2n1 n2 (H + 2q2 ) 2n1 I H + 2q1 = Since (2n1 n2 (H + 2q2 )) = 2n1 n2 (H) 6 (H + 2q1 ) = (H), n1 n2 6 0. If n1 = n2 , then H + 2q2 2n1 I , contradicts the hypothesis that (H + 2q2 ) _ 2n1 I is a null set. Thus, n1 n2 ? 0. Construct a sequence of measurable sets {Jl }l5N as follows. Let J0 = 2n1 I \ 2n1 n2 (H + 2q2 ) = 2n1 I \ 2n1 n2 (H + 2q1 + 2(q2 q1 )) (H + 2q1 ) \ V(H + 2q1 )= Let Jl = V l (J0 ) V l (H + 2q1 )\V l+1 (H + 2q1 ), for each l 5 N. Notice that the measure of Jl is bounded by 2l·(n1 n2 ) · (H), which will approach H + 2q1 , and it 0 as l approaches infinity. Furthermore, S V(H + 2q1n) 1 follows that the Jl ’s are disjoint and 4 l=0 Jl 2 I . Also by definition, n2 n1 Jl+1 = Jl + 2(q2 q1 ), ;l > 0. Let ^4 l=1 Jl V(H + 2q1 ) and 2 J=(

4 [

Jl ) ^˙ (H + 2q2 )\(

l=0

4 [

2n2 n1 Jl )=

l=1

S4 n2 n1 S4 Jl )) = V(H + 2q1 )\( l=1 Jl ), Then, since 2n1 n2 ((H + 2q2 )\( l=1 2 S 4 and {Jl }4 l=0 ^ {V(H + 2q1 )\( l=1 Jl )} constitutes a measurable partition n1 of 2 I , it is clear that J is 2-dilation congruent to I . On the other hand, since 4 4 [ [ 2n2 n1 Jl = (Jl + 2(q2 q1 ))> l=1

l=0

J is 2-translation congruent to H.

t u

34 33 34 Proof of Theorem 10.4. Let H1 = [ 33 16 > 16 )> H2 = [ 8 > 8 ) + 12> H3 = 33 34 33 34 34 [ 4 > 4 ) + 8 and H4 = [ 2 > 2 ) + 96. Let I1 = [ 33 4 > 4 )> I2 = 33 34 33 34 33 34 [ 16 > 16 )+6> I3 = [ 2 > 2 )+16 and I4 = [ 8 > 8 )+24. Notice that

H1 > H2 > H3 > H4 are both 2-translation congruent and 2-dilation congruent to disjoint pieces of [2> ) ^ [> 2), and so are I1 > I2 > I3 > I4 . Furthermore, H1 ^ H2 ^ H3 ^ H4 2 I1 ^ I2 ^ I3 ^ I4 2 [

> )> 16

H1 ^ H2 ^ H3 ^ H4 2 I1 ^ I2 ^ I3 ^ I4 2 7 7 > + ) ^ [ + > + )= [ + 128 16 4 128 4 64 Let J be the measurable set determined by Lemma 10.5 and the following two containment relations: 7 7 1 ([> + ) ^ [ + > + ) ^ [ + > 2)) [0> )> 32 128 16 4 128 4 64 16

10 Interpolation Maps and Congruence Domains for Wavelet Sets

205

7 7 ) + 8 8 · ([> + ) ^ [ + > + ) ^ [ + > 2))= 16 128 16 4 128 4 64 Then, J is both 2-translation congruent to [0> 16 ) and 2-dilation congruent 7 ) ^ [ + 16 > 4 + 128 ) ^ [ 74 + 64 > 2). Hence, to [> + 128 [0>

H := [2> ) ^ J ^ H1 ^ H2 ^ H3 ^ H4 > I := [2> ) ^ J ^ I1 ^ I2 ^ I3 ^ I4 I are both wavelet sets. The associated interpolation map H is defined by: ; v + 6 if v 5 H1 A A A A ? v + 12 if v 5 H2 I (v) = v 8 if v 5 H3 H A A v 80 if v 5 H4 A A = v if v 5 [2> ) ^ J=

Straightforward computation shows that I == (4> 0) ^ DH

and DIH == (4> 0) ^

[

2n J ^

[

2n J ^

n5Z

[

1 1 1 (H1 ^ H2 ^ H3 ^ H4 ) 2 4 8

[

1 1 1 (I2 ^ I4 ^ I1 ^ I3 )= 2 4 8

n> 0

n5Z

n> 0

Observe that I2 = 12 H2 > 12 I4 = 18 H4 > 14 I1 I = DIH . However, since implies that DH

= H1 , and

1 8 I3

=

1 4 H3 ,

which

1 I 1 I 2 I I 1 ) (H1 ) = H (H1 +6) = H ( H2 ) = H (H2 ) = (H2 +12) = H1 +12> (H 2 2 2 (H> I ) is not an interpolation pair.

t u

It turns out that if one of H> I is the Shannon set, then the equality of integral domains does imply that (H> I ) is an interpolation pair.

Proposition 10.6. Let H> I be wavelet sets. If H = [2> )^[> 2), then I = DIH implies that (H> I ) is an interpolation pair. DH

Proof. Let Hq = H _ (I 2q), q 5 Z. Then {Hq }q5Z is a measurable partition of H and {Hq + 2q} is a measurable partition of I . Let Hq = Hq _ [2> )> Hq+ = Hq _ [> 2). Then, we have the following diagram. Hq Hq+ H: £ ¢£ ¢£ ¢£ ¢ 2 0 2 + + 4 H0 H1 2H1+2 H0+ H2+4 H1++2 · · · I : · · · ¢£H12H ¢£ 2 ¢£ ¢£ ¢£ ¢£ ¢£ ¢£ ¢£ 4 3 2 0 2 3 4

206

Xiaofei Zhang and David R. Larson

Observe that I DH =

[ ˙

2m (Hq ^ Hq+ ) ^˙

[ ˙

2m H0 =

m5Z

q6=0>m> [q]

Then I DH _ [0> ) =

=

[ ˙

2m Hq+ ^˙

[ ˙

2m H0+

m?0

q6=0>[q]6 m?0

[ ˙

1 + 1 + 1 + 1 + 1 + 1 + 2m H0+ ^˙ H±2 ^˙ H±4 ^˙ H±4 ^˙ H±6 ^˙ H±8 ^˙ H±8 ^˙ · · · > 2 2 4 2 2 4 m?0

I here H±q := Hq ^ Hq . DH _ [> 2) = [> 2). Similarly,

DIH =

[ ˙

2m ((Hq ^ Hq+ ) + 2q) ^˙

[ ˙

2m H0 =

m5Z

q6=0>m> [q]

Then, following from the diagram, DIH _ [0> ) = (H1 + 2) ^˙ DIH _ [> 2) = H0+ ^˙ (

[ ˙

[ ˙

2m H0+ >

m?0

2m (H1 + 2) _ [> 2))

m> 1

1 1 ^˙ (H2 + 4) ^˙ (H4 + 8) ^˙ · · · = 2 4

I _ [0> ) and DIH _ [0> ), we have Comparing DH

1 + 1 1 1 1 1 H ^˙ H + ^˙ H + ^˙ H + ^˙ H + ^˙ H + ^˙ · · · = (10.1) 2 ±2 2 ±4 4 ±4 2 ±6 2 ±8 4 ±8 S S Then, it is easy to see that ( m> 1 2m (H1 + 2)) _ [> 2) = q52Z\0 Hq+ . I Comparing DH _ [> 2) and DIH _ [> 2), we have H1 + 2 =

[ ˙

Hq+

q52Z+1

1 1 (H + 4) ^˙ (H4 + 8) ^˙ · · · = 2 2 4

(10.2)

By symmetry, we also have + 2 = H1

[ ˙

Hq

q52Z+1

1 1 1 1 1 1 H ^˙ H ^˙ H ^˙ H ^˙ H ^˙ H ^˙ · · · > (10.3) 2 ±2 2 ±4 4 ±4 2 ±6 2 ±8 4 ±8 1 + 1 + (H 4) ^˙ (H4 8) ^˙ · · · = 2 2 4

(10.4)

10 Interpolation Maps and Congruence Domains for Wavelet Sets

207

+ Equations (10.1) and (10.4) imply that H1 +2 = 12 H2 , and Hq has measure + zero for each odd integer q except 1, and Hq has measure zero for each even + 2 = integer q except 2> 0. Equations (10.2) and (10.3) imply that H1 1 + H , and H has measure zero for each odd integer q except 1, and Hq q 2 2 has measure zero for each even integer q except 0> 2. Thus, I must have the following form + + 4) ^ H0 ^ (H1 2) ^ (H1 + 2) ^ H0+ ^ (H2 + 4)= I = (H2

Using the fact that H1 + 2 = I 2 ) = lgR . verify that (H

1 + 2 H2

+ and H1 2 =

1 2 H2 ,

it is easy to t u

10.5 Some Examples of Wavelet Sets Wavelet sets are useful as examples and counterexamples. Many examples of them in the real line were given in [DL] exactly for that purpose, for experimentation and testing hypotheses, and many open questions still remain in that setting. The existence of wavelet sets in the plane, and more generally in Rq , was first proved by Dai, Larson, and Speegle [DLS1] in the summer of 1994 during a course at Texas A&M taught by the second author using the manuscript of [DL] as a text. The proof in [DLS1] was abstract and covered dual congruence results for more general types of dual-dynamical systems. But given the definitions, the proof was constructive, implicitly giving an algorithm (a casting-outward technique) for constructing examples of wavelet sets for arbitrary expansive matrices and full rank translation lattices on Rq . However, this method produced only wavelet sets that were unbounded and had 0 as a limit point and were not very well described by diagrams or pictures. In other words, they were wavelet sets but not “nice” wavelet sets. In 1995, concrete examples of dyadic wavelet sets in the plane, which were bounded and bounded away from 0, and could be nicely diagrammed, were constructed by Soardi and Weiland [SW]. At about the same time, Dai and Larson constructed two R2 dyadic wavelet sets (denoted the “four-corners set” and the “wedding cake set”) for inclusion in a final version of [DL] in response to a referee’s suggestion. In 1996—1997, a number of authors constructed many other concrete examples of wavelet sets in the plane, and in higher dimensions. Included are the wavelet sets computed and diagrammed in the articles [BMM, BL1, BL2, DLS2, Za, C, GH, Gu2, Han2, S2]. The last three are the Ph.D. theses of Gu, Han, and Speegle, respectively. Importantly, Baggett—Medina—Merrill [BMM], and Benedetto—Leon [BL2], independently found two interesting completely dierent constructive characterizations of all wavelet sets for expansive matrices in Rq . Open questions remain, especially questions concerning the existence of wavelet sets with special properties, and algorithms for constructing special classes of such sets.

208

Xiaofei Zhang and David R. Larson

In this section, we will apply the techniques introduced in Lemma 10.5 to construct an unbounded symmetric wavelet set and counterexamples to two related questions on wavelet sets, in addition to new constructions of some known wavelet sets. q

q+1

2 1 q+2 2)> q A 0, then Example 10.7. Let Jq = [ 2 21 q > 2q+1 ) + (2

2q 1 2q+1 1 > + )= 2q 2q+1 S S S 1 Jq ˙ qA0 [ + Observe that ˙ qA0 (Jq (2q+2 2)) = [ 2 > ) and ˙ qA0 2q+1 S 2q 1 2q+1 1 3 1 ˙4 2q > + 2q+1 ) = [ 2 > 2). Let H0 = [0> 2 ), let I0 = [> 2) \ ( q=1 2q+1 3 Jq ). Notice that since I0 [> 2 ), Jq [(2q+2 2)> (2q+2 1)) = 2q+1 [ +

H0 + 2 2I0

and

1 I0 H0 > 4

and (H0 + 2) _ 14 I0 = >. Let J0 be the set obtained by applying Lemma S4 10.5. Then ˙ q=0 Jq is both 2-translation congruent to [0> ) and 2-dilation S4 S4 congruent to [> 2). Hence, by symmetry, ( ˙ q=0 Jq ) ^˙ ( ˙ q=0 Jq ) is an unbounded symmetric wavelet set. t u We note that the first example of an unbounded symmetric wavelet set was given in Proposition 2.14 of [FW]. The construction involved some significant computations and explanation. Example 4.5 (xi) of [DL] gave a dierent unbounded wavelet set whose construction required little explanation, but it was not symmetric. The following is a counterexample to a question posed by the second author in [La1]: Let H be a wavelet set, and suppose that J 2H ^H ^ 12 H is also a wavelet set. Is (H> J) an interpolation pair? Example 10.8. Let H = [2> ) ^ [> 2), and J 2H ^ H ^ 12 H = [4> 2 )^ [ 2 > 4) be a wavelet set. Suppose that J1 = J _ [3> 4) has J 2 ) (v) = positive measure, then J1 2 [> 2) H. ;v 5 J1 2, (H 1 J J 1 H (v + 2) = 2 · H ( 2 v + ) = 2 · ( 2 v + + 2n) = v + 2 + 4n, for some J 2 ) (v) 6= v for each v 5 J1 2. Therefore, (H> J) is not an n 5 Z. Thus, (H interpolation pair if J _ [3> 4) has positive measure. Since 7 1 [> ) [0> ) 2 4

and

7 [0> ) + 2 2[> )> 4

and 12 [> 74 ) _ ([0> ) + 2) = >, by Lemma 10.5, there exists J0 [ 2 > 78 ) ^ [2> 3), such that J0 is 2-translation congruent to [0> ) and 2-dilation congruent to [> 74 ). Let J = [> 2 ) ^ J0 ^ [ 72 > 4). Then J is a wavelet t u set and J 2H ^ H ^ 12 H, but (H> J) is not an interpolation pair. The next example answers another question from [La1]:

10 Interpolation Maps and Congruence Domains for Wavelet Sets

209

Let H be a wavelet set, and suppose that J (H 2) ^ H ^ (H + 2) is also a wavelet set. Is (H> J) an interpolation pair? Example 10.9. Let H = [2> )^[> 2) and J (H 2)^H ^(H +2) = [4> 3) ^ [2> 2) ^ [3> 4) be a wavelet set. Suppose that J _ [> ) is not a null set. Without loss of generality, assume that J1 = J _ [0> ) has J 2 J ) (v) = H (v + 2) = positive measure. Then, for each v 5 J1 2 H, (H n J n n 2 · H (2 (v + 2)), for some n A 0 such that 2 (v + 2) 5 [> 2). Then J 2 ) (v) = 2n ·(2n (v+2)+2q) = v+2+2n ·2q for some q 5 {1> 0> 1}. (H J 2 ) (v) 6= v for each v 5 J1 2. Therefore, if J _ [> ) is not a Thus, (H null set, (H> J) is not an interpolation pair. Based on the above observation, let J0 be the measurable set given by Lemma 10.5 and the following two containment relations: 11 9 3 [2> ) [2> ) ^ [ > )> 2 8 8 ([2>

9 3 11 ) ^ [ > )) 2 2[2> )= 8 8 2

Then, J := [ 58 > 78 ) ^ [> 54 ) ^ [ 72 > 4) ^ [ 34 > 2 ) ^ J0 is a wavelet set, since 5 7 9 11 ) + 2) ^˙ [ > ) ^˙ ([ > ) + 2) 8 8 8 8 3 7 5 ^˙ [> ) ^˙ ([ > ) + 2) ^˙ ([ > 4) 2)> 4 4 2 2

[0> 2) = ([2>

and 5 5 3 3 [2> ) ^ [ > ) = [2> ) ^˙ 2 · [ > ) 8 4 2 4 2 1 7 5 5 7 ^˙ [ > ) ^˙ · [ > 4) ^˙ [> ) 8 8 4 2 4 is a 2-dilation generator of a measurable partition of R. However, J _ [> ) J 2 ) 6= lgR . t u is not a null set, hence (H Lemma 10.5 can be useful in constructing special wavelet sets and, in particular, parametric families of wavelet sets. Example 10.10. Let o> p> q 5 N be given. Let H+ (p) = [0> 2p+1 ), I+ = [> 2). Observe that 1 I+ H+ (p) 2p

and

H+ (p) + 2o 2o I+ >

and 21p I+ _(H+ (p) +2o ) = >. Let J+ (o> p) be the measurable set obtained by applying Lemma 10.5. Straightforward computation shows that J+ (o> p) = (

2o 2o+p

1

> 2p+1 ) ^ [2o > 2o +

2o 2o+p

1

)=

210

Xiaofei Zhang and David R. Larson

Similarly, let H (p) = [2 + 2p+1 > 0), I (p) = [2 + 2p+1 > + 2p ). Observe that I (p) H (p)

and

H (p) 2p+q + 2q 2p+q I (p)=

By applying Lemma 10.5 again, we get 2p+q 2q > 2p+q + 2q ) 2p+q 1 2p+q 2q )= ^ [2 + 2p+1 > p+q 2 1

J (p> q) = [2p+q + 2q

Thus, for fixed o> p> q, J (p> q) ^ J+ (o> p) is both 2-translation congruent to H (p) ^ H+ (p) = [2 + 2p+1 > 2p+1 ) and 2-dilation congruent to t u I (p) ^ I+ , hence is a wavelet set.

Notice that J (o> p> q) ^ J+ (o> p> q)> o> p> q > 1 is exactly the family of wavelet sets No>p>q > o> p> q > 1 introduced by X. Fang and X. Wang in [FW] (see [FW, Example (5)]). In the following examples, we will construct some known and new wavelet sets in R2 with dilation matrix D given by 2 · lgR2 . It is known (see [DLS1] and [SW]) that H R2 is a wavelet set for 2 · lgR2 if and only if H is both 2-translation congruent to [> ) × [> ) and 2-dilation congruent to [2> 2) × [2> 2) \ [> ) × [> ). Wavelet sets in higher-dimensional spaces with respect to arbitrary real expansive matrices can also be obtained in the similar way. Example 10.11. Consider the first quadrant. We have the following two relations: [0> ) × [0> ) \ [0> ) × [0> ) [0> ) × [0> )> 2 2 [0> ) × [0> ) + (2> 2) 4 · [0> ) × [0> ) \ [0> ) × [0> )> 2 2 and [0> ) × [0> ) \ [0> 2 ) × [0> 2 ) _ ([0> ) × [0> ) + (2> 2)) = >. Applying Lemma 10.5, we can construct a set Z1 that is both 2-translation congruent to [0> ) × [0> ) and 2-dilation congruent to [0> ) × [0> ) \ [0> 2 ) × [0> 2 ). Symmetrically, construct sets Z2 > Z3 > Z4 in second, third, and fourth quadrants, respectively. Then Z1 ^ Z2 ^Z3 ^ Z4 is a wavelet set. Straightforward compuation shows that it is exactly the “four corners set” in [DLS2]. t u Example 10.12. Consider the right half plane. We have the following two relations: [0> ) × [> ) \ [0> ) × [ > ) [0> ) × [> )> 2 2 2 [0> ) × [> ) + (2> 0) 4 · [0> ) × [> ) \ [0> ) × [ > )> 2 2 2 and [0> ) × [> ) \ [0> 2 ) × [ 2 > 2 ) _ ([0> ) × [> ) + (2> 0)) = >. By Lemma 10.5, we can construct a set Z1 that is both 2-translation congruent

10 Interpolation Maps and Congruence Domains for Wavelet Sets

211

to [0> )×[> ) and 2-dilation congruent to [0> )×[> ) \ [0> 2 )×[ 2 > 2 ). Symmetrically, construct the set Z2 in the left half plane. Then Z1 ^ Z2 is a wavelet set. Straightforward computation shows that it is exactly the “wedding cake set” (Example 6.6.2 in [DL] and also Figure 2 in [DLS2].) u t Example 10.13. Consider the left-top half plane (above the line | = { in the left half plane). Let H1 = {({> |)|{ > > | 6 > | > {}, and I1 = {({> |)|{ > > | 6 > | > {} \ {({> |)|{ > 2 > | 6 2 > | > {}. Then we have the following two relations: I1 H1 >

H1 + (2> 2) 4 · I1 >

and I1 _ (H1 + (2> 2)) = >. Now use Lemma 10.5 to construct a set Z1 that is both 2-translation congruent to H1 and 2-dilation congruent to I1 . Symmetrically, construct a corresponding set Z2 in the right-bottom half plane. Then Z1 ^ Z2 is a wavelet set. The diagram for this is given in Figure 10.1. t u

3

3

Fig. 10.1 Pine tree.

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Xiaofei Zhang and David R. Larson

Example 10.14. Consider the first quadrant. Let H1 = [ 2 > 0) × [ 2 > 0) ^ [> 34 ) × [> 34 ) and I1 = [ 32 > 2) × [ 32 > 2). Then we have the following two relations: I1 H1 + (2> 2)>

H1 + (4> 4) 2 · I1 >

and I1 _ (H1 + (4> 4)) = >. By Lemma 10.5, we can construct a set Z1 which is both 2-translation congruent to H1 and 2-dilation congruent to I1 . Symmetrically, we can define H2 > H3 > H4 and I2 > I3 > I4 , and construct sets Z2 > Z3 > Z4 in the second, third, and fourth quadrants, respectively. Let E = [> ) × [> ) \ ([ 12 > 12 ) × [ 12 > 12 ) ^ [ 34 > ) × [ 34 > ) ^ [> 34 ) × [ 34 > ) ^ [ 34 > ) × [> 34 ) ^ [[> 34 ) × [> 34 )). Then since E ^ H1 ^ H2 ^ H3 ^ H4 = [> ) × [> ) and 2E ^ I1 ^ I2 ^ I3 ^ I4 = [2> 2) × [2> 2) \ [> ) × [> ), Z1 ^ Z2 ^ Z3 ^ Z4 ^ E is a wavelet set. Computation shows that it is one of the wavelet sets introduced in [SW]. The diagram for this is given in Figure 10.2. t u Remarks. (1) The idea of Lemma 10.5 was used in constructing a covering of R by symmetric wavelet sets, and this was a key to subsequent work of the first author with Rzeszotnik in [RZ].

34

32

3

Fig. 10.2 A wavelet set in [SW].

2

4

10 Interpolation Maps and Congruence Domains for Wavelet Sets

213

(2) Further Directions: Interpolation maps, and interpolation pairs and more general interpolation families of wavelet sets, make sense and have been studied for matrix dilations in Rq . Congruence domains also make sense for matrix dilations. Lemma 10.5 in this chapter was stated and proved for Rq , and was applied to solve Question B in R1 , and also used to study examples of dyadic (i.e., for matrix dilation 2L) wavelet sets in the plane. The results in this chapter suggest some directions for further research. In particular, does Theorem 10.1 extend to matrix dilations in Rq , especially for the cases where interpolation pairs are known to exist? It might be useful to try to extend Lemma 10.5 and also Theorem 10.3 to matrix dilations. Acknowledgment The second author was partially supported by NSF grant DMS0139386.

References [BMM]

L. Baggett, H. Medina and K. Merrill, Generalized multi-resolution analyses and a construction procedure for all wavelet sets in Rq , J. Fourier Anal. Appl. 5 (1999), 563—573. [BL1] J. J. Benedetto and M. T. Leon, The construction of multiple dyadic minimally supported frequency wavelets on Rg , Contemp. Math. 247 (1999), 43—74. [BL2] J. J. Benedetto and M. T. Leon, The construction of single wavelets in ddimensions, J. Geom. Anal. 11 (2001), 1—15. [C] A. Calogero, A characterization of wavelets on general lattices, J. Geom. Anal. 10 (2000), 597—622. [DL] X. Dai and D. Larson, Wandering vectors for unitary systems and orthogonal wavelets, Mem. Amer. Math. Soc. 134 (1998), no. 640. [DLS1] X. Dai, D. Larson and D. Speegle, Wavelet sets in Rq , J. Fourier Anal. Appl. 3 (1997), 451—456. [DLS2] X. Dai, D. Larson and D. Speegle, Wavelet sets in Rq II, Contemp. Math. 216 (1998), 15—40. [FW] X. Fang and X. Wang, Construction of minimally supported frequency (MSF) wavelets, J. Fourier Anal. Appl. 2 (1996), 315—327. [GH] Q. Gu and D. Han, On multiresolution analysis (MRA) wavelets in Rq , J. Fourier Anal. Appl. 6 (2000), 437—447. [Gu1] Q. Gu, On interpolation families of wavelet sets, Proc. Amer. Math. Soc. 128 (2000), 2973—2979. [Gu2] Q. Gu, Ph.D. Thesis, Texas A&M University, 1998. [Han1] D. Han, Interpolation operators associated with sub-frame sets, Proc. Amer. Math. Soc. 131 (2003), 275—284. [Han2] D. Han, Ph.D. Thesis, Texas A&M University, 1998. [HWW1] E. Hern´ andez, X. Wang, and G. Weiss, Smoothing minimally supported frequency wavelets I, J. Fourier Anal. Appl. 2 (1996), 329—340. [HWW2] E. Hern´ andez, X. Wang, and G. Weiss, Smoothing minimally supported frequency wavelets II, J. Fourier Anal. Appl. 3 (1997), 23—41. [IP] E. Ionascu and C. Pearcy, On subwavelet sets, Proc. Amer. Math. Soc. 126 (1998), 3549—3552. [La1] D. Larson, Lecture notes from the VIGRE Seminar Course “Wavelet theory and matrix analysis”, Texas A&M University, 2000—2001.

214 [La2]

[La3]

[La4] [La5]

[RZ] [SW] [S1] [S2] [Wut] [Za] [Zh]

Xiaofei Zhang and David R. Larson D. Larson, Unitary systems and wavelet sets, “Wavelet Analysis and Applications.” Applied and Numerical Harmonic Analysis, Birkh¨ auser Verlag, Basel, 2006, pp. 143—171. D. R. Larson, Von Neumann algebras and wavelets, “Operator Algebras and Applications (Samos, 1996),” NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 495, Kluwer, Dordrecht, 1997, pp. 267—312. D. R. Larson, Frames and wavelets from an operator-theoretic point of view, Contemp. Math. 228 (1998), 201—218. D. R. Larson, Unitary systems, wavelet sets, and operator-theoretic interpolation of wavelets and frames, Tutorial notes given at the National University of Singapore Aug. 2004, WSPC/Lecture Notes Series, to appear, 48pp. Z. Rzeszotnik and X. Zhang, Unitary operators preserving wavelets, Proc. Amer. Math. Soc. 132 (2004), 1463—1471. P. Soardi and D. Weiland, Single wavelets in n-dimensions, J. Fourier Anal. Appl. 4 (1998), 299—315. D. Speegle, The s-elementary wavelets are path-connected, Proc. Amer. Math. Soc. 127 (1999), 223—233. D. Speegle, Ph.D. Thesis, Texas A&M University, 1997. Wutuam Consortium, Basis properties of wavelets, J. Fourier Anal. Appl. 4 (1998), 575—594. V. Zakharov, Nonseparable multidimensional Littlewood—Paley like wavelet bases, Centre de Physique Th´eorique, CNRS Luminy 9 (1996) X. Zhang, Ph.D. Thesis, Texas A&M University, 2001.

Chapter 11

Orthogonal Exponentials for Bernoulli Iterated Function Systems Palle E. T. Jorgensen, Keri Kornelson, and Karen Shuman Dedicated with respect and fondness to Larry Baggett

Abstract We investigate certain spectral properties of the Bernoulli convolution measures on attractor sets arising from iterated function systems (IFSs) on R. In particular, we examine collections of orthogonal exponential functions in the Hilbert space of square-integrable functions on the attractor. We carefully examine a test case = 34 in which the IFS has overlap. We also determine rational = de for which infinite sets of orthogonal exponentials exist.

11.1 Introduction This work examines the spectral properties of a class of measures on fractals that arise from ane iterated function systems (IFSs) on R. The fractals [ are compact subsets of R, which may or may not have Hausdor dimension less than 1. Even though such sets do not have a group structure or Haar measure, we are able to identify a substitute. Associated with each compact set [ is a measure , often called a Hutchinson measure, with reference to [15]. The measure is a probability measure having support [. By analogy to a Haar measure, is uniquely determined by the maps that characterize the iterated function system and exhibits an invariance property arising from these maps. The Bernoulli ane IFS on R is given by two functions

Palle E. T. Jorgensen Department of Mathematics, University of Iowa, Iowa City, IA 52242 e-mail: [email protected] Keri Kornelson, Karen Shuman Department of Mathematics & Statistics, Grinnell College, Grinnell IA 50112-1690 e-mail: [email protected], [email protected]

217

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Palle E. T. Jorgensen, Keri Kornelson, and Karen Shuman

e1 ({) = ({ e1 )>

e2 ({) = ({ + e2 )>

where E = {e1 > e2 } and the parameter 5 (0> 1). The measures arising from a Bernoulli IFS are called Bernoulli convolution measures. These measures have a long history that can be studied, for example, in [21]. We will examine orthogonality of exponential functions in the Hilbert space O2 ([> ), in order to better understand the Bernoulli convolution measures. Jorgensen and Pedersen [19, 20] recently found examples of Bernoulli measures for which the Hilbert space O2 ([> ) has an orthonormal basis of exponential functions. In such cases, we say that is a spectral measure. These known examples have certain properties in common. The Bernoulli IFS maps are of the form e1 ({) = ({ e1 ) and e2 ({) = ({ + e2 )> where e1 and e2 are integers, and = q1 is the reciprocal of a natural number. These conditions on and E describe what we will call the rational case of a Bernoulli ane IFS. In recent papers [7, 20], an additional condition to the rational case–Hadamard duality–is discovered, which will guarantee that an orthonormal Fourier basis exists corresponding to the rational cases of and E. In this chapter, we consider the nonrational cases for which the scaling factor is not thus restricted. In such examples, we also find that the values e1 and e2 are not integers. The arguments from dynamics and random-walk theory that can be used in the rational cases no longer apply, so the problem of determining whether Fourier bases exist, and constructing them if they do, are much harder.

11.1.1 Motivation A leading theme in harmonic analysis in general [12], and in the work of Larry Baggett and co-authors in particular, is that of building basis decompositions, or direct integral decompositions, out of geometric composite structures. Such processes generally involve a group structure, as in the case of semidirect product groups, and the use of induced representation constructions, both for discrete groups and continuous groups. This setting accommodates a variety of applications, including wavelet analysis, which relates to d{ + e -like groups and time-frequency analysis, which involves Heisenberg-like groups. For examples of results in these areas, see [1, 2, 3]. There is recent evidence [17, 8] that the question of basis decompositions can be examined on a class of fractals, which are attractors of so-called ane iterated function systems (IFSs), including Cantor’s middle-third example; see also [15]. As it turns out, a number of the classical tools from the group case still work, but there is no longer a group structure to work with. Without the groups, we must first look for a substitute for Haar measure. Thanks to a construction of Hutchinson [15], this is available in the form

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of equilibrium measures with support on the fractal attractor set [ of the IFS. We can therefore describe a precise notion of an orthogonal Fourier expansion in the Hilbert space O2 ([> ). It turns out in fact that the equilibrium measures will typically be singular with respect to Lebesgue measure; see [18]. Although there is already some work on expansion problems in the context of ane IFS fractals [8, 19], in these papers the class of admissible IFSs with equilibrium measures is restricted in several ways such that the Hilbert space O2 ([> ) will carry an orthogonal Fourier expansion based on complex exponentials. However, the systems considered so far have a rather restricted and rigid Diophantine structure, and they do not admit overlap. They also do not admit continuous deformations. The ane IFSs considered here are not restricted to the conditions imposed in the earlier studies [8, 19].

11.1.2 Related Topics The analysis of fractal measures is motivated by and has influences on a variety of areas outside fractal theory itself. Some such fields include tiling spaces (see, e.g., [4]), lacunary expansions ([26, 22]), and random Fourier series ([21, 23]). Each of these subfields oers a variation of the general theme of recursive constructions based on some notion of self similarity. Moreover, each area invites an approach that mixes tools from operator algebra theory and from probability. Together this part of mathematics stands at the crossroads of operator algebras, basis constructions, and dynamics. This study also interacts with parallel developments in the study of wavelets (including wavelets on fractals [5, 8, 17]) and of iterated function systems; see, e.g., [9]. We summarize here a few relevant facts to illustrate the connections and common themes to these diverse topics. 1. Tiling spaces [4] are important for our understanding of diraction in molecular structures that form quasi crystals. The simplest tiles in Rg come from translations by rank g lattices; and others involve both translations and matrix operations. A pioneering paper by Fuglede [13] suggested a close connection between the Fourier bases and translation tiles, and this connection was clarified in later works such as [25]. 2. Lacunary Fourier series on the line refers to Fourier expansions where there are infinitely many gaps in the frequency variable that separate powers of a selected finite set of numbers. By inspection, one checks that the “fractals in the large” described in [19] which arise in the known examples as the spectra of ane IFSs fit this pattern. 3. Random series of functions (as in [21, 23]) naturally generalize the first ane IFS systems — those constructed as infinite convolutions of independent Bernoulli variables as in [10]. The notion “random” here refers to the study of Fourier series where the Fourier coecients are random variables.

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11.1.3 Overview In Section 11.2, we carefully describe the Bernoulli ane iterated function system for parameter 5 (0> 1) and the resulting attractor set [ and Hutchinson measure . We also describe some of the early results about this measure, by Erd˝os and others. Next, we establish the notation and elementary results that will be used in the following sections. In Section 11.3, we determine that there do exist infinite collections of orthogonal exponentials for the special case where = 34 . We find that some of these collections are maximally orthogonal, in the sense that such a collection is not properly contained in another orthogonal collection. In Section 11.4, we determine for which rational values of there exist infinite orthogonal collections of orthogonal exponentials. Our main result is the following, which is stated in the chapter as Theorem 11.9. Given = de , if e is even, then there exist infinite families of orthogonal exponentials in the corresponding Hilbert space. If e is odd, then every collection of mutually orthogonal exponentials must be finite. In Section 11.5, we conjecture that none of these collections of orthogonal exponentials for = 34 are actually orthonormal bases for the Hilbert space. This conjecture is based on numerical evidence. It would support a conjecture in [7] that no orthonormal bases exist in the nonrational cases.

11.2 Ane IFSs and Their Associated Invariant Measures We study families of exponentials that are mutually orthogonal with respect to invariant measures associated with ane iterated function system (IFS) on the real line. As in [6], we will consider ane IFSs that are determined by two ane maps on R + ({) := { + 1

and ({) := { 1>

(11.1)

where 5 (0> 1). Because = 34 arises as the first value in a special family of values of considered by Dutkay and Jorgensen ([6, Theorem 4.5]), we first explore families of exponentials associated with the ane IFS with parameter = 34 . It turns out that this value of provides good intuition for what happens in the more general case 5 ( 12 > 1) _ Q. We then generalize our results for = 34 to = de , where d> e 5 N. When two ane maps as in (11.1) generate an IFS, the resulting measure is called an infinite Bernoulli convolution measure (Lemma 11.1). Infinite Bernoulli convolution measures are characterized by the infinite product structure of their Fourier transforms. To be precise, when 5 (0> 1), the

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221

Q4

product q=0 cos(2q w) is the Fourier transform of an infinite Bernoulli convolution measure. The term infinite convolution is not mysterious at all, as the Fourier transform converts convolutions to products. However, one might ask why these measures have the name Bernoulli attached to them. These measures arise in the of Erd˝os and others via the study of the random geometric seP work ries ±q for 5 (0> 1), where the signs are the outcome of a sequence of independent Bernoulli trials. In other words, we could consider the signs to P be determined by a string of fair coin tosses. This makes ±q a random variable, i.e., a measurable function from a probability space into the real numbers R. In Erd˝os’s language, is the distribution of the random variable [ is defined on the probability space of all infinite sequences of ±1. The measure on is the infinite-product measure resulting from assigning ±1 equal probability 12 . The random variable [ takes on a specific real value for each sequence from . This distribution , then, is the familiar Bernoulli distribution from elementary probability theory. The infinite Bernoulli convolution measure is determined by the distribution G of the random variable P ±q , which can be constructed from infinite convolution of dilates of : G := ({) (1 {) (2 {) = = = (q {) = = = =

These Bernoulli convolution measures have been studied from at least the mid-1930s in various contexts. There seems to have been a flurry of activity in the 1930s and 1940s surrounding these measures. Jessen and Wintner study these measures in their study of the Riemann zeta function in their 1935 paper [16]; in 1939 and 1940, Erd˝os published two important papers about these measures [10, 11]. In the 1939 paper, Erd˝os proved that if is a Pisot number (that is, is a real algebraic integer greater than 1 all of whose conjugates ˜ satisfy |˜ | ? 1), then the infinite Bernoulli convolution measure associated with = 1 is singular with respect to Lebesgue measure. However, more recently, Solomyak proved that for almost every 5 ( 12 > 1), the measure is absolutely continuous with respect to Lebesgue measure [24]. Bernoulli convolution measures arise in the study of ane iterated function systems because they possess a special invariance property–a property that is described in Hutchinson’s 1981 theorem [15, Theorem 2, p. 714]. In the cases studied here, the invariance property of the measure can be written =

1 1 1 ( + + )= 2

(11.2)

We will now see that the measure satisfying this invariance property is a Bernoulli convolution measure. We note that the following lemma is not a new result; we state and sketch a proof of the lemma to show the connection between ane IFSs and Bernoulli convolution measures. See [6, Lemma 4.7]. Lemma 11.1. Given a fixed 5 (0> 1), the Fourier transform of satisfies the equation

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Palle E. T. Jorgensen, Keri Kornelson, and Karen Shuman

c (w) =

4 Y

q=0

³ ´ cos 2q w =

(11.3)

That is, the measure arising from the IFS (11.1) is a Bernoulli convolution measure. Proof. We use Hutchinson’s invariance property in Equation (11.2). This invariance property has integral form Z ¸ Z Z 1 i ( ({)) d ({) + i (+ ({)) d ({) = (11.4) i ({) d ({) = 2 We use this relation to show that the Fourier transform of has the structure of an infinite Bernoulli convolution measure. Z (w) = h2l{w d ({) c Z 1 h2l({1)w + h2l({+1)w d ({) by Eq. (11.4) = 2 Z 1 2lw 2lw +h ) h2l{w d ({) = (h 2 = cos(2w)c (w) = cos(2w) cos(2w)c (2 w) .. .. . . 4 ³ ´Y q = lim c cos(2q w) ( w) q$4

q=0

q c The limit limq$4 c ( w) exists and is equal to (0) = 1, since it is known that has no atoms and is a probability measure. Dutkay and Jorgensen show that has no atoms in [6, Corollary 6.6 ]; the fact that is a probability measure follows from Hutchinson’s theorem [15, Section 4.4]. u t

Let [ be the support of –that is, [ R is the attractor for the IFS (11.1). For R, we determine conditions under which the set of exponentials {h2lc· : c 5 } is an orthonormal basis in the Hilbert space O2 ([ > ). In other words, we explore whether is a spectral set for the Hilbert space. It was shown in [20] that when = 13 , there is no orthonormal basis of exponentials, but when = 14 there is such an ONB. In this chapter, we are particularly interested in values of A 12 , for which the IFS has overlap. After some general formulas, we will examine the special case = 34 . In Section 11.4, we state some generalizations to other rational values of . We wish to show that the infinite product in Equation (11.3) is zero if and only if one of the factors is zero. In the following lemma, we have omitted the 2 for notational convenience.

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Lemma 11.2. Suppose w is a fixed real number and that 5 (0> 1). There exists Q 5 N and f A 0 such that 4 Y

cos(q w) f=

(11.5)

q=Q

In other words, if for some w0 5 R, 4 Y

cos(q w0 ) = 0>

q=0

then one of the factors of the product must be 0. Proof. To start, we note that 4 Y

4 X

cos(q w) f /

q=Q

q=Q

³ ´ ln cos(q w) ln(f)=

The Taylor expansion of cosine around 0 yields cos(q w) = 1

4 X 2nq 2n w = (1)n+1 (2n)! n=1

Define %q (w) :=

4 X 2nq 2n w ; (1)n+1 (2n)!

n=1

since 1 1 %q (w) 1 for all q 5 N, we know that %q (w) 0 for all q 5 N. 2q 2 w , and then we We can choose Q1 such that for all q A Q1 , %q (w) 2 2q 2 w ? 1. We now consider the can choose Q2 such that for all q A Q2 , 2 Taylor expansion of ln(1 %q (w)), which is valid when |%q (w)| ? 1: 4 ³ ´ X %q (w)n = ln cos(q w) = ln(1 %q (w)) = n n=1

P4 %q (w)n Finally, we choose Q3 such that for all q A Q3 , 2%q (w). n=1 n Now, for Q A max{Q1 > Q2 > Q3 }, we have Ã 4 ! 4 4 4 ³ ´ X X X %q (w)n X q ln cos( w) = 2%q (w) n q=Q

q=Q

n=1

q=Q 4 X q=Q

2q w2 =

224

But

Palle E. T. Jorgensen, Keri Kornelson, and Karen Shuman

P4

q=Q

2q is a convergent geometric series, so set 4 ³ ´ X 2q = f = exp w2 q=Q

We have now found Q and f such that 4 Y

cos(q w) f=

t u

q=Q

Our goal in this chapter is to study collections of orthogonal exponentials in the Hilbert space O2 ([ > ). We now observe that the Fourier transform c of the Bernoulli measure arises in the inner product of exponential functions in the Hilbert space O2 ([ > ). Z 2lcw 2lc0 w 0 >h i = h2lcw h2lc0 w g (w) = c (11.6) hh (c c )

Using Lemmas 11.1 and 11.2, we can conclude that the exponential functions hc and hc0 are orthogonal if and only if cos(2q (c c0 )) = 0 for some q 5 N0 . (To simplify notation, we will use hc to denote the exponential function h2lc· .)

11.3 Orthogonal Exponentials with Respect to 34 In order to understand the issues involved with the study of orthogonal exponentials, and the possible existence of orthonormal bases, we study the special case = 34 . Lemma 11.3. The function c34 (w) is equal to zero if and only if w 5 2Z) for some q 5 N0 .

Proof. Choose q 5 N0 and c 5 Z, and let w =

4q1 3q (2c

4q1 3q (1 +

+ 1). Then

´ ³ ³ 3 ´q ´ ³ 3q 4q1 ´ ³ cos 2 w = cos 2 q · q (2c + 1) = cos (2c + 1) = 0= 4 4 3 2

Since this is one of the factors in the infinite product

we see that c34 (w) is 0.

c34 (w) =

³ ³ 3 ´q ´ cos 2 w > 4 q=0 4 Y

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225

Conversely, suppose that for some w0 5 R, the product c34 (w0 ) = 0. By ³ ³ ´q ´ Lemma 11.2, this implies that for some q 5 N0 , the factor cos 2 34 w0 is ³ ´q equal to zero. Therefore, the quantity 2 34 w0 must be an odd multiple of 2,

so there exists c 5 Z such that 2

which then gives

³ 3 ´q 4

w0 =

(2c + 1)> 2

4q1 (2c + 1)= 3q As a result, we have established that c34 (w0 ) = 0 if and only if one of the terms in the infinite product defining c34 is 0. In other words, w0 =

c34 (w0 ) = 0 / w0 5

(

) 4q1 (1 + 2Z) : q 5 N = 3q

t u

We will begin our examination of the sets of orthogonal exponentials for O2 ([ 34 > 34 ) by finding sets R such that {hc : c 5 } is a mutually orthogonal set of functions with respect to 34 . By the preceding discussion, we have the following proposition. Proposition 11.4. The exponential functions {hc : c 5 } are pairwise orthogonal if and only if for c> c0 5 , we have c c0 5 O, where we define ( ) 4q1 (1 + 2Z) : q 5 N0 = O= (11.7) 3q Proof. This follows from Equation (11.6), Lemma 11.2, and Lemma 11.3. u t Theorem 11.5. There exist infinitely many infinite sets such that {hc : c 5 } is a mutually orthogonal set of functions with respect to the measure 34 . Proof. Define the set n for each n 5 N as follows. ( ) 4m : m 5 N> m n 1 ^ {0} n := 3n If c = then

4s 3n

and c0 =

4t 3n

(11.8)

(s> t n 1, s A t) are two nonzero elements of n , c c0 =

4s 4t 4t st = (4 1)> 3n 3n 3n

and 4st 1 is an odd integer. Now, multiply through by

(11.9) 3tn+1 : 3tn+1

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Palle E. T. Jorgensen, Keri Kornelson, and Karen Shuman

c c0 =

4t 3tn+1 st 4t (4 1) = t+1 (3tn+1 )(4st 1)> n tn+1 3 3 3

(11.10)

and (3tn+1 )(4st 1) is still an odd integer. Therefore, c34 (c c0 ) = 0. If c 6= 0 and c0 = 0, then 4s 3sn+1 4s 4s = n · sn+1 = s+1 3sn+1 > n 3 3 3 3

and 3sn+1 is an odd integer since s n 1.

t u

If we draw a diagonal diagram of the n sets in Theorem 11.5, we have j: 0 1 : 0 13 2 : 0 3 : 0 .. . 0

1

2

3

4 3 4 9

16 3 16 9 16 27

64 3 64 9 64 27

..

4 ··· ··· ··· ···

128 3 128 9 128 27

.

These sets are certainly not the only infinite sets satisfying the condition of Proposition 11.4. Since our condition tests the dierences between elements, any of the above can be translated by a real number . We choose representative sets by making the requirement that 0 be an element of each n . Similarly, if every element in a n set is multiplied by the same odd integer, the dierences remain elements of the set O. One natural question is whether some of these n sets can be combined to form larger collections of orthogonal exponentials. We find, however, that the orthogonality condition from Proposition 11.4 is lost if we take the union of dierent n sets. Proposition 11.6. Suppose c 6= 0, c 5 n where n A 1. Then the set 1 ^{c} does not form a mutually orthogonal family of exponential functions. m

Proof. Let c = 34n , where n A 1 and m n. We can show that hc and h1@3 are not orthogonal, and therefore we cannot add c to the set 1 to build a larger family of orthogonal exponential functions. c

4m 4m 3n1 1 1 = n = 3 3 3 3n

The numerator in the last expression is still an integer since n 1 A 0, and we also observe that 4 does not divide the numerator. This means that if we are to write this dierence in the form of set O from (11.7), our power of 4 must be zero, and therefore the power of 3 must be 1. µ ¶ 4m 3n1 40 4m 3n1 1 = 1 c = 3 3n 3 3n1

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Since m n A 1, the second fraction cannot be an integer, and therefore this dierence is not an element of the set O. This proves that the two t u exponentials hc and h1@3 are not orthogonal. A similar argument shows that nonzero elements from n cannot be combined with m for m 6= n while maintaining orthogonality. We can merge a finite number of n ’s and still form an orthogonal set, as long as we are willing to throw out finitely many terms. For example, (2 ^ 1 ) \ { 13 } forms an orthogonal set. However, experimental evidence indicates that (2 ^ 1 ) \ { 13 } is not total (see Section 11.5). Rather than merging the n sets, we can expand each of them to a larger collection of orthogonal exponentials. Theorem 11.7. Define the set n for each n 5 N as follows: ; < s ? X @[ dm 4m {0} n = : s finite> d 5 {0> 1} m = > 3n m=n1

Each set {h : 5 n } is an orthonormal family in O2 ([ 34 > 34 ). Proof. We will demonstrate the proof for the set 1 . The argument is similar for each n . First, we show that each nonzero element in 1 does belong to the set O, which P proves that the dierences with 0 are in O. Let d 5 1 > d 6= 0. Then d = 13 sl=0 dl 4l , where dl 5 {0> 1}. Let u be the smallest integer such that du 6= 0, so 0 u ? s. We can then write d in the form: Ã ! µ u¶ s s s X 3 4u 4u X 1X l lu u lu = u+1 3 dl 4 = dl 4 dl 4 d= 3 l=u 3 l=u 3u 3 l=u Since du = 1, we know that the sum in the last two expressions above is an odd integer, which when multiplied by 3u yields another odd integer. Therefore, d 5 O. Next, we must show Ps that the dierence Ptof any two nonzero elements in 1 is in O. Let d = 13 l=0 dl 4l and e = 13 l=0 el 4l . In order to combine these, assume without loss of generality P that s t and let el = 0 for l = t + 1> = = = s. Therefore, we have d e = 13 sl=0 (dl el )4l , where dl el 5 {1> 0> 1}. As above, let u be the smallest integer such that du eu 6= 0, so 0 u ? s. We can then write d e as follows: µ u¶ s s 3 4u X 1X l lu (dl el )4 = (dl el )4 de = 3 l=u 3 l=u 3u Ã ! s X 4u = u+1 3u (dl el )4lu 3 l=u

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Since du eu = ±1, the sum in the last two expressions above is an odd integer, and so is its product with 3u . Therefore, d e 5 O. A parallel argument works for the other sets n , and in fact, demonstrates why each set must start with powers of 4 no more than one less than the power n of 3 in the denominator. t u For the same reasons that the sets n cannot be combined while retaining orthogonality of the exponentials (Proposition 11.6), the n sets also cannot be combined. In fact, we find that the sets n are maximal in a stronger sense as well. For each n, n is not strictly contained in another set for which all the exponentials are pairwise orthogonal. We will state the proof here for the set 1 for ease of notation, but remark that a parallel argument holds for each n . Theorem 11.8. {h : 5 1 } is a maximally orthogonal collection of exponentials for O2 ([ 34 > 34 ). In other words, given { 5 R \ 1 there exists 5 1 such that h{ and h are not orthogonal. @ O, where O is the set given by Proof. First, note that since 0 5 1 , if { 5 Equation (11.7) in Proposition 11.4, then h{ is not orthogonal to h0 . Thus, q1 we can restrict to { 5 O, so { = 4 3(2n+1) for some choice of q 1 and q n 5 Z. @ 1 . Then 2n +1 can be written in Case 1, (q = 1)= Take { = 2n+1 3 s , but { 5 P a base-4 expansion 2n+1 = l=0 dl 4l , where at least one of {d0 > d1 > = = = ds } is eitherP 2 or 3. Let u be the smallest index P such that du = 2 or 3. If du = 2, u1 u1 l l u ( d 4 l l=0 dl 4 ) + 4 and if du = 3, let = . let = l=0 3 3 In both cases above, we have Ps 2 · 4u + l=u+1 dl 4l { = = 3 Since the numerator cannot be written as an odd multiple of a power of 4, we have { 5 @ O, which proves that the exponentials h{ and h are not orthogonal by Proposition 11.4. q1

where q A 1. It is possible that Case 2, (q A 1). Let { = 4 3(2n+1) q powers of 3 divide the odd integer 2n + 1, so we can cancel some of these q1 where either p A 1 and 2c + 1 is not if they exist to write { = 4 3(2c+1) p divisible by 3 or we have p = 1. q1 p1 . Since If p A 1, then let = 13 . We find that { = 4 (2c+1)3 3p there is no way to cancel more powers of 3, the numerator would need to be divisible by 4p1 if { were to be an element of O. We see, however, that the numerator is not divisible by 4. Therefore, the exponential functions h and h 13 are not orthogonal.

11 Orthogonal Exponentials for Bernoulli IFSs

229 q1

If p = 1, then we have a situation similar to Case 1. Since { = 4 (2c+1) 3 but { 5 @ 1 , we have that 2c + 1 has a base-4 expansion that includes at coecient that is not a 0 or a 1. As above, given 2c + 1 = Psleast one l d 4 , let u be the smallest index for which du = 2 or 3. If du = 2, let l=0 l Pu1 P q1 l l u 4q1 ( u1 4 l=0 dl 4 l=0 dl 4 + 4 ) , and if du = 3, let = . This = 3 3 gives in both cases { =

4q1 (2 · 4u + du+1 4u+1 + · · · + ds 4s ) = 3

The numerator cannot be expressed as an odd multiple of a power of 4, so { 5 @ O and therefore, h and h{ are not orthogonal. This proves that the set 1 is maximal, in the sense that there is no @ 1 that can be added to {h : 5 1 } to form exponential function h{ > { 5 t u an orthogonal collection properly containing 1 .

11.4 Rational Values of We outlined in Section 11.2 our rationale for focusing on the Bernoulli IFSs and on the specific value = 34 for the scaling constant. It is natural to next explore whether these results are typical for other rational values of . In this section, we find that the orthogonality results from Section 11.3 do indeed extend to rational values of other than 34 . Theorem 11.9. Let 5 Q _ (0> 1) and let = de be in reduced form. If e is odd, then any collection of pairwise orthogonal exponential functions in the Hilbert space O2 ([ > ) can have only finitely many elements. If e is even, then there exists a countably infinite collection of orthogonal exponentials in O2 ([ > ). Before we prove this theorem, we must find a new set O corresponding to the set (11.7) in Proposition 11.4 that will identify the zeros of c (w) and thereby serve as a test for orthogonality of exponential functions. We will assume that the collections under consideration all contain 0. Lemma 11.10. A set of real numbers containing 0 has the property that {h : 5 } is an orthogonal collection of exponentials in O2 ([ > ) if and only if for each > 0 5 , we have 0 5 O, where we define ¾ ½ ³ ´ 1 e q (11.11) (1 + 2Z) : q 5 N0 = O := 4 d Proof. We showed in Section 11.2 that

230

Palle E. T. Jorgensen, Keri Kornelson, and Karen Shuman 0 hh > h 0 i = c ( ) =

4 Y

cos(2q )

q=0

0 for any choice of 5 (0> 1). By Lemma 11.2, we know that c ( ) is zero if and only if for some q, the expression cos(2q ( 0 )) is zero. We reproduce the computations from Lemma 11.3 to find our new set O corresponding to the set in Equation 11.7. If we take = de , we find that

2q ( 0 ) = (2n + 1)

1 eq / 0 = (2n + 1) 2 4 dq

n 5 Z=

Since we are taking 0 to always be an element in our sets of frequencies , this means each element of must already be an element of O, and the dierences between any two elements must also be in O. t u Proof (Theorem 11.9). Let = de , where d and e are relatively prime. Let be a collection of real numbers containing 0 such that each nonzero element is in O. We can accomplish much of this proof using parity arguments, so we consider individually the cases for which d> e are even/odd. 1. d, e odd: Let > 0 5 be distinct and nonzero, so that = 0

and =

0

eq dq0

0

eq dq (2n

+ 1)

0

(2n + 1). Without loss of generality, let q q . # " 0 0 1 eq (2n + 1) eq dqq (2n0 + 1) = 4 dq 0

Since the numerator above is an even number, it cannot be written as a product of es and some odd integer. Multiplying this expression in both numerator and denominator by powers of d will not resolve this. Therefore, we find that in the case where d and e are both odd, an orthogonal set can only contain one nonzero frequency. 2. (d even, e odd) Let 6= 0 be an element of O, so that = 14 ( de )q (2n +1). We will show that any collection containing 0 and and having the property that the dierence between any two elements is an element of O 0 must be a finite set. Let 0 5 , so that 0 0 = 14 ( de )q (2n 0 + 1) in O. Lemma 11.11. Let e be odd and d be even. Fix 5 O. For every 0 5 O with 0 6= , exactly one of the following is true. a. q = q0 and 0 = 14 ( de )s (2p + 1) with s ? q; b. q 6= q0 and 0 = 14 ( de )s (2p + 1) with s = max{q> q0 }; @ O. c. 0 5

Proof. Let q A q0 . We have ! Ã 0 0 1 eq (2n + 1) + eq dqq (2n 0 + 1) 0 = = 4 dq

11 Orthogonal Exponentials for Bernoulli IFSs

231

Since d is even, it does not divide the odd integer eq (2n + 1), and therefore does not divide the numerator. We therefore have 0 in one of the following rational forms: µ ¶s qs 0 0 e (2n + 1) + eq s dqq (2n0 + 1) 1 e for s q0 4 d dqs or 1 4 or 1 4

µ ¶s qq0 0 e (2n + 1) + dqq (2n 0 + 1) e d dqs esq0

µ ¶s qq0 sq 0 e d (2n + 1) + dsq (2n 0 + 1) e d esq0

for q0 ? s ? q

for q0 ? q ? s=

In the first two cases, the final factor in the expression cannot be an odd integer since d does not divide the numerator. In the third case, the numerator of the final fraction is even while the denominator is odd. This fraction could be an integer but cannot be an odd integer. Therefore, the only remaining possibility is s = q. An identical argument holds for q ? q0 . Therefore, if q 6= q0 , either s = @ O. max{q> q0 } or 0 5 Let q = q0 . Then µ ¶q 1 e (2n 2n0 ) 0 = 4 d If s q, the rational form of 0 is µ ¶s 2(n n0 )dsq 1 e = 4 d esq

As above, since the numerator of the last factor is an even integer and the denominator is odd, their ratio cannot be an odd integer. Therefore, when @ O. t u q = q0 , either s ? q or 0 5 Lemma 11.11 shows that given 5 O, we need only consider elements 0 5 O that satisfy q = q0 or for which q 6= q0 and s = max{q> q0 }. We will show that we can only find finitely many such 0 that also have their pairwise dierences in the set O. ¡ ¢q a. (q = q0 ) Suppose 0 = 14 de (2n 0 + 1). We seek 0 for which 1 = 4 0

µ ¶s qs e 2(n n 0 ) e > d dqs

where the final factor is an odd integer, and we have shown in Lemma 11.11 that s ? q is a necessary condition. Since d and e are relatively

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prime, we see that the expression 2(nn0 ) dqs

eqs 2(nn0 ) dqs

is an odd integer only if

is an odd integer. Suppose that n and n0 are congruent modulo dq+1 . Then dq+1 divides 0 ) nn 0 , and therefore 2(nn dqs is an even integer. Therefore, given our fixed 5 , there can be only one n 0 from each congruence class modulo dq+1 to make up a 0 in . b. (q A q0 0) Since s = q is our only option, by Lemma 11.11, we seek 0 such that # µ ¶q " 0 1 e (2n0 + 1)dqq 0 = (2n + 1) 4 d eqq0 where the final factor on the right-hand side is equal to an odd integer. 0 This requires that eqq divide 2n0 + 1 since d and e are relatively prime. For each choice of q0 ? q, every 2n 0 + 1 that is a multiple of 0 eqq satisfies this property. This gives an infinite set for each q0 ? q, but by the previous q = q0 argument, only finitely many of them for each q0 can be included in the set . Since q is a fixed finite integer, we still have a finite collection . c. (q0 A q 0) We seek 0 such that # µ ¶q0 " q0 q 1 e (2n + 1) d 0 0 (2n + 1) > = 4 d eq0 q where the last expression is an odd integer. Similar to the above argu0 ment, this requires that eq q divides 2n + 1. Since n and q are fixed 0 with , there can only be finitely many q0 A q such that eq q divides 2n + 1. Then for each such q0 , there can only be a finite selection of values for n 0 to include in . This completes the proof that any set containing 0 and elements of O such that the dierence between any two elements again is in O must be a finite set in the case where d is even and e is odd. 3. d odd, e even Now, let = de , where e is even. Since d and e are relatively prime, d must be odd. We will show that the set ¾ ½ l e : l5N = {0} ^ 4d has the property that the exponentials {h : 5 } are pairwise orthogonal in O2 ([ > ). We first show that the nonzero elements are all in O. µ ¶ 1 el el = dl1 4d 4 dl

11 Orthogonal Exponentials for Bernoulli IFSs

233

Since dl1 is an odd integer for any choice of l 1, each nonzero element of is in O. Next, we verify that dierences of distinct nonzero elements of are in O. el em and 0 = 4d , and assume without loss of generality that l ? m. Let = 4d em el 4d 4d el (1 eml ) = 4d µ ¶l 1 e (1 eml )dl1 = 4 d

0 =

Since 1 l ? m, the expression (1 eml )dl1 is an odd integer. This t u proves that 0 5 O.

11.5 Experimental Evidence for Nontotal Sets Next, we consider whether any of our collections of orthogonal exponentials form orthonormal bases (ONB). We can use Equation (11.3) and Parseval’s identity for orthonormal bases to determine whether a collection of orthogonal exponentials is total. From Parseval, we know that if {h : 5 } is an ONB, then for any i 5 O2 ([ 34 > 34 ), we have X |hi> h i|2 = ki k2 3 = 4

5

If we apply this to an exponential function, we find X 2 khw k2 3 = |hhw > h i| 4

5

=

X

5

|c 34 (w )|2

by Equation (11.3).

(11.12)

Using Stone—Weierstrass to show the density of exponentials, we find that our collection is an ONB if and only if the expression (11.12) is a function of w identically equal to 1. X

5

[c 34 (w )]2 =

4 XY

5 n=0

µ ³ ´ ¶ 3 n cos2 2 (w ) 1= 4

(11.13)

In Section 11.3, we observed that we could remove one element from a n set and replace it with countably many other elements from another m set. For example, the set 1 ^ 2 \{1@3} corresponds to a mutually orthogonal

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set of exponentials. However, when we graph the corresponding versions of Equation (11.12), we see that the sum is far from being 1. See Figures 11.1, 11.2, and 11.3 where we notice that as we omit a frequency, we lose a “peak” to 1 at that frequency, but when we include a frequency, we gain a “peak” to 1. Remark 11.12. An interesting corollary to Theorem 11.8 above is that when 1 is used in the summation in Equation (11.13), then as a function of w, this 1 summation is strictly positive on U. (See Figure 11.4.) We now turn to the sets n , which expand the n sets in a dierent way from the way we discussed above. We have two reasons to suspect that none of the sets n can be used to construct ONBs. First, numerical approximations from Mathematica have provided evidence that the sets n are not total in O2 ([ 34 > 34 ). The graph in Figure 11.4 shows that the expression (11.12) for

1

1 3

4 3

Fig. 11.1 1 : Equation (11.12) for w M [0> 2].

1

4 9

4 3

16 9

Fig. 11.2 (1 2 )\{1@3}: Equation (11.12) for w M [0> 2]. We gain a peak at 4@9 but lose a peak at 1@3.

11 Orthogonal Exponentials for Bernoulli IFSs

235

1 is far from being identically 1 after going out 40 terms in the sum with 40 terms in each product. In contrast, the analogous approximation of the sum and product for = 1@4 from [20], for which there is an orthonormal basis, appears to be identically 1 after fewer than 40 terms in each. Second, Dutkay and Jorgensen have a conjecture [7, Conjecture 6.1] which implies that in the case 5 (1@2> 1) it is impossible to have an ONB for O2 ([ > ). In the conjecture, the existence of an ONB requires that a certain Hadamard duality condition is fulfilled. If the conjecture by Dutkay and Jorgensen is true, then none of the sets n (and therefore none of the sets n ) for 5 (1@2> 1) can possibly be ONBs. Figure 11.4 provides graphical evidence for this conjecture in the case of 1 when = 34 .

1

16 27

16 9

Fig. 11.3 (1 2 3 )\{1@3> 4@3> 16@3}: Equation (11.12) for w M [0> 2]. We gain a peak at 16@27 but lose peaks at 4@9 and 4@3.

1

1 9

2 9

Fig. 11.4 The first 40 terms of the sum

1 3

5 9

³ ³ 3 ´q ´ cos2 2 (c 3 w) = The elements of 4 q=0

40 X Y

cM1

4 9

1 are summed in increasing order: 0, 1@3, 4@3, 5@3, . . . , 1045@3.

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Acknowledgments The authors would like to express their appreciation to Professor Kathy Merrill and Professor Judy Packer for organizing the excellent conference Current Trends in Harmonic Analysis and Its Applications: Wavelets and Frames (agectionately known as C’estLarrybration) in honor of Larry Baggett. Two of us attended this conference, with support from the conference funds, and our discussions there motivated some of the early stages of this work. The first named author was supported in part by a grant from the National Science Foundation. Also, the first named author is pleased to acknowledge helpful discussions with Dorin Dutkay. The second two named authors wish to express their appreciation of Grinnell College’s contributing support for this work. At the conclusion of our work on this chapter, we were advised that existence results similar to those in Theorems 11.5 and 11.9 were found simultaneously and independently by Hu and Lau in [14].

References 1. Baggett, L. W., Carey, A. L., Moran, W. and Ramsay, A., Nonmonomial multiplier representations of Abelian groups, J. Funct. Anal. 97 (1991), no. 2, 361—372. 2. Baggett, L.W., Carey, A.L., Moran, W. and Ohring, P., General existence theorems for orthonormal wavelets, an abstract approach, Publ. Res. Inst. Math. Sci. 31 (1995) no. 1, 95—111. 3. Baggett, L. W., Medina, H. A. and Merrill, K. D., Generalized multi-resolution analyses and a construction procedure for all wavelet sets in Rq , J. Fourier Anal. Appl. 5 (1999) no. 6, 563—573. 4. Barge, M. and Diamond, B., Proximality in Pisot Tiling Spaces, Available at http://arxiv.org/abs/math.DS/0509051. 5. Dutkay, D. E. and Jorgensen, P. E. T., Wavelets on fractals, Rev. Mat. Iberoamericana 22 (2006), no. 1, 131—180. 6. Dutkay, D. E. and Jorgensen, P. E. T., Harmonic analysis and dynamics for ane iterated function systems, Houston J. Math. 33 (2007), no. 3, 877—905. 7. Dutkay, D. E. and Jorgensen, P. E. T., Analysis of orthogonality and of orbits in ane iterated function systems, Math. Z. 256 (2007), no. 4, 801—823. 8. Dutkay, D. E. and Jorgensen, P. E. T., Methods from multiscale theory and wavelets applied to nonlinear dynamics, Wavelets, multiscale systems and hypercomplex analysis, 87—126, Oper. Theory Adv. Appl., 167, Birkh¨ auser, Basel, 2006. 9. Dutkay, D. E. and Jorgensen, P. E. T., Iterated function systems, Ruelle operators, and invariant projective measures, Math. Comp. 75 (2006), no. 256, 1931—1970. 10. Erd˝ os, P., On a family of symmetric Bernoulli convolutions, American Journal of Mathematics 61(4) 1939, pp. 974—976. 11. Erd˝ os, P., On a family of symmetric Bernoulli convolutions. American Journal of Mathematics 62(1/4) 1940, pp. 180—186. 12. Folland, G. B., A course in abstract harmonic analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. 13. Fuglede, B., Commuting self-adjoint partial digerential operators and a group theoretic problem, J. Funct. Anal. 16 (1974), 101—121. 14. Hu, T.-Y. and Lau, K.-S., Spectral property of the Bernoulli convolutions, preprint, 2006. 15. Hutchinson, J. E., Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713—747. 16. Jessen, B. and Wintner, A., Distribution functions and the Riemann zeta function, Trans. Amer. Math. Soc. 38 (1935), no. 1, 48—88.

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17. Jorgensen, P. E. T., Analysis and probability: wavelets, signals, fractals, Graduate Texts in Mathematics, 234, Springer, New York, 2006. 18. Jorgensen, P. E. T., Kornelson, K. A., and Shuman, K. L., Erd˝ os measures and their higher dimensional analogues, preprint. 19. Jorgensen, P. E. T. and Pedersen, S., Harmonic analysis of fractal measures, Constr. Approx. 12 (1996), no. 1, 1—30. 20. Jorgensen, P. E.T. and Pedersen, S., Dense analytic subspaces in fractal O2 -spaces, Journal d’Analyse Math´ ematique 75 (1998), 185—228. 21. Kahane, J.-P., Some random series of functions, Second edition, Cambridge Studies in Advanced Mathematics, 5, Cambridge University Press, Cambridge, 1985. 22. Kovrizhkin, O., A version of the uncertainty principle for functions with lacunary Fourier transforms, J. Math. Anal. Appl. 288 (2003), no. 2, 606—633. 23. Marcus, M. B. and Pisier, G., Random Fourier series with applications to harmonic analysis, Ann. of Math. Studies, 101, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1981. P os problem), Ann. of Math. (2) 24. Solomyak, B., On the random series ±q (an Erd˝ 142 (1995), no. 3, 611—625. 25. Tao, T., Fuglede’s conjecture is false in 5 and higher dimensions, Math. Res. Lett. 11 (2004), no. 2—3, 251—258. 26. Triebel, H., Lacunary measures and self-similar probability measures in function spaces, Acta Math. Sin. (Engl. Ser.) 20 (2004), no. 4, 577—588.

Chapter 12

A Survey of Projective Multiresolution Analyses and a Projective Multiresolution Analysis Corresponding to the Quincunx Lattice Judith A. Packer Dedicated to Larry Baggett, colleague, mentor, and friend Abstract We give a survey of the concept of projective multiresolution analyses as introduced by M. Rieel and studied further by M. Rieel and the author. We give examples of projective multiresolution analyses corresponding µ ¶ 01 to the nondiagonal 2 × 2 integer dilation matrix that has determinant µ 2 0¶ 11 2> and also to the nondiagonal 2 × 2 matrix having determinant 2 1 1 related to the quincunx lattice. The method of construction follows that given by Rieel and the author in their earlier work but also poses new problems. In both examples given here, the one-dimensional initial F(T2 )-modules are not free, but the in the quincunx case, the one-dimensional wavelet module is free, whereas in the case corresponding to the dilation matrix whose determinant is negative, the one-dimensional wavelet module is not free either.

12.1 Introduction Let D be an q × q integer dilation matrix. In previous papers ([20], [17]), M. Rieel and the author, and then the author alone, studied the existence of projective multiresolution analyses in corresponding to dilation by certain q × q matrices D= In [20], the general abstract construction was given, and the examples of cases where D was a 2 × 2 diagonal matrix was emphasized, and in [17], examples of projective multiresolution analyses corresponding to diagonal dilation matrices with q 3 were studied in detail. A very important module used in both [20] and [17] was the module > where we recall that for q 5 N> the right F(Tq ) module is defined as the completion of Ff (Rq ) Judith A. Packer Department of Mathematics, University of Colorado at Boulder, Boulder, CO 80309-0395 e-mail: [email protected]

239

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Judith A. Packer

under the norm determined by the following F(Tq )-valued inner product: X h> iF(Tq ) (w) := ()(w s)= (12.1) s5Zq

for w 5 Rq = We recall further P that 5 if and only if is a bounded continuous function on Rq and s5Zq |(ws)|2 defines a continuous function on Tq = The right module action of F(Tq ) on is given by pointwise multiplication, and one can verify that Z |(w)|2 gw> (12.2) kh> iF(Tq ) k Rq

so that O2 (Rq )= The matrices studied in [20] and [17] were those that were either diagonal or similar to a diagonal matrix by an element V 5 JO(q> Z)= Our approach used the definition of projective multiresolution analysis for a general q × q dilation matrix given by M. Rieel and the author in [20]: Definition 12.1. Fix q 5 N> let D be a q × q integer dilation matrix, and let be the right-rigged Hilbert F(Tq ) module defined above. A sequence {Vm }m5Z of subspaces of is called a projective multiresolution analysis for dilation by D if: (i) V0 is a finitely generated projective F(Tq ) submodule of ; (ii) Vm = Gm (V0 )> ; m 5 N> where G is as defined below, (iii) S Vm Vm+1 > ; m 5 Z> 4 (iv) m=0 Vm is dense in > in the Hilbert F(Tq )-module topology, T4 (v) 4 Vm = {0}=

Here G is defined to be the Fourier transformed version of GD > G = F GD F > where the dilation operator GD is defined on O2 (Rq ) by GD ()({) = |det(D)1@2 | (D({))> 5 O2 (Rq )> and the Fourier transform F is defined by Z i (w)h(w · {) gw> F(i )({) =

(12.3)

(12.4)

Rq

where h is the exponential function defined on R by h(u) = h2lu > and i 5 O1 _ O2 (R2 )= Extend the Fourier transform to all of O2 (R2 ) in the usual fashion. An easy calculation shows that G = GE 1 >

12 Projective Multiresolution Analysis for the Quincunx Lattice

241

for E = Dw = It was shown in [20] that condition (y) in the definition above is implied by conditions (l) and (ll) ([20], Proposition 13), and Proposition 14 of [20] showed that if V were a submodule of satisfying conditions (l)> and if upon setting Vm = Gm (V)> the family {Vm } satisfied (ll) and (lll) of Definition 12.1, and if in addition there existed 5 V such that (0) 6= 0> then the family {Vm } would satisfy condition (ly) as well. Indeed, this last fact is very much related to the familiar condition on a scaling function ! 5 O2 (R)> ˆ ˆ that !(0) = 1 (so that !(0) 6= 0). In [17], following earlier work in [20], we were able to construct projective multiresolution analyses corresponding to dilation matrices in JO(q> Z) that were of the special type described in the first paragraph. In this chapter, we survey these earlier results, and remark how in the case where the 2 × 2 dilation matrix is diagonal, whether or not the “wavelet module” W0 = V1 ª V0 is a free F(T2 )-module depends only on the sign of the determinant. We conjecture that in the 2 × 2 nondiagonal case, if one starts with an “initial module” V0 that is not free, whether or not W0 is free depends only on the sign of the determinant as well. In the final two sections of the chapter, we study the two special cases of the 2×2 integer dilation matrices that are not similar to µ diagonal ¶ integer dilation 11 matrices. One of these matrices is the matrix T = associated to the 1 1 quincunx lattice, and the other is a 2 × 2 matrix S having determinant 2= It is interesting to note that the results in these two seemingly special cases mirror those in [20] and support our conjecture stated above; that is, in the projective multiresolution analyses we construct, if the corresponding dilation matrix has positive determinant, the wavelet module W0 is free even if the initial module V0 is not free, whereas in the case where the dilation matrix has a negative determinant, when the initial module V0 is not free, the wavelet module W0 is also not free. There are some technical diculties that need to be overcome in these cases that can be generalized to a wider setting, and hence we believe these examples are of some interest.

12.2 Projective Multiresolution Analyses: Some Motivation and a Summary of Known Results We first review the notion of ordinary multiresolution analyses as applied to wavelet theory, as developed by S. Mallat and Y. Meyer. Using the Fourier transform on this concept leads one naturally to consider the projective multiresolution analysis theory. We concentrate on the two-dimensional case here and in the rest of the chapter. We consider ordinary multiresolution analyses on O2 (R2 ) corresponding to dilation matrices 2 × 2 integer dilation matrices D> so that both of the eigenvalues of D have modulus greater than 1= We remark at this point that

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Judith A. Packer

thoughout the chapter, we shall denote closed subspaces of Hilbert spaces with ordinary font, rather than the calligraphic font that we have reserved to denote F -modules. Indeed, it will often be the case that Ym denotes the closure of a subspace Vm coming from a projective multiresolution analysis in the Hilbert space norm, or the Hilbert space closure of the range of Vm under the inverse Fourier transform. Definition 12.2. A singly generated multiresolution analysis (MRA) for dilation by D is a nested sequence {Yl }l5Z of closed subspaces of O2 (R2 ) satisfying the following conditions: (i) There exists ! 5 Y0 such that {Wn (!) : n 5 Z2 } is an orthonormal basis for Y0 > where Wn is translation by n 5 Z2 = (ii) Ym = (GD )m (Y0 ) for all m. (iii) Ym1 Ym for all m. S S4 2 2 2 2 (iv) T4 4 Ym = O (R ), i.e., 4 Ym is dense in O (R )= 4 (v) 4 Ym = {0}. Here Wn is the unitary operator defined by translation on O2 (R2 ) : Wn (i )(w) = i (w n)> n 5 Z2 > and GD is the dilation operator defined on O2 (R2 ) in Equation (12.3). The main aim of multiresolution analyses is to construct orthonormal wavelets: Definition 12.3. The set {#1 > #2 > · · · > #u } O2 (R2 ) is called an orthonormal wavelet family for dilation by the 2 × 2 matrix D if the set of functions [ [ m {GD Wn (#l : 1 l u} m5Z n5Z2

forms an orthonormal basis for O2 (R2 )= We also recall the definition of (normalized) low-pass filter function p0 in this setting, as it is crucial in the construction of the scaling function. Definition 12.4. Let D be a 2 × 2 dilation matrix with integer entries and suppose |det(D)| = g= A low-pass filter function p0 for dilation by D is a Z2 periodic function p0 : R2 : $ C that satisfies the conditions (i) p0 is continuously dierentiable at (0> 0) and p0 (0> 0) = 1= P g1 2 (ii) g1 m=0 |p0 ((v> w) + vj )| = 1> where {vj }m=0 is a collection of coset representatives in [Dw ]1 (Z2 ) for the g-element group [Dw ]1 (Z2 )@Z2 = (iii) p0 satisfies some form of Cohen’s criterion for low-pass filters as outlined in Equation (P’) of page 212 of [6], for example. Given a low-pass filter p0 defined as above, the scaling function ! can be defined as the inverse Fourier transform of the function

12 Projective Multiresolution Analysis for the Quincunx Lattice

({) =

4 Y

p0 ([Dw ]m ({))=

243

(12.5)

m=1

We remark that if one has a multiresolution analysis for dilation by D in the sense described in Definition 12.2, it is possible to find a wavelet family {#1 > #2 > · · · > #g1 } for g = |det(D)|> where {#1 > #2 > · · · > #g1 } Z0 = Y1 ª Y0 = We describe the procedure briefly. By means of the Fourier transform, one can transfer the MRAs discussed above over to the frequency domain, and one first builds wavelets there. This was a key insight behind M. Rieel’s initial ideas on projective multiresolution analyses. So, given an ordinary multiresolution analysis {Ym }4 m=4 > consider Y0 in the frequency domain via F, i.e., let us denote all the of the subspaces F(Ym ) by Ym also. Let = F(!)> and suppose that is associated to a lowpass filter p0 as defined above. Using the fact that g = |det(D)|> it is possible to find g 1 functions in O4 (T2 )> denoted by p1 > p2 · · · > pg1 > where {G(p1 )> G(p2 )> · · · > G(pg1 )} are the Fourier transforms of the wavelet family. Here G = F GD F 1 > and the original wavelet family can be recovered from the formulas {#l = F 1 (G(pl )) : 1 l g 1}=

(12.6)

Since F 1 (G(pl )) = F 1 F GD F 1 (pl F(!)) = GD (F 1 (pl ) !)> it is clear that our wavelet family is in the wavelet subspace Z0 = Y1 ª Y0 = This is the observation that motivated Rieel’s original ideas in 1997: under appropriate circumstances, a dense subspace V of the Fourier transform of the “initial space,” F(Y0 )> can be viewed as a free, singly generated projective F(T2 )-module, where we view T2 as R2 @Z2 = Since F(T2 ) has finitely generated projective modules that are not free, the first way to generalize the above construction is to take as the “initial” F(T2 )-module a non-free finitely generated projective F(T2 )-module, rather than a free one. Under the Fourier transform, the main operators we consider are the dilation in the frequency domain G (defined above) and translation conjugated by the Fourier transform, which turns into modulation in the frequency domain, Pn = F Wn F 1 > n 5 Z2 > which by standard calculation satisfies Pn (i )({) = i ({)h(n · {)> i 5 O2 (R2 )= The action of the operators {Pn : n 5 Z2 } generates a (right) action of the F -algebra F(T2 ) of continuous functions on the 2-torus, on a Hilbert module > which we define as follows:

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Definition 12.5. Let F(T2 ), considered as continuous functions on R2 that are periodic modulo the lattice Z2 > act on the right on Ff (R2 ) as follows: · i ({) = ({)i ({)> 5 Ff (R2 )> i 5 F(T2 )=

We define a norm on Ff (R2 ) by setting X kk2 = sup{5T |({ s)|2 = s5Z2

If we complete Ff (R2 ) in this norm, we obtain the Hilbert F(T2 )-module = Inequality (12.2) shows that O2 (R2 )= Recall in the classical MRA case, the initial space Y0 is formed by taking the closed linear span in O2 (R2 ) of translates by elements of Z2 of a scaling function != We have just observed that when we move to the frequency domain, the operation of taking integer translates of a function transforms into multiplying the Fourier transform of the scaling function by a complex exponential function. In the non-free case, our initial module V will be formed by taking pointwise multiples of a single function 5 > by elements of a certain finitely generated and projective F(T2 )-module. If our initial module is free and singly generated, we will just be multiplying the function > which plays the role played by the Fourier transform of the scaling function, by elements in F(T2 )> and the resulting family of functions is a singly generated free projective F(T2 )-module over itself. Before proceeding further, we consider the structure of the F(T2 )-module defined above. Proposition 12.6. Let be the F(T2 )-module defined in Definition 12.5. Then is equal to the set of bounded continuous functions on R2 for which there is some constant N A 0 such that X |({ s)|2 N> ;{ 5 R2 = s5Z2

The periodization of that appears on the left-hand side of the above equation is very familiar, both to wavelet theorists, and to those who construct equivalence bimodules. It also allows us to define the F(T2 )-valued inner product on > as follows: X ({ s)({ s)> { 5 R2 @Z2 = (12.7) h> iF(T2 ) ({) = s5Z2

The definition of projective multiresolution analysis, first due to M. Rieel and reviewed in Definition 12.1, specialized to the case q = 2> shows how one can take an increasing union of finitely generated F(T2 )-modules, each one formed from the one preceding it, by applying dilation, and construct as the norm closure of the union.

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We remind the reader of some of the redundancy inherent in Definition 12.1. In particular, it is not necessary to assume the separation condition (v): Assuming only parts (l) and (ll) of Definition 12.1, we obtain the following result: Proposition 12.7. ([20], Proposition 13) Let V be any finitely generated projective F(T2 )-submodule of > let D be a 2 × 2 integer dilation matrix with associated dilation operator G> and set Vm = Gm (V) for all m. Then T 4 4 Vm = {0}.

We note also that the subspaces Vm for m 0 will each themselves be F(T2 )modules, but the spaces Vm for m ? 0 will not be F(T2 )-modules, but only subspaces of = For the purposes of constructing frames in O2 (R2 )> however, all of the Vm are useful, for both positive and negative m= We move on to the problem of constructing nontrivial initial modules V= We first show that the density condition (ly) of Definition 12.1 holds under a natural condition that is closely related to the familiar condition on scaling ˆ functions that !(0) = 1.

Proposition 12.8. ([20], Proposition 14) Let V be a projective F(T2 )-submodule of that satisfies conditions (l)> (ll) and (lll) of Definition 12.1. If there is at least one 5 V such that (0) 6= 0, then V satisfies condition (ly) of Definition 12.1. The proof of this result is fairly technical; however it is an extremely convenient result to use, because the main hypothesis is fairly easy to verify in most of our examples. Let Am = F(R2 @Dm (Z2 )) for m 5 Z= Given a projective multiresolution analysis {Vm } for dilation by the diagonal dilation matrix D, note that for m 0> and n m> each Vn is a Am module. Moreover, we observe that the dilation operator G = FGD F 1 carries Vm onto Vm+1 = Let Wm be the orthogonal complement of Vm in Vm+1 viewed as a Am -module. Definition 12.9. The A0 = F(T2 )-module W0 constructed as above is called the wavelet module for dilation by D associated to the projective multiresolution analysis {Vm }= From now on, we denote our initial module by V . In straightforward cases, V will coincide with the set !ˆ · F(T2 ), for a suitably chosen scaling ˆ For function !, i.e., it will be a free F(T2 )-module with one generator, = != example, if one has an ordinary multiresolution analysis in the time domain generated by one scaling function ! whose Fourier transform is an element of > then taking V = { · j : j 5 F(T2 )} > and we obtain the desired projective multiresolution analysis. Given an ordinary multiresolution analysis in the time domain where the initial space Y0 has more than one scaling function {!1 > !2 > · · · > !u } all of whose Fourier transforms lie in > using the same procedure as above, we can construct a free F(T2 )-module V with u generators. This module approach

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has proved useful in studying various problems in ordinary wavelet theory. However, one of the aims of this chapter is to give a survey of methods of constructing projective multiresolution analyses whose initial modules V are not free. For any finitely generated projective F(T2 )-module, and for any diagonal dilation matrix ¶ µ g1 0 > g1 > g2 5 Z> |gl | A 1> D = 0 g2 Rieel and the author have shown that it is possible to construct an embedding of this module into so as to construct a projective multiresolution analysis for dilation by D= We review the standard construction non-free finitely generated projective F(T2 )-modules, as described by Rieel in [22]: Example 12.10. For t 5 N and d 5 Z> let [(t> d) denote the right F(T2 )module consisting of the space of continuous complex-valued functions k on T × R that satisfy k(v> w t) = h(dv)k(v> w)> (12.8) with module action given by k · I (v> w) = k(v> w)I (v> w)>

(12.9)

for k 5 [(t> d) and I 5 F(T2 )= Then [(t> d) is a finitely generated, projective F(T2 )-module, of rank t and twist d= There is a F(T2 )-valued inner product defined on [(t> d) that is compatible with the right F(T2 )-action defined by hk1 > k2 iF(T2 )(v>w) =

t1 X

k1 (v> w n)k2 (v> w n)>

(12.10)

n=0

k1 > k2 5 [(t> d)= The module [(t> d) is closed in the norm induced by this inner product. Moreover, Rieel has proved the following essential fact concerning his modules {[(t> d) : t 5 N> d 5 Z} : Proposition 12.11. For t 5 N and d 5 Z> let [(t> d) denote the right F(T2 )-module defined above. Then [(t> d) is a finitely generated, projective F(T2 )-module. The set {[(t> d) : t 5 N> d 5 Z} parameterizes the isomorphism classes of finitely generated projective F(T2 )-modules, in the sense that if [ is a finitely generated projective F(T2 )-module, there exist unique values of t and d such that [ = [(t> d)= Proof. For the proof of this result, refer to [22], Theorem 3.9. The additive structure of the {[(t> d)} is as follows: given t1 > t2 5 N and d1 > d2 5 Z>

12 Projective Multiresolution Analysis for the Quincunx Lattice

[(t1 > d1 ) [(t2 > d2 ) = [(t1 + t2 > d1 + d2 )=

247

(12.11)

Using these results, Rieel was able to explicitly describe the isomorphism between N0 (F(T2 )) and Z2 = {(t> d) : t> d 5 Z} ([22]). It was also shown in [22] that with respect to the above parameterization, N0 (F(T2 ))+ > the positive cone of N0 (F(T2 ))> is equal to {[[(t> d)] : (t> d) 5 N × Z} ^ {[0> 0]} Z2 = Moreover, Rieel showed that cancellation holds for finitely generated projective F(T2 )-modules, as follows: if [(t> d) and [(t 0 > d0 ) are given, and if W is a finitely generated projective F(T2 )-module such that [(t> d) W = [(t 0 > d0 )> then W and [(t 0 t> d0 d) are isomorphic as F(T2 )-modules. We now review the main construction of [20]: we want to take any one of the F(T2 )-modules [(t> d) described above and construct a F(T2 )-module monomorphism R : [(t> d) $ > so as to obtain the initial module of a projective multiresolution analysis for dilation by a diagonal dilation matrix. Theorem 12.12. ([20], Theorem 4) Let g1 > g2 be integers whose absolute value is greater than one, and let ¶ µ g1 0 = D= 0 g2 Let G denote the operator of dilation by D in the frequency domain, i.e., G = F GD F 1 = For every positive integer t and each d 5 Z there is a F(T2 )-module monomorphism R : [(t> d) $ that satisfies (1) hR(k1 )> R(k2 )iF(T2 ) = hk1 > k2 iF(T2 ) > for all k1 k2 5 [(t> d)> and (2) R([(t> d)) G(R([(t> d))= Proof. We give only a brief sketch of the proof. The key idea, due to M. Rieel, is to find construct an appropriate function 5 and then define R(k) = k(v> w)(v> w)> k 5 [(t> d)=

(12.12)

Conditions (1) and (2) impose restrictions on the choice of the function ; here to simplify the description, we restrict ourselves to the study of t = 1> d = 1= The more general conditions are similar, up to change of scale. In this case, one checks that in order that condition (1) be satisfied, it is necessary to have X |((v> w) s)|2 = 1> s5Z2

i.e., h> iF(T2 ) 1= Then, using calculations with the dilation operator G> one checks that if one wants (2) to hold, there must exist p e 0 5 [(1> (1 g1 g2 )) such that e 0 (v> w)(v> w)> (g1 v> g2 w) = p

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(again this looks like ordinary wavelet theory, except for the condition p e0 5 [(1> (1 g1 g2 )) and hence, if one wants (1) and (2) to hold simultaneously, e 0 (0> 0) = 1 and the function p e 0 must satisfy p X

X

m5{0>1···|g1 |1} n5{0>1···|g2 |1}

¯ µ ¶¯2 ¯ m n ¯¯ ¯p ¯ e 0 v + g1 > w + g2 ¯ = 1=

We do this by finding an ordinary “low-pass” filter p0 > and then multiplying p0 by a discontinuous jump function of constant modulus one so that the product will be an element p e 0 of [(1> (1g1 g2 ))= The points of discontinuity of the jump function will come along the line w = 1@2 (where p0 is equal to 0=) The function is constructed in a familiar way: µ ¶ v w 4 > (12.13) (v> w) := m=1 p e0 > gm1 gm2

e0 and one can choose p e 0 so that lies inside = Indeed one can choose p related to the filters of Y. Meyer so that will be continuous and have compact support, or, one can choose p e 0 similar to Haar filters, and the paragraphs following Remark 2.2 of [17] show that will lie in in this case as well. Proposition 12.8 then guarantees that setting V = R([(t> d))> we obtain the initial module of a projective multiresolution analysis. Corollary 12.13. ([20], Theorem 6) Let D be the diagonal 2 × 2 integer dilation matrix and R : [(t> d) $ be the module map given in the statement of Theorem 12.12. Let V = R([(t> d)) and define Vm = Gm (V)> m 5 Z= Then the family {Vm : m 5 Z} is a projective multiresolution analysis for dilation by D with initial module V0 = [(t> d)= The modules {Vm : m 0} will all be finitely generated projective F(T2 )-modules. In addition, it is possible to write = V l0 Wl (topologically)> where, as before, Wl = Vl+1 ª Vl = We now discuss the structure of W0 = V1 ª V0 > as in ordinary wavelet theory. In fact, we will compute the isomorphism class of W0 as a F(T2 )-module using structure results about F(T2 )-modules described above. (The isomorphism class of Wl can be determined from knowledge about W0 =)

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249

Theorem 12.14. ([20], Theorem 8) Let the 2 × 2 diagonal dilation matrix D with diagonal entries g1 and g2 > and the module map R : [(t> d) $ be as in the statement of Theorem 12.12, and let {Vm : m 5 Z} be the projective multiresolution analysis with initial module V0 isomorphic to [(t> d) discussed in Corollary 12.13. Let W0 denote the wavelet module V1 ª V0 = Then W0 is isomorphic to [((|det(D)| 1)t> (sign(det(D)) 1)d) as a F(T2 )-module. In particular, if det(D) is positive, then W0 is a free module of dimension (|det(D)| 1)t= For the proof, we refer the reader to [20], where detailed calculations allow one to prove that V1 = [(|det(D)|t> sign(det(D))d)= Cancellation of finitely generated projective F(T2 )-modules is then used to compute the isomorphism class of the wavelet module. In [17], the above construction is generalized to give results on diagonal dilation matrices acting on Rq for q 3= We can still form the Hilbert F(Tq )module by completing Ff (Rq ) in the appropriate Hilbert module norm, and we construct projective multiresolution analyses in q = For q 5> one can no longer use cancellation to calculate the isomorphism class of the wavelet module, but3if q = 4 3 or q = 4> cancellation still holds. For example, let q = 3, 200 and D = C 0 2 0 D = In this case, a projective multiresolution analysis was 002 constructed with both initial module and wavelet module non-free in [17], even though the determinant of D is positive. It should be noted that I. Raeburn has mentioned to the author that N. Larsen and he have developed an alterntive approach to the theory of projective multiresolution analyses from the “direct limit of Hilbert F -modules” point of view, just as they developed the theory of ordinary multiresolution analyses from direct limits of Hilbert space ([12]). It will be interesting to consider any new developments that arise from their approach; Larsen and Raeburn have recently posted a preprint on the ArXiv ([13]). Remark 12.15. For completeness, we briefly discuss the relationship between projective multiresolution analyses and the “multiplicity function” p first constructed by L. Baggett, H. Medina, and K. Merrill in [1]. Given an increasing nested sequence of closed subspaces {Yq }q5Z in O2 (Rq )> if Y0 is invariant under the unitary representation of Zq on O2 (Rq ) induced by translation, and if under the dilation operator G corresponding toTa q × q dilation matrix S D one has Yq = Gq (Y0 )> q5Z Yq = O2 (Rq ) and q5Z Yq = {0}> we say we have a generalized multiresolution analysis with core subspace Y0 = Using the spectral theory of M. Stone and G. Mackey, the unitary representation of Zq on Y0 both gives rise to and is completely determined by a multiplicity cq function p : Z = Tq $ {0> 1> 2> = = = > 4}> which, roughly speaking, counts

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the number of times each character " 5 Tq occurs in the representation of Zq on Y0 = Given a projective multiresolution analysis {Vm }m5Z of subspaces of with respect to the q×q dilation matrix D in the sense of Definition 12.1, one easily checks by use of the inverse Fourier transform that one obtains a generalized multiresolution analysis in the sense defined above by setting Ym = F 1 (Vm )> for all m 5 Z= It is of interest to determine what the multiplicity function p will be corresponding to the representation of Zq on Y0 = F 1 (V0 )= Indeed, by the discussion in [3], 1.7.1, one can find a projection (self-adjoint idempotent) S 5 P (u> F(Tq )) such that V0 is isomorphic as a projective module to the module (H) of continuous sections from Tq into H> where (H> 1 > Tq ) is the vector bundle constructed from S as follows: H = {(z> y) : z 5 Tq > y 5 S (z)[Cu ]}> and 1 : H $ Tq is projection in the first variable. Since Tq is connected, H will have constant dimension t for some positive integer t u= It follows that V0 can be viewed as a subspace of the Hilbert space O2 (Tq > Cu )= If we now close up (H) under the Hilbert space norm Z 2 ko(z)k2 gz> kok = Tq

we obtain the Hilbert space of all square-integrable Borel cross-sections from Tq into H, which we denote by E (H), i.e., Y0 = E (H) can be identified with {square-integrable Borel i : Tq $ H : 2 (i (z)) 5 rangeS (z)> ; z 5 Tq }= This construction is independent of the projection operator S chosen up to conjugacy by a unitary operator. It follows from facts about the Fourier transform that translation by v in Y0 O2 (Rq ) corresponds to multiplication by a complex exponential function on E (H)= Since the fibers (1 )1 (") have the same dimension t for every " 5 Tq > it follows that the multiplicity of " is equal to t> for every " 5 Tq = Thus our projective multiresolution analyses give rise to generalized multiresolution analyses with multiplicity functions that are everywhere constant. There do exist generalized multiresolution analyses with nonconstant multiplicity functions; however, the discussion above shows that the topological obstructions that give rise to nontrivial vector bundles disappear when we take the closure in the Hilbert space setting, and that the more unusual nonconstant multiplicity functions will not arise from the projective multiresolution analysis setting. This is not all that surprising, as our complex vector bundles over Tq will be Borel isomorphic to Cartesian product bundles of the form Tq × Ct for some integer t 1= To date, known results about projective multiresolution analyses from [20] and [17] concern diagonal matrices with integer entries, and their conjugates

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251

by elements of JO(q> Z)= In the next sections, we discuss two dierent examples where the dilation matrices are not similar to diagonal matrices with integer entries.

12.3 A Projective Multiresolution Analysis for a Nondiagonalizable Matrix Having Determinant 2 We now construct a projective multiresolution analysis for the matrix S> which is not similar via an element of JO(2> Z) to a diagonal matrix with integer entries. Therefore, such a matrix cannot be studied using the methods of Section 4 of [17]. µ ¶ 01 We consider the matrix given by S = = This matrix has eigenvalues 20 s s 2l and 2l> so is a dilation matrix, and cannot be similar by an element of VO(2> Z) to a diagonal matrix. Since the determinant of S is equal to 2> if we tried to generalize the results of Theorem 7 of [20], and construct a projective multiresolution analysis in O2 (R2 ) with initial module V0 isomorphic to [(1> 1)> the conjecture that was stated in the last paragraph of the Introduction would be that the dilated module G(V0 ) = V1 would be isomorphic to [(2> 1)> so that by cancellation, the wavelet module would be isomorphic to [(1> 2) = [(1> sign(det(S )) 1)= We shall verify the conjecture in this case by constructing the projective multiresolution analysis and computing the class in N0 (F(T2 )) of the wavelet module W0 . Although the methods of [20] and [17] cannot be applied directly, we will use some of the same general principles: first we will construct the filter functions, corresponding to an ordinary multiresolution analysis constructed for the matrix S , and then we will modify these filter functions so as to construct the desired embedding of a non-free finitely generated projective F(T2 )-module into = This will give us the initial space V0 in the desired projective multiresolution analysis. Some of the techniques employed in constructing the isomorphisms that follow in this section and in Section 12.4 have been used before in a previous paper by the author dealing with the construction of projective modules for F -algebras related to the discrete Heisenberg group [16]. We first construct a low-pass filter for dilation by S as defined in Definition 12.4. Since µ ¶ 1 02 > [S w ]1 = 2 10

we choose for our coset representatives of the two-element group [S w ]1 (Z2 )@Z2 the elements in the set {(0> 0)> (0> 1@2)}= Thus if we take any of the standard low-pass filter functions of one variable w corresponding to dilation on by 2 and extend this to a function, denoted

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by p0 > defined on R2 by letting it be constant in the v variable, one obtains an ordinary scaling function ! corresponding to dilation by S defined in the usual way by 4 Y ˆ w) = (v> w) = p0 ([S w ]m (v> w))= !(v> m=1

Thus, for example, we can take the low-pass filter corresponding to the Haar wavelet 1 p0 (v> w) = (1 + h(w))> 2 or any other low-pass filter in the variable w, e.g., we could take a low-pass filter in the w variable giving rising to a Meyer wavelet, whose scaling function is compactly supported in the frequency domain. Now as in Equation (12.12), we want to find a function 5 and then set (12.14) R (k) = k · > k 5 [(1> 1)> and we want to choose in such a way that (1) hR (k1 )> R (k2 )iF(T2 ) = hk1 > k2 iF(T2 ) > for all k1 > k2 5 [(1> 1)> and (2) R ([(1> 1)) G[R ([(1> 1))]> where µ ¶now G denotes the Fourier transformed version of GS > for S = 01 = We denote these maps with “” superscripts, as the matrix S un20 der consideration has negative determinant 2> and in the next section we shall carry out a similar exercise for a matrix T corresponding to the quincunx lattice that has positive determinant 2; the maps in that section will be decorated with “+” superscripts. One checks that, as in Theorem 4 of [20], in order to have (1) it is necessary and sucient to have h > iF(T2 ) = 1> where, as in Equation (12.13), we will set

(v> w) =

4 Y

m=1

p e 0 ([S w ]m (v> w))>

(12.15)

for an appropriately chosen function p e 0= Recall from Section 4 of [20] the jump function Mt>d> : R2 $ C defined by Mt>d> (v> w) = h(mdv) for +mt w ? +(m +1)t> t> d> m 5 Z> A 0= (12.16)

We have Mt>d> (0> 0) = 1 and Mt>d> (v> w t) = h(dv)Mt>d> (v> w)= (This formula is slightly dierent from that given in [20], Section 4, but a slight change in subscripts in formulas involving the jump functions in [20] is all that is needed to modify the proofs there.) Note that Mt>d> has jump discon-

12 Projective Multiresolution Analysis for the Quincunx Lattice

253

tinuities in the w-variable along the lines w = + mt> m 5 Z> but is continuous elsewhere. Hence, if one multiplies Mt>d> by a function in the variables v and w that vanishes along all of those lines, the resulting product function will be continuous in v and w simultaneously. Calculations lead us to choose p e 0 (v> w) = h(2vw)M1>3>1@2 (v> w)p0 (w)>

(12.17)

where the jump function M1>3>1@2 (v> w) is defined as stated above: M1>3>1@2 (v> w) = h(3mv) for m1@2 w ? m+1@2> m 5 Z= As before, one checks that M1>3>1@2 (0> 0) = 1 and M1>3>1@2 (v> w 1) = h(3v)M1>3>1@2 (v> w)> so that (v> w) = Q 4 e 0 ([S w ]m (v> w))> and m=1 p (2w> v) = ([S w ]1 (v> w)) = p e 0 (v> w) (v> w)=

Then, using the fact that G = G[S w ]1 > we see that in order for (2) to hold, it is necessary that GS w (R ([(1> 1))) R ([(1> 1))>

(12.18)

that is, s { 2 (2w> v)k(2w> v) : k 5 [(1> 1)} { (v> w)i (v> w) : i 5 [(1> 1)}= Thus, given k 5 [(1> 1)> we want to find j 5 [(1> 1) such that s 2 (2w> v)k(2w> v) = (v> w)j(v> w)= But s

s 2p e 0 (v> w) (v> w)k(2w> v) s e 0 (v> w)k(2w> v)]= = (v> w)[ 2p

2 (2w> v)k(2w> v) =

Thus we take the obvious choice j(v> w) =

s 2p e 0 (v> w)k(2w> v)>

and we now show that j as defined above is in [(1> 1)= We first note s

2p e 0 (v 1> w)k(2w> v 1) s = h(2w)h(2w) 2h(2vw)M1>3>1@2 (v> w)p0 (w)h(2w)k(2w> v) = j(v> w)>

j(v 1> w) =

so that j is Z-periodic in the v-variable. Now we check what happens under Z-translation in the w-variable.

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s j(v> w 1) = 2p e 0 (v> w 1)k(2(w 1)> v) s = 2h(2v)h(3v)M1>3>1@2 (v> w)p0 (w)k(2w> v) = h(v)j(v> w)= Therefore, j 5 [(1> 1)> and we have shown that the inclusion (12.18) holds, so that Condition (2) is verified. The above results have proved most of the following: Theorem 12.16. There is a F(T2 )-module monomorphism R : [(1> 1) $ that satisfies (1) hR (k1 )> R (k2 )iF(T2 ) = hk1 > k2 iF(T2 ) > for all k1 > k2 5 [(1> 1)> and (2) R ([(1> 1)) G[R ([(1> 1))]> where G denotes the Fourier transformed version µ of G ¶S > and S is the matrix 01 of determinant 2 discussed above, i.e., S = = Furthermore, setting 20 Vm = Gm (R ([(1> 1))) for m 0> the resulting family of subspaces {Vm }m5Z of is a projective multiresolution analysis for dilation by T with initial module V0 isomorphic to [(1> 1)= Proof. All we have left to show is that {Vm = Gm (R ([(1> 1)))}4 m=4 is a projective multiresolution analysis in the sense of [20]. But this follows from Proposition 12.8, since obviously R ([(1> 1)) contains an element with (0> 0) 6= 0= We now consider V1 = G(R ([(1> 1))) = G[S w ]1 (R ([(1> 1))) n o = s12 (w> v@2)k(w> v@2) : k 5 [(1> 1) =

We will show there is an isomorphism between V1 and [(2> 1)> thus giving an example where Theorem 12.14 generalizes to a nondiagonal matrix, supporting our conjecture as stated at the end of the introduction of this chapter. We define the map : V1 = G(R ([(1> 1))) $ [(2> 1) as follows:

³

s1 ( 2

´ · k) [S w ]1 (v> w)

= u(v> w)k(w> v@2) + u(v 1> w)k(w> (v 1)@2)> (12.19)

where u : R2 $ C is a continuous function to be determined. In order that ( s12 ( · k) [S w ]1 )(v> w) 5 [(2> 1)> we need to have Z-periodicity in the v variable, i.e., we need ³ ´ ³ ´ s12 ( · k) [S w ]1 (v1> w) = s12 ( · k) [S w ]1 (v> w)> (12.20) and we need a functional equation satisfied in the w-variable:

12 Projective Multiresolution Analysis for the Quincunx Lattice

³

s1 ( 2

255

´

· k) [S w ]1 (v> w 2) ³ ´ = h(v) s12 ( · k) [S w ]1 (v> w)> (12.21)

One checks that Equation (12.20) will hold only when

u(v 2> w) = h(w)u(v> w);v> w 5 R>

(12.22)

and Equation (12.21) will hold if and only if u(v> w 2) = h(v)u(v> w) ;v> w 5 R=

(12.23)

Thus, we want to find continuous u : R2 $ C that satisfy the two conditions given in Equations (12.22) and (12.23). A simple calculation shows that if we take u(v> w) =

1 h(vw@2)> 2

(12.24)

Equations (12.22) and (12.23) will be satisfied. The constant multiple 12 has been introduced to make future inner product calculations work out correctly. Thus we define ³ ´ s12 ( · k) [S w ]1 (v> w) ³ ¶¸ µ v´ h(vw@2) v1 k w> > (12.25) = + h(w@2)k w> 2 2 2 for all k 5 [(1> 1)> and we have shown that the image of is contained in [(2> 1)> as desired. We now are ready to complete the proof of the following: Theorem 12.17. Let V0 = R ([(1> 1)) be the projective F(T2 ) submodule of isomorphic to [(1> 1) constructed in Theorem 12.16, so that V1 = G(V0 ) V0 > µ ¶ 01 for the dilation corresponding to S = . Then V1 = [(2> 1), so that 20 W0 = V1 ª V0 = [(1> 2)= i.e., W0 is a finitely generated projective F(T2 )-module of dimension 1 and twist 2= Proof. We have constructed a map of V1 = G(R ([(1> 1))) into [(2> 1) above, and clearly is a right F(T2 )-module homomorphism. We need to show that this map preserves the F(T2 )-valued inner products and is bijective. We consider the inner product in V1 viewed as a submodule of first:

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hG(R (k1 ))> G(R (k2 ))iF(T2 ) (v> w) 1 X = ( · k1 ) [S w ]1 ((v> w) (p> q))( · k2 ) [S w ]1 ((v> w) (p> q)) 2 2 (p>q)5Z

=

1 2

X

¯ ³³ v ´ ´¯2 ´ ³³ v ´ ¯ ¯ w> (p0 > q0 ) ¯ k1 · k2 w> (p0 > q0 ) = ¯ 2 2 2

(p0 >q0 )5[S w ]1 (Z )

We break this sum up into two terms, using the two coset representatives (0> 0) and (0> 12 ) for the two-element quotient group (S w )1 (Z2 )@(Z2 )> obtaining as our inner product ¶¶ µ³ µ ³ v´ 1 v´ 1 1 h · k1 > · k2 iF(T2 ) w> > + h · k1 > · k2 iF(T2 ) w> + 0> 2 2 2 2 2 which is equal to " ¶ µ ¶# µ ³ v´ ³ v´ v1 v1 1 k2 w> k1 w> k2 w> + k1 w> = 2 2 2 2 2

(12.26)

We now consider the inner product of (G(R (k1 ))) and (G(R (k2 ))) in [(2> 1)= Recall if i1 > i2 5 [(2> 1)> the inner product is given by hi1 > i2 iF(T2 ) (v> w) =

1 X

i1 (v> w m)i2 (v> w m)=

m=0

Thus ´ ´E ³ D ³ s12 ( · k1 ) > s12 ( · k1 )

³ v´ w> 2 F(T2 )

¶¶ µ 1 µ ³ v1 v´ 1X h(v(w m)@2)k1 w m> + h((v 1)(w m)@2)k1 w m> 4 m=0 2 2 ¶¶ µ µ ³ v1 v´ × h(v(w m)@2)k2 w m> + h((v 1)(w m)@2)k2 w m> 2 2 ¶ ¶¸ µ µ ³ ³ ´ ´ 1 v1 v v1 v (k1 · k2 ) w> + h(w@2)k1 w> k2 w> = + (k1 · k2 ) w> 4 2 2 2 2 ¶ µ ¸ ³ 1 v1 v´ h(w@2)(k1 · k2 ) w> + (k1 · k2 ) w 1> + 4 2 2 ¶ ¶¸ µ µ ³ v1 v1 v´ 1 (k1 · k2 ) w> + h((w 1)@2)k1 w 1> k2 w 1> + 4 2 2 2 ¶ µ 1 v1 > + h((w 1)@2)(k1 · k2 ) w 1> 4 2 =

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which, using the fact that k is Z-periodic in the first variable, and the fact that h(1@2) = h(1@2) = 1> gives us ¶¸ µ ³ v´ v1 1 (k1 · k2 ) w> = (12.27) + (k1 · k2 ) w> 2 2 2 By comparing the quantities in Expressions (12.26) and (12.27), it may be observed that we have shown hG(R (k1 ))> G(R (k2 ))iF(T2 ) = h (G(R (k1 )))> (G(R (k2 )))iF(T2 ) > (12.28) so that preserves the F(T2 )-valued inner product. Finally, we need to show that is surjective. Define a map : [(2> 1) $ Fe (T> R) by

(j)(v> w) = h((vw))j(2w> v) + h((v 1)w)j(2w> v 1)= Typical calculations show that in fact : [(2> 1) $ [(1> 1)> and that ³³ ´ ¡ ¢´ s1 [S w ]1 · (j) [S w ]1 (v> w) = j(v> w)> (12.29) 2

and since j was an arbitrary element of [(2> 1), we have shown that is surjective. The fact that W0 is isomorphic to [(1> 2) follows directly from our identification of V1 and the cancellation property for finitely generated projective F(T2 )-modules.

12.4 A Projective Multiresolution Analysis for the Matrix Associated to the Quincunx Lattice µ

¶ 01 The matrix discussed in the previous section, S = , is of interest in 20 that it has integer determinant 2 yet is not similar via a matrix with integer entries diagonal matrix. We now consider another matrix of this type, µ to a ¶ 11 T= = The matrix T is also not similar via an element of JO(2> Z) to 1 1 a diagonal matrix with integer entries, so it too cannot be studied using the methods of Section 4 of [17]. We will construct a projective multiresolution analysis corresponding to dilation by the unitary operator constructed from T by means similar to those used in Section 12.3. Recall the quincunx lattice refers to the lattice in R2 formed from the columns of the 2 × 2 dilation matrix

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T=

µ

11 1 1

¶

>

¶ µ ¶ 1 1 i.e., the lattice in R with basis vectors q1 = and q2 = = The 1 1 eigenvalues of T are 1 ± l> so that T cannot be similar to a matrix with integral diagonal entries via an element of JO(2> C)> thus certainly not through an element of VO(2> Z)= We now construct a projective multiresolution analysis in O2 (R2 ) corresponding to this dilation matrix with V0 = [(1> 1)> and calculate that the corresponding wavelet module W0 is isomorphic to [(1> 0) = F(T2 )= Although the methods of [20] and [17] cannot be applied directly, we will use some of the same general principles that were used in the previous section. First we will construct the filter functions, or masks, corresponding to an ordinary multiresolution analysis coming from the quincunx dilation matrix, and then we will modify these filter functions so as to construct the desired embedding of a non-free finitely generated projective F(T2 )-module into = This will give us the initial space V0 in the projective multiresolution analysis that we want. We first construct a low-pass filter for dilation by T as defined in Definition 12.4. Since µ ¶ 1 11 w 1 > = [T ] 2 1 1 2

µ

we choose for our coset representatives of [Tw ]1 (Z2 )@Z2 the two element set {(0> 0)> (1@2> 1@2)}= Then the equation in (ll) of Definition 12.4 becomes |p0 (v> w)|2 + |p0 (v + 1@2> w 1@2)|2 = 1>

and Cohen has verified on p. 213 of [6] that if we take any of the standard low-pass filter functions of one variable v corresponding to dilation on by 2 and extend this to a function, denoted by p0 > defined on R2 by letting it be constant in the w-variable, one obtains an ordinary scaling function ! corresponding to dilation by T defined in the usual way by ˆ w) = (v> w) = !(v>

4 Y

p0 ([Tw ]m (v> w))=

m=1

Thus, as in Section 12.3, we can take the low-pass filter corresponding to the Haar wavelet 1 p0 (v> w) = (1 + h(v))> 2 or any other low-pass filter in the variable v whose corresponding infinite product lies in = The most important fact in what follows is that the function p0 vanishes all along the line v = 12 = As in the previous section, we want to find a function + : R2 $ C such that the map

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259

R+ : [(1> 1) $ defined by R+ (k) = + · k> k 5 [(1> 1)>

preserves the F(T2 )-module structure and the F(T2 )-valued inner product. As mentioned in the previous section, we denote these maps with “+” superscripts, as the matrix corresponding to the quincunx lattice has positive determinant +2= Keeping close to our notation of the previous section, we will denote the image of R+ > R+ ([(1> 1))> by V0 > and since we want to have V0 G(V0 )> the usual calculations involving Fourier transforms lead us to choose + so that GTw (V0 ) V0 = Here, as in the discussion following Definition 12.1, G denotes the Fourier transformed version of GT = We look for conditions on the function + that will guarantee that + (v w> v + w)i (v w> v + w) 5 + · [(1> 1)> ; i 5 [(1> 1)= As in [20], we choose + so that it satisfies a dilation equation, this time the equation e 0 (v> w) + (v> w)> + (v w> v + w) = p

where p e 0 is a modified form of the low-pass filter p0 multiplied by jump functions of various sorts that have constant modulus one on R= We state the theorem:

Theorem 12.18. There is a F(T2 )-module monomorphism R+ : [(1> 1) $ that satisfies (10 ) hR+ (k1 )> R+ (k2 )iF(T2 ) = hk1 > k2 iF(T2 ) > for all k1 > k2 5 [(1> 1)> and (20 ) R+ ([(1> 1)) G[R+ ([(1> 1))]> where G denotes the Fourier transformed version µ of G ¶T > and T is the matrix 11 associated to the quincunx lattice, i.e., T = = Furthermore, setting 1 1 Vl = Gl (R+ ([(1> 1))) for l 5 Z> the resulting family of subspaces {Vl }l5Z of is a projective multiresolution analysis for dilation by T with initial module V0 isomorphic to [(1> 1)= Proof. As mentioned above, we need to find an appropriate function + 5 and then set R+ (k) = k · + > k 5 [(1> 1)=

As in the previous section, in order to have (10 ), it is necessary and sucient to have

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h + > + iF(T2 ) = 1=

(12.30)

GTw (R+ ([(1> 1))) R+ ([(1> 1))>

(12.31)

Then, using the fact that G = G[Tw ]1 > we see that in order for (20 ) to hold, it is necessary that

so that s { 2+ (v w> v + w)k(v w> v + w) : k 5 [(1> 1)}

{ + (v> w)i (v> w) : i 5 [(1> 1)}=

If k 5 [(1> 1)> then k will satisfy the two equalities k(v 1 w> v 1 + w) = h(v w)k(v w> v + w)> and k(v (w 1)> v + (w 1)) = h(v w)k(v w> v + w)= A routine calculation shows that a sucient condition for s j(v> w) := 2 + (v w> v + w)k(v w> v + w) to be an element of R+ ([(1> 1)) for every k 5 [(1> 1) is that there should exist p e 0 : R2 $ C such that and satisfying

e 0 (v> w) p e 0 (v> w 1) = h(w)p

e 0 (v> w) p e 0 (v 1> w) = h(w v)p

(12.32)

(12.33)

+ (v w> v + w) = p e 0 (v> w) + (v> w)=

Calculations similar to those done in [20] show that in order for (10 ) to hold, it is necessary and sucient that p e 0 (0> 0) = 1> e 0 (v + 1@2> w + 1@2)|2 = 1> |p e 0 (v> w)|2 + |p

and the filter p e 0 satisfy some form of Cohen’s nonvanishing condition around e 0 will the origin in R2 = Keep in mind that our proposed candidate for p be a function of modulus one times any of the low-pass filter functions p0 for dilation by 2> which were shown by Cohen to satisfy his nonvanishing condition, so that p e 0 will certainly satisfy the nonvanishing condition also. Recall from Equation (12.16) that the jump function M1>1>1@2 (v> w) is given by M1>1>1@2 (v> w) = h(mv) for m 1@2 w ? m + 1@2> m 5 Z= We know that

12 Projective Multiresolution Analysis for the Quincunx Lattice

261

M1>1>1@2 (0> 0) = 1 and M1>1>1@2 (v> w 1) = h(v)M1>1>1@2 (v> w)= Define

e w) = M1>1>1@2 (w v> v)= M(v>

e 1> w) = h(w v)M(v> e w) and Then standard calculations show that M(v e e ˜ M(v> w 1) = M(v> w)= The function M has discontinuities along the line v = ± 12 = We now use a technique similar to one used in Lemma 1.3 of [16] and define ¶ µ [w2 + w] = (12.34) t(w) = h 2 Observe that t(0) = 1 and t(w 1) = h(w)t(w)= Finally, set e w)p0 (v)> p e 0 (v> w) = t(w)M(v>

(12.35)

where p0 is any continuously dierentiable low-pass filter function for dilation by 2> such that the Fourier transform of the corresponding scaling function lies in = We note that p e 0 is continuous, because along the line of discontinuity v = ± 12 of M˜ we have p0 = 0= Also, by construction, p e 0 (0> 0) = 1>

e 0 (v> w)> p e 0 (v> w 1) = h(w)p

e 0 (v> w)> p e 0 (v 1> w) = h(w v)p

e 0 satisfies Cohen’s nonvanishing and since p e 0 has the same modulus as p0 > p condition and ¯ µ ¶¯2 ¯ 1 ¯¯ 1 = 1= |p e 0 (v> w)|2 + ¯¯p e0 v + >w + 2 2 ¯ Again, as in Equation (12.13) we set + (v> w) =

4 Y

m=1

p e 0 ([Tw ]m (v> w));

(12.36)

it follows that + (v> w) is continuous and

e 0 (v> w)+ (v> w)= + (Tw (v> w)) = p

Furthermore, we know that

h + > + iF(T2 ) (v> w) = 1 everywhere on R2 > and not just almost everywhere, by our choice of p0 that has the same modulus as p e 0 = Defining

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Judith A. Packer

R+ (k)(v> w) = + (v> w)k(v> w)> we thus obtain the desired F(T2 )-module monomorphism of [(1> 1) into , which preserves the F(T2 )-valued inner products, and which, in addition, satisfies the condition R+ ([(1> 1)) G[R+ ([(1> 1))]= Setting Vm = Gm [R+ ([(1> 1))] for m 5 Z> an application of Theorem 6 of [20] shows that the family {Vm }m5Z is the desired projective multiresolution analysis. Our aim now is to compute the isomorphism type of the wavelet module W0 = V1 ª V0 which arises from Theorem 12.18. We note that the determinant of T is 2> which is positive. Our conjecture as stated in the last paragraph of the introductory section leads us to believe that V1 = [(2> 1)> and hence that W0 = [(1> 0)> a free module of rank 1 over F(T2 )= This turns out to be true, although the proof is somewhat technical in nature: Theorem 12.19. Let V0 = R+ ([(1> 1)) be the projective F(T2 )-submodule of isomorphic to [(1> in Theorem 12.18, so that V1 = µ 1) constructed ¶ 11 G(V0 ) V0 > for T = = Then V1 = [(2> 1)> so that 1 1 W0 = V1 ª V0 = [(1> 0)> i.e., W0 is a finitely generated free F(T2 )-module of dimension 1= Proof. Recall that ¶ ¾ ½ µ 1 v + w v + w + > : k 5 [(t> d) > V1 = s ( · k) 2 2 2

(12.37)

where f0 (v> w) + (v> w)> + (v w> v + w) = p

for p f0 : R2 $ C constructed as in Theorem 12.18, so that + is defined as in Equation (12.36). We will construct an isomorphism V1 = [(2> 1)=

This allows us to deduce the desired result using the cancellation property for finitely generated projective modules over F(T2 )= We first compute for k1 > k2 5 [(1> 1) the value of hG(R+ (k1 ))> G(R+ (k2 ))iF(T2 ) :

12 Projective Multiresolution Analysis for the Quincunx Lattice

263

hG(R+ (k1 ))> G(R+ (k2 ))iF(T2 ) (v> w) ¯ µµ ¶ ¶¯2 X ¯ + ¯ 1 v + w v + w 0 0 ¯ ¯ × ··· > (p = > q ) ¯ ¯ 2 2 2 (p0 >q0 )5[Tw ]1 (Z2 )

· · · × (k1 · k2 )

µµ

v + w v + w > 2 2

¶

¶ (p0 > q0 ) =

In the standard fashion, we split this sum into two portions, using the two cow 1 (Z2 )@(Z2 )> set representatives (0> 0) and ( 12 > 1 2 ) for the quotient group (T ) obtaining ¶ µ v + w v + w 1 + + h · k1 > · k2 iF(T2 ) > 2 2 2 ¶ µ v + w + 1 v + w 1 1 + + > + h · k1 > · k2 iF(T2 ) 2 2 2 ¶ ¶¸ µ µ v + w v + w v + w + 1 v + w 1 1 (k1 · k2 ) > + (k1 · k2 ) > = = 2 2 2 2 2 Define a function + : V1 $ [(2> 1) by

h i + [G(R+ (k))](v> w) = + s12 + [Tw ]1 k [Tw ]1 (v> w) ¶ µ ¶ µ v + w + 1 v + w 1 v + w v + w > + ˜o(v + 1> w)k > > = ˜o(v> w)k 2 2 2 2 where

˜o(v> w) = 1 h 2

µ

vw 4

¶ µ 2 ¶ µ 2 ¶ v + 2v w + 2w h h = 8 8

(12.38)

Note the modulus of ˜o is identically equal to 12 = The calculations of the previous paragraph show that in order to have h+ [G(R+ (k1 ))]> + [G(R(k2 ))]iF(T2 ) (v> w)

= hG(R+ (k1 ))> G(R+ (k2 ))iF(T2 ) (v> w)

for any choice of k1 > k2 5 [(1> 1)> and + [G(R+ (k))](v> w) 5 [(2> 1) ;k 5 [(1> 1)> it is necessary that h+ [G(R+ (k1 ))]> + [G(R+ (k2 ))]iF(T2 ) (v> w) ¶ ¶¸ µ µ v + w v + w v + w + 1 v + w 1 1 (k1 · k2 ) > + (k1 · k2 ) > = = 2 2 2 2 2

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We first compute using Equation (12.38) that ˜o satisfies the conditions µ ¶ ˜o(v> w 2) = h v w ˜o(v> w)> (12.39) 2 and ˜o(v 2> w) = h

µ

v+w 2

¶

˜o(v> w)

(12.40)

for all v> w 5 R= Now note that for every k 5 [(1> 1)> + [G(R+ (k))](v 1> w) ¶ µ ¶ µ v + w v + w v 1 + w v 1 + w + 2 > + ˜o(v> w)k > = ˜o(v 1> w)k 2 2 2 2 ¶ µ v + w v + w > = ˜o(v> w)k 2 2 ¶ ¶ µ µ v1+w ˜ v 1 + w v 1 + w > +h o(v 1> w)k 2 2 2 = + [G(R+ (k))](v> w)> by Equation (12.40). Also, computing further, we see that + [G(R+ (k))](v> w 2) ¶ µ v + w 2 v + w 2 ˜ > = o(v> w 2)k 2 2 ¶ µ v + w 1 v + w 3 ˜ > + o(v + 1> w 2)k 2 2 ¶ ¶ µ µ v + w v+w v + w ˜o(v> w 2)k > = h 2 2 2 ¶ ¶¸ µ µ 1 ˜ v + w + 1 v + w 1 > > +h o(v + 1> w 2)k 2 2 2 which, by Equations (12.39) and (12.40), = h(v)+ [G(R+ (k))](v> w)= Hence, + [G(R+ (k))] 5 [(2> 1)> ; k 5 [(1> 1)= We must finally show that the inner product identity h+ [G(R+ (k1 ))]> + [G(R+ (k2 ))]iF(T2 ) (v> w)

= hG(R+ (k1 ))> G(R+ (k2 ))iF(T2 ) (v> w)

12 Projective Multiresolution Analysis for the Quincunx Lattice

265

is satisfied. Using the definition of + [G(R+ (k1 ))] and the inner product on [(2> 1)> one computes: h+ [G(R+ (k1 ))]> + [G(R+ (k2 ))]iF(T2 ) (v> w) ¶ ¶¸ ¸2 µ µ 1 v + w v + w v + w 1 v + w 1 > + (k1 · k2 ) > (k1 · k2 ) = 2 2 2 2 2 2 ¶ µ ¶ µ v + w v + w v + w 1 v + w 1 + P (v> w)k1 > k2 > 2 2 2 2 ¶ µ ¶ µ v + w 1 v + w 1 v + w v + w + P (v> w)k1 > k2 > > 2 2 2 2 for ¶ µ v+w ˜ P (v> w) = ˜o(v> w)˜o(v + 1> w) + h o(v + 1> w 1)˜o(v> w 1)= (12.41) 2 We now verify the identity P (v> w) P (v> w) 0>

(12.42)

and the proof of the equality of the inner products will follow, since we have already shown that hG(R+ (k1 ))> G(R+ (k2 ))iF(T2 ) (v> w) ¶ ¶¸ µ µ 1 v + w v + w v + w + 1 v + w 1 (k1 · k2 ) > + (k1 · k2 ) > = = 2 2 2 2 2 Using the definition of P given in Equation (12.41) and the definition of ˜o given in Equation (12.38), we calculate µ ¶ µ ¶ ˜o(v + 1> w)˜o(v> w) = 1 h v + w h 3 > 4 4 8 as ˜o(v> w) has constant modulus 12 = A similar calculation yields ¶ µ ¶ µ ¶ µ 1 1 v+w v+w ˜ ˜ h = h o(v> w 1)o(v + 1> w 1) = h 2 4 4 8 Hence ¶ µ v+w ˜ P (v> w) = ˜o(v + 1> w)˜o(v> w) + h o(v> w 1)˜o(v + 1> w 1) 2 µ ¶ µ ¶ µ ¶ µ ¶ 1 1 1 v+w 3 v+w h + h h = 0> = h 4 4 8 4 4 8

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and we have verified Equation (12.42), as we desired to show. The only thing left to do is to show that the map + from V1 to [(2> 1) is surjective, but this is a calculation similar to that done in Section 12.3, which we leave to the reader. As mentioned in the beginning of the proof, the fact that W0 = F(T2 ) = [(1> 0) follows immediately from the isomorphism between V1 and [(2> 1) and the cancellation property for finitely generated projective F(T2 )-modules. Corollary 12.20. Let V0 4 l=0 Wl be decomposition of corresponding to the projective multiresolution analysis related to the quincunx matrix T given above. Then for each l 5 N ^ {0}> Wl is a free F(T2 ) module of rank 2l = Proof. This follows immediately from Theorem 12.19 and Theorem 5.1 of [17]. An important fact to note in the proofs of both Theorems 12.18 and 12.16, as noted in the discussion given in Section 5 of [20], is that in order to construct projective multiresolution analyses for an integer dilation matrix D> we need the existence of ordinary low-pass filter functions p0 for the corresponding dilation matrix D such that: (i) p0 satisfies the usual wavelet conditions, i.e., p0 is continuously dierentiable at 0 with p0 (0> 0) = 1> g1 X m=0

|p0 ((v> w) + vj )|2 = 1>

w 1 where {vj }g1 (Z2 ) for m=0 is a collection of coset representatives in [D ] the g-element group [Dw ]1 (Z2 )@Z2 > and p0 satisfies Cohen’s condition; Q w m b (ii) p0 is suciently regular that 4 m=1 p0 ([D ] (v> w)) = !(v> w) lies in , i.e., !b needs to have sucient decay properties; (iii) p0 vanishes along some sort of line or curve in R2 @Z2 > so that we can multiply it by a jump function that has a jump discontinuity along that very same curve to obtain a new continuous function p e 0 satisfying certain necessary functional relations that will ensure GDw (R([(t> d))) R([(t> d))= Here, as previously, R is multiplication by 5 cone 0 so that structed from p e 0 > so the method to date has been to find p p e 0 (v> w) · k Dw (v> w) 5 [(t> d)> for all k 5 [(t> d)=

Once having obtained an ordinary low-pass filter of this type, the construction of the embedding of the projective F(T2 )-module into as the core projective module V0 is not too dicult. The identification of the isomorphism type of V1 = G(V0 ) as carried out in the proofs of Theorems 12.17 and 12.19 at this point seems to require various delicate maneuvers in calculation, which vary somewhat depending on the matrix being considered.

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12.5 Module Frames and Ordinary Frames in L2 (R2 ) We now discuss how to use the results of the previous sections to construct normalized tight frames (Parseval frames) for O2 (R2 )= We first review a definition due in its greatest generality to M. Frank and D. Larson ([9]). Definition 12.21. Let A be a unital F -algebra, let V be a finitely generated projective D-submodule that is a right Hilbert F -module over A= Then the set {!1 > !2 > · · · > !p } V is called a module-frame for V if for all 5 V> the following reconstruction formula is satisfied. X !m h!m > iA = = 1 m p

In the case where A = F(T2 ) or even F(Tq ) for q A 2> it is quite easy to come up with some explicit constructions of module frames, given the explicit form of the projective module used as initial module. They are a replacement for ordinary frames in the Hilbert space setting. For completeness, we give the definition of Hilbert space frames as well: Definition 12.22. Recall that a sequence {in : n 5 N} of elements in a Hilbert space H is said to be a frame for H if there are real constants F> G A 0 such that 4 4 X X |hi> in i|2 ki k2 G |hi> in i|2 F n=1

n=1

for every i 5 H where here the inner product is Hilbert space inner product. If F = G = 1> the frame is said to be a normalized tight frame or Parseval frame for H= Parseval frames {in : n 5 N} for H give perfect reconstruction in the Hilbert space setting: 4 X hi> in iH in = i = n=1

Example 12.23. We construct a module frame for [(1> d) : Let p0 be a continuous normalized low-pass filter defined on R for dilation by 2> so that p0 (0) = 1> and

Define

¶¯2 µ µ ¶ 1 ¯ X ¯ ¯ ¯p0 w + m ¯ = 1> so that p0 1 = 0= ¯ 2 ¯ 2 m=0 k0 (v> w) = p0 (w)M0 (v> w)>

and

(12.43)

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Judith A. Packer

µ

k1 (v> w) = p0 w + where

1 2

¶

M1 (v> w)>

(12.44)

¶ 1 1 > q 5 Z> M0 (v> w) = h(qdv)> w 5 q > q + 2 2

and M1 (v> w) = h(qdv)> w 5 [q> q + 1)> q 5 Z= Then k0 > k1 5 [(1> d)> and one can calculate that they give a F(T2 )module frame for [(1> d)= The idea of strong Morita equivalence bimodules is essential to this calculation. We note that the jump functions M0 and M1 are just relabeled forms of the jump functions Mt>d> : R2 $ C reviewed in Equation (12.16). Using this notation, we see that the functions M0 and M1 of Example 12.23 are M1>d>1@2 and M1>d>0 > respectively. By using the above construction and the isomorphism [(t> d) = [(1> d) [(t 1> 0) for positive integers t A 1> we note that for the non-free finitely generated projective modules [(t> d) of dimension t described earlier, the number of elements in any module frame will be at least t + 1 (this is an illustration of redundancy inherent in frame theory). Hence, if d 6= 0> the least number of elements in the module frame for the initial modules V0 that are isomorphic to [(t> d) constructed in earlier sections is t + 1= Example 12.24. We now use Example 12.23 and Theorem 12.12 to construct first a module frame, and then a Parseval frame, for the initial module V0 and its closure in O2 (R2 )> respectively, corresponding to the projective multiresolution µ ¶analysis in the special case where t = 1 and the dilation matrix 20 D = = Recall in this case our module map is given by 02 R(k)(v> w) = (v> w)k(v> w)> k 5 [(1> d)> for

µ ¶¸ 4 Y v w p e0 > > (v> w) = 2m 2m m=1

where p e 0 5 [(1> 3d) and in addition p e 0 satisfies the normalized low-pass µ ¶ 20 filter equations corresponding to the matrix = Indeed, the proof of 02 Theorem 5 of [20] shows how we can take p e 0 (v> w) = p0 (v)p0 (w)M1>3>1@2 (v> w)>

where p0 is a continuous low-pass filter for dilation by 2 defined on R@Z such that the corresponding function lies in = It follows that the two element F(T2 )-module frame for V0 in this case will be

12 Projective Multiresolution Analysis for the Quincunx Lattice

269

½ R(k0 ) := p0 (w)M1>d>1@2 (v> w)(v> w)> ¶ µ ¾ 1 M1>d>0 (v> w)(v> w) = (12.45) R(k1 ) := p0 w + 2 By Proposition 12 of [20], {R(k0 )·(v> w)·h(p·(v> w)) : p 5 Z2 }^{R(k1 )·(v> w)·h(p·(v> w)) : p 5 Z2 } is a normalized tight frame for V0 = Y0 O2 (R2 )= A result of Frank and Larson tells us that if J is a countably generated right Hilbert F -module over a unital F -algebra A> then J will have a countable standard module-frame, from which one can deduce that there will exist a countable set {!1 > !2 > · · · > } J such that X = !n h!n > iA = (12.46) n5 N

Thus in particular, should have a module frame over F(T2 )= Given a projective multiresolution analysis {Vm } in for dilation by D> explicit formulas for module frames for can be obtained as follows. Step 1: We first prove that = V0 4 m0 Wm (topological direct sum)= We remark here that the topological direct sum V0 4 m0 Wm can be identified with {y + {zm }m0 : zm 5 Wm > and

X

? zm > zm AF(T2 ) converges}=

m0

Step 2: Find a module frame for the initial module V0 and each of the modules Wm > by using an isomorphism between the modules Wm and [(tm > dm )> and finding a family of module frames for [(tm > dm ) that can be explicitly written down for each m. The benefits of this method are that by using dierent projective multiresolution analyses for > we can construct dierent module frames for = Then, using these module frames for > we can construct ordinary normalized tight frames in the Hilbert space setting for O2 (R2 )> as in Example 12.45. Using the fact that is dense in O2 (R2 ) in the O2 norm, and that each Hilbert module V0 and Wm is dense in the closed subspaces Y and Zm of O2 (R2 )> respectively, arising as in ordinary multiresolution analysis theory, it is possible to use families of module frame for V0 and each Wm to construct

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a normalized tight frame for Y0 and each Zm = Since O2 (R2 ) = Y0 [4 m=0 Zm ]> (Hilbert space sum) the union of the normalized tight frames for Y0 and each Zm as m runs over N ^ {0} gives a normalized tight frame for O2 (R2 )= Example 12.25. If {#1 > #2 > · · · > #g } is a module frame for the wavelet module W0 , it is possible to show that ^gn=1 {#n · h(p·(v> w)) : p 5 Z2 } is a Parseval frame for W 0 = Z0 O2 (R2 ); see [20] Proposition 12 for details.

12.6 Conclusion and Open Problems The methods we have used so far may seem fairly specialized. The question then arises as to how to construct examples beyond the special cases of diagonal matrices with integer entries, their conjugates by elements of VO(q> Z)> and the special matrices S and T discussed above. We hope that this chapter indicates a general method at the end of Section 12.4 for the 2 × 2 case. Indeed, the construction of an ordinary multiresolution analysis for a general q×q dilation matrix with integer entries was first accomplished in 2001 by M. Bownik in [4]; his construction would presumably prove useful as the scaling functions and wavelets he constructs have arbitrary degrees of smoothness. A special case of our construction gives the Fourier transform of a dense subspace of certain ordinary multiresolution analyses, with this dense subspace lying inside the Hilbert F(T2 ) module mentioned in the first section. In the case where q = 2> Bownik and Speegle have recently shown that for any 2 × 2 dilation matrix, it is possible to construct a scaling function whose Fourier transform is smooth and has compact support in R2 > hence will lie in the Hilbert F(T2 )-module = For F(Tq )-modules with q 3> the problem of constructing an ordinary multiresolution analysis for an arbitrary dilation matrix such that the initial space Y0 has a dense subspace invariant under translation by Zq whose Fourier transform lies inside remains an open one. Acknowledgments The author would like to thank Professor Marc Riegel of UC Berkeley, who first initiated the study of projective multiresolution analyses, and who gave the author access to his unpublished notes [23]. She also would like to thank Professor Larry Baggett for his listening ear; while she was thinking about the general topic of this chapter he listened carefully, made many valuable suggestions, and corrected many misconceptions.

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References 1. L. Baggett, H. Medina, and K. Merrill, Generalized multi-resolution analyses and a construction procedure for all wavelet sets in Rq , J. Fourier Anal. Appl. 5 (1999), 563—573. 2. L. Baggett, J. Packer, The primitive ideal space of two-step nilpotent group F W algebras, J. Funct. Anal. 124 (1994), 389—426. 3. B. Blackadar, “N-Theory for Operator Algebras,” Second Edition, Mathematical Science Research Institute Publications, Vol. 5, Cambridge University Press, Cambridge, U.K., 1998. 4. M. Bownik, The construction of u-regular wavelets for arbitrary dilations, J. Fourier Anal. Appl. 7 (2001), 489—506. 5. M. Bownik, D. Speegle, Meyer type wavelet bases in R2 > J. Approx. Theory 116 (2002), 49—75. 6. A. Cohen, “Wavelets and Multiscale Signal Processing,” translated by R. Ryan, Chapman and Hall, London, 1995. 7. I. Daubechies, “Ten Lectures on Wavelets,” American Mathematical Society, Providence, RI, 1992. 8. G. Elliott, The N-theory of the F W -algebra generated by a projective representation of a torsion-free discrete abelian group, in “Operator Algebras and Group Representations,” vol. 1. Pitman, London, 1984, pp. 157—184. 9. M. Frank, D. R. Larson, A module frame concept for Hilbert C*-modules, in “The functional and harmonic analysis of wavelets and frames,” pp. 201—233, L. Baggett and D. Larson, Eds., Proceedings of AMS Special Session on the Functional and Harmonic Analysis, January 13—14, 1999, San Antonio, TX, American Mathematical Society, Providence, RI, 1999. 10. D. Husemoller, “Fibre Bundles,” Second Edition, Springer-Verlag, New York, Heidelberg, Berlin, 1974. 11. E. C. Lance, “Hilbert F W -modules — a Toolkit for Operator Algebraists,” London Mathematical Society Lecture Note Series, vol. 210, Cambridge University Press, Cambridge, England, 1995. 12. N. S. Larsen and I. Raeburn, From filters to wavelets via direct limits, in “Operator theory, operator algebras, and applications,” pp. 35—40, D. Han, P. E. T. Jorgensen and D. Larson, eds., Contemp. Math. 414, Amer. Math. Soc., Providence, RI, 2006. 13. N. S. Larsen and I. Raeburn, Projective multi-resolution analyses arising from direct limits of Hilbert modules, Math. Scand. 100 (2007), 317—360. 14. Y. Meyer, “Wavelets and Operators,” Cambridge Studies in Advanced Mathematics vol. 37, Cambridge University Press, Cambridge, England, 1992. 15. S. T. Lee, J. A. Packer, N-theory for F W -algebras associated to lattices in Heisenberg Lie groups, J. Operator Theory 41 (1999), 291—319. 16. J. Packer, Strong Morita equivalence for Heisenberg F W -algebras and the positive cones of their N0 -groups, Canad. J. Math. 40 (1988), 833—864. 17. J. Packer, Projective multiresolution analyses for dilations in higher dimensions, J. Operator Theory 57 (2007), 147—172. 18. J. Packer, I. Raeburn, On the structure of twisted group F W -algebras, Trans. Amer. Math. Soc. 334 (1992), 685—718. 19. J. Packer, M. A. Riegel, Wavelet filter functions, the matrix completion problem, and projective modules over F(Tq )> J. Fourier Anal. Appl. 9 (2003), 101—116. 20. J. Packer, M. A. Riegel, Projective multi-resolution analyses for O2 (R2 ), J. Fourier Anal. Appl. 10 (2004), 439—464. 21. M. A. Riegel, F W -algebras associated with irrational rotations, Pacific J. Math. 93 (1981), 415—429. 22. M. A. Riegel, The cancellation theorem for projective modules over irrational rotation F W -algebras, Proc. London Math. Soc. 47 (1983), 285—302.

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23. M. A. Riegel, Multiwavelets and operator algebras, talk at a special session, Joint Mathematics Meeting, San Diego, January 8—11, 1997, private communication / cf. Abstracts Amer. Math. Soc. 18 (1997), p. 107, 918-46-722. 24. R. Strichartz, Construction of orthonormal wavelets, in “Wavelets: Mathematics and Applications,” pp. 23—50, J. Benedetto and M. Frazier, Eds., CRC Press, Boca Raton, FL, 1994. 25. N. E. Wegge-Olsen, “K-theory and C*-algebras — a friendly approach,” Oxford University Press, Oxford, England, 1993.

Chapter 13

Sampling and Time-Frequency Localization of Band-Limited and Multiband Signals Jerey A. Hogan and Joseph D. Lakey Dedicated to Larry Baggett on the occasion of his retirement

Abstract This chapter develops aspects of previous work ([10] and [11]) that pertain distinctly to band-limited functions. In particular, we discuss some representation formulas for band-limited functions in terms of periodic nonuniform samples. In the case of multiband signals, periodic nonuniform sampling is often valid at a lower sampling rate than is uniform sampling, as will be discussed. Finally, we will consider some related questions about optimally time- and multiband-limited signals.

13.1 Introduction This chapter addresses some connections between sampling and time-frequency localization. Specifically, we are concerned with periodic nonuniform sampling (PNS) and with properties of time-frequency localization operators of the form S TW defined as follows: For i 5 O2 (R), S i = (iˆ11 )b , where iˆ and j b denote the Fourier and inverse Fourier R 4 transforms of i and j (normalized so that if i 5 O1 (R) then iˆ() = 4 i (w)h2lw gw) and where 11 is the characteristic function of the set . Also, TW i = 11[W @2>W @2] · i is the operation of multiplication of i by the characteristic function of the interval of length W centered at zero. The operator S TW is the Pcomposition of these two. The classical Shannon sampling formula, i (w) = n5Z i (n) sinc(w n), (see, e.g., [2, 3]) is a fundamental fact about band-limited functions, as it holds whenever i 5 O2 (R) lies in the image of S[1@2>1@2] . More reJegrey A. Hogan Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR 72701; e-mail: [email protected] Joseph D. Lakey Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003-8001; e-mail: [email protected]

275

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cently, connections between sampling and time-frequency localization operators S[ @2> @2] TW have been established, for example by Walter and X. Shen [21] and by Khare and George [14]. On the other hand, PNS sampling formulas for multiband signals have also been established by Herley and Wong [8] and by Bresler and Venkataramani [20], among others. The latter results point to the possibility of reconstructing signals in the Paley—Wiener space PW( ) = S (O2 (R)) (i.e., the class of those i 5 O2 (R) whose Fourier transforms vanish outside ) when PNS sampling is carried out at an average rate of | | samples per unit time. When samples are taken uniformly, this is possible only when is congruent to an interval, in an appropriate sense. More generally, one can ask what information about localization of a signal i in time and frequency can be coded in a discrete set of samples of i and vice versa. This naive question brings up a number of issues for which a satisfying, unified theory seems unlikely. It is our goal here, rather, to give some indication of potential benefits that attempts to answer this question can yield. The chapter is outlined as follows. In Section 13.2, we illustrate the possibility of interpolating certain band-limited functions from their samples with better interpolating functions, provided the functions lie in a certain proper subspace of the corresponding Paley—Wiener space (see also [11]). Although PNS does not appear explicitly in this section, the use of multiple interpolating functions–as in the PNS case–does. In Section 13.3, we establish a variant of Yen’s theorem on interpolation from PNS samples ([23], see also [7]). In Section 13.4, we review some formulas for samples of prolate spheroidal wave functions (PSWFs)–eigenfunctions of S[ @2> @2] TW –established independently by Shen and Walter [21] and Khare and George [14]; we then illustrate how PNS samples of PSWFs can be obtained as eigenfunctions of a certain matrix of time-localized approximants of a Gram matrix for interpolating functions defined in Section 13.3. In Section 13.5, we review a method for interpolating multiband signals from PNS samples that was developed in work of Bresler and Venkataramani [20]. We also outline there a few problems relating this method with time-frequency localization. This raises the question of how one determines the eigenfunctions of S TW when is something other than a single interval. In Section 13.6, we outline one possible method for computing eigenfunctions for S 1 ^ 2 TW when the eigenfunctions for S 1 TW and S 2 TW are known.

13.2 Sampling of Band-Limited Signals in Shift-Invariant Subspaces According to the Shannon sampling theorem, any function i band-limited to an interval of length P can be expanded in terms of its samples {i (n@P )}P5Z . The shifted sinc functions that interpolate these functions have nice properties, but their decay rate of R(1@|{|) at infinity is lousy.

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When signals live in certain proper subspaces of the Paley—Wiener space PW( ), better interpolation properties might be possible, as will be illustrated now. For * 5 O2 (R), the principal shift-invariant (PSI) P space Y (*) generated by * consists of all i 5 O2 (R) of the form i = n5Z fn *(· n) where {fn } 5 c2 (Z). The Fourier transform of such an i has the form P P 2ln 2ln = F()*() ˆ where F() = deiˆ() = *() ˆ n5Z fn h n fn h notes the Fourier series of the sequence {fn }. The Zak transform plays a convenient role in the study of PSI spaces, particularly P in their sampling theory ([1, 13, 12, 10, 11, 22]). One defines ]i (w> ) = n5Z i (w + n)h2ln . Then for i 5 Y (*), one has ]i (w> ) = F() ]*(w> ). One can establish a simple relationship between the integer samples {i (n)} of i 5 Y (*) and its coecients {fn } provided that * possesses a simple Tauberian property that () = ]*(0> ) has no zeros. In this case, one has F() = ]i (0> )@(). However, the function that interpolates i from its samples then is the inverse Zak transform of 1@, which, generally, will not have particularly nice decay properties. Oversampling might be considered as a possible means of producing better localized interpolating functions. By using a system of Zak transforms, one might hope for better decaying interpolating functions. One particular incarnation of this principle is embodied in the following theorem and corollary (cf. [10]). Theorem 13.1. Suppose that * 5 PW ([0> P ]) and that the integer translates of * are orthonormal. Then for all and all | 5 [0> 1], P1 X¯ c=0

´¯2 ³ c ¯ ¯ + |> ¯ = P= ¯]* P

Corollary 13.2. For * as in the theorem, and any i 5 Y (*), one has i (w) =

P1 XX c=0 n5Z

where Vc (w) =

i

´ ³ c + n Vc (w n) P

´ 1 X ³ c + n *(w + n)= * P P n

As an element of PW ([0> P ]), the rate of P samples per unit time for i in Corollary 13.2 appears to be the critical sampling rate. However, as an element of Y (*), only one sample per unit time is required of i . Therefore, the corollary is actually an example of P times uniform oversampling. Seen from this point of view, it is not surprising that localization properties of the interpolating functions Vc are on par P with those2 of *. The orthogonality ˆ + n)| 1, which is simple condition on * is equivalent to (·) = n |*(· p to ˆ˜ impose by replacing * by *˜ defined on the Fourier side by *() = *()@ ˆ () provided has no zeros (e.g., [6], p. 139).

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c c Proof. To prove Corollary 13.2, first, since ]i ( P > ) = F()]*( P > ), mulc tiplying both sides of this equation by ]*( P > ) and summing over c yields P1 X c=0

P1 ³ c ´ ³ c ´ X ¯¯ ³ c ´¯¯2 > ]* > = F() > ¯ = P F()= ]i ¯]* P P P c=0

The inversion formula for the Zak transform, i (w) = i (w) =

Z

1

]i (w> ) g = 0

Z

1

R1 0

]i (w> ) g, then gives

F()]*(w> ) g

0

P 1 Z ³ c ´ ³ c ´ 1 X 1 > ]* > ]*(w> ) g = ]i P P P c=0 0 P1 ´Z 1 ³ c ´ 1 XX ³ c +n > ]*(w> ) g = i h2ln ]* P P P 0 c=0 n5Z

=

=

=

1 P 1 P

P1 XX

i

c=0 n5Z

P1 XX c=0 n5Z

P1 XX c=0 n5Z

i

´X ³ c ´Z 1 ³ c +n + * h2l(n) ]*(w> ) g P P 0 5Z

i

´X ³ c ´ ³ c +n + *(w + n) * P P 5Z

´ ³ c + n Vc (w n)= P

Proof of Theorem 13.1. We apply the Poisson summation formula in the ˆ w) and the orthogonality condition on *, which form ]*(w> ) = h2lw P ] *(> ˆ + n)|2 1. Thus, for any and | we have can be expressed as n5Z |*( ¶¯2 ¶¯2 µ µ P1 ¯ P1 ¯ ¯ ¯ 1 X ¯¯ 1 X ¯¯ c c ¯ ¯ + |> + | = ]* ] * ˆ > ¯ ¯ ¯ ¯ P P P P c=0

c=0

P1 X X c 1 X = *( ˆ +n)*( ˆ +) h2l(n)( P +|) = |*( ˆ +n)|2 1= P n>5Z

c=0

n5Z

This proves the theorem.

13.3 Periodic Nonuniform Sampling in PW([ /2, /2]) Here we continue to study representations of band-limited functions in terms of multiple interpolating functions. However, the emphasis on behavior of

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the interpolating functions will be replaced by emphasis on the sampling pattern. Yen [23] first formulated a modification of the classical sampling theorem inSwhich samples are taken along a recurrent sampling lattice of the 2 form = Q =1 w + OZ. If i 5 O (R) is band-limited to an interval of length

, then it can be recovered from its samples along provided the average sampling rate is at least samples per unit time, that is, Q@O . The critical sampling rate of Q@O = will be assumed in what follows.

13.3.1 Interpolation from Samples As a notational convenience, we will denote @P as P and v (w) = sinc( w). We will denote the integer part of { by b{c.

Theorem 13.3. Let P 1 be an integer and suppose that {w0 > = = = > wP1 } are such that the remainders {wc P bwc P c} are distinct so that the P ×P Vandermonde matrix H with (m> n)-th entry Hmn = h2lwm n P is invertible. For s = 0> = = = > P 1, set Vs (w) = hl(wws )( P ) v P (w ws )

P1 X

1 2lwt P Hts h =

(13.1)

t=0

Then each i 5 PW([ @2> @2]) satisfies i (w) =

P1 X s=0

³ p ´ ³ p ´ i ws + Vs w =

P

P

(13.2)

Proof. To facilitate matters, for i 5 PW([ @2> @2]) we will set j(w) = hl w i (w), which is band-limited to [0> ]. It will be convenient to obtain a sampling formula for j, then convert it to one for i . We have j(w) =

Z

2lw

jˆ()h

g =

0

P1 X Z (c+1) P c=0

=

jˆ()h2lw g

c P

P1 X

2lwc P

h

c=0

Z

P

jˆ( + c P )h2lw g= (13.3)

0

In particular, Z P P 1 ³ X p ´ jˆ( + c P )h2lws c P h2l(ws +p@ P ) g = j ws +

P 0 c=0

=

³P1 ´ X p

P FB jˆ(· + c P )h2lws c P h2lws · (p) c=0

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where FB k(p) s denotes the p-th Fourier coecient of k with respect to B = {h2lp@ P @ P }p5Z , which forms an orthonormal basis for O2 ([0> P ]). Writing the expansion of this Fourier series, we have, in turn, that P1 X

jˆ( + c P )h2lws c P h2lws =

c=0

or

P1 X

Hsc jˆ( + c P ) =

c=0

1 X ³ p ´ 2lp@ P j ws + h

P p

P

h2lws X ³ p ´ 2lp@ P j ws + = h

P

P p

This implies that jˆ can be recovered from the samples of j via jˆ( + t P ) =

P1 1 X 1 2lws X ³ p ´ 2lp@ P Hts h j ws + h

P s=0

P p

(13.4)

whenever 5 [0> P ]. Substituting (13.4) into (13.3) we then obtain j(w) =

=

=

Z P1 P1 p ´ 2l(wws p ) 1 X 2lwt P P X 1 X ³ P h Hts j ws + g h

P t=0

P 0 p s=0 Z P1 P1 1 XX ³ p ´ X 1 2lwt P P 2l(wws p ) P j ws + Hts h h g

P s=0 p

P t=0 0 P1 XX s=0

p

P1 ³ p ´ X 1 2lwt P j ws + H h

P t=0 ts ³ p ´ ×(1)p hl(wws ) P v P w ws =

P

Consequently, with Vs as in (13.1), i (w) = hl w j(w) =

P1 XX s=0

p

³ p ´ ³ p ´ i ws + Vs w

P

P

as claimed.

13.3.2 Properties of the Interpolating Functions Sp The interpolating functions Vs in Theorem 13.3 are dierent and in some ways preferable to those considered by Yen [23] (cf. [7]). Nevertheless, they still fail some of the key properties of the sinc function. For example, they

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need not be orthogonal to their shifts. Nonetheless, the interpolating and correlation properties can be quantified in a useful way, as we consider here. In order to quantify truncation errors, aliasing errors, et cetera for functions in PW([ @2> @2]) in terms of PNS samples, it is useful first to compute the correlations hVs (· p@ P )> Vt (· c@ P )i. Recalling that

P = @P , one has hVs (· p@ P )> Vt (· c@ P )i = hl( P )(ws wt +(pc)@ P )

P1 1 XP X

1

1 Hus H vt

u=0 v=0

Z

³ ³ p ´ c ´ × h2l P (uv)w v P w ws v P w wt gw

P

P R l( P )(ws wt +(pc)@ P )

=h

P1 1 XP X

1

1 Hus H vt

u=0 v=0

×

Z

1@ P

h2l P (uv)w

0

X n

P

³ ³ n p´ n c´ v P w ws + v P w wt + gw=

P

P

However, since n sinc({ n) sinc(| n) = sinc({ |), the sum in the last integral collapses to X n

³ ³ ¡ n p´ n c´ c p´ v P w ws + v P w wt + = v P wt ws +

P

P

P

and as a consequence we have hVs (· p@ P )> Vt (· c@ P )i = hl( P )(ws wt +(pc)@ P ) ×

P1 X P1 X u=0 v=0

³ c p ´ uv 1 1 Hus H vt v P wt ws + =

P P

Proposition 13.4. Let Vs be defined as in Theorem 13.3. Then for 0 s> t ? P and p> c 5 Z, the Gram matrix entry Js>p;t>c = hVs (· p@ P )> Vt (· c@ P )i satisfies Js>p;t>c =

³ (1)(1P)(cp) c p´ l(wt ws )( P ) (HH )1 h v w + w =

t s ts P

P

P

We also have the following interpolation/cardinality property of the Vs .

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Jegrey A. Hogan and Joseph D. Lakey

Proposition 13.5. Let Vs and w0 > = = = > wP1 be defined as in Theorem 13.3. Then for all p 5 Z and 0 s> t ? P , Vs (wt + p@ P ) = p st . Proof. One has ³ ³ p ´ p ´ Vs wt + = hl(wt ws +p@ P )( P ) v P wt ws +

P

P P1 X 1 2 P uwt × Hus h u=0

l(wt ws +p@ P )( P )

=h

= hlp(P1) v P

³ p ´ v P wt ws + ts

P

³ p ´ ts = p ts =

P

13.4 Sampling and Time-Frequency Localization 13.4.1 Samples of Prolate Spheroidal Wave Functions There are no simultaneously time- and band-limited functions. Nonetheless, as Landau, Slepian, and Pollack quantified in multiple ways (see [18, 19] for nice expositions of their work), for a single time interval, say [W @2> W @2] and frequency interval [ @2> @2], the space of functions i having most of their energy in [W @2> W @2] in time and in [ @2> @2] in frequency is essentially bW c dimensional (see [9], [16]). The operator S[ @2> @2] TW is compact and self-adjoint as an operator on PW([ @2> @2]). Its eigenfunctions are the prolate spheroidal wave functions (PSWFs), which form an orthonormal basis for PW([ @2> @2]). Their restrictions to [W @2> W @2] also form a complete orthogonal set (e.g., [9]). Let q , q = 0> 1> 2> = = = denote the q-th eigenvalue of S[ @2> @2] TW expressed in decreasing order of magnitude, and let *q denote the corresponding eigenfunction (the eigenvalues are nondegenerate). Mercer’s theorem gives sinc( ({ |)) =

4 1 X *q ({)*q (|)=

q=0

Using the eigenfunction property together with the classical sampling theorem, one has q *q ({) = =

Z

Z

W @2

sinc( ({ |))*q (|) g| W @2 W @2

sinc( ({ |)) W @2

X

n5Z

*q

³n´

sinc( | n) g|=

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283

Evaluating at { = p@ , one then has a discrete eigenvector equation for *q as was observed by Khare and George [14], who also noted the possibility of extending their approach to other band-limited kernels that admit sampling series, as would be the case for the spaces Y (*) in Section 13.2. Proposition 13.6. Let *q be the eigenfunction of the q-th largest eigenvalue q of the operator S[ @2> @2] TW considered as an operator on PW([ @2>

@2]). Then ³p´ X ³n´ Dpn *q = q *q

n

where the doubly infinite matrix D has entries Dpn given by Dpn =

Z

W @2

sinc( | p) sinc( | n) g|

W @2

=

Z

W @2

sinc(w p) sinc(w n) gw=

(13.5)

W @2

That is, the sample values of *q form the q-th eigenvector of {Dpn }. Before seeking a generalization of this observation to the case of periodic nonuniform sampling, we mention a couple of related identities that were established independently by Khare and George [14] and Shen and Walter [21]. These identities are 4 X

*q

p=4 4 X

q=0

*q

³p´ ³p´ *c = q>c

³p´

*q

and

³c´ = p>c =

(13.6)

(13.7)

Equation (13.6) simply states that, just as the {*q } form an orthonormal basis for PW([ @2> @2]), their samples also form an orthonormal basis for the isomorphic sample space. Equation (13.7) says that, in eect, the samples of the reproducing kernel for the Paley—Wiener space forms the reproducing kernel (the Dirac delta) for the sample space. An interesting problem, which we state imprecisely here, is to quantify the extent to which these identities approximately hold for the space of essentially time- and band-limited signals. Problem 13.7. Obtain precise asymptotic estimates for the quantities D1 X

p=D1

*q

³p´ ³p´ *c

and

D2 X

q=0

*q

³p´ ³ c ´ *q

with D1 and D2 regarded in terms of the area W .

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Jegrey A. Hogan and Joseph D. Lakey

13.4.2 Localization Eigenvectors for Periodic Nonuniform Sampling Since the notation will get a bit cumbersome, we will work with a variation of Theorem 13.3 in which P = 2O + 1 and the sample osets ws are indexed by s = O> = = = > O. In this case, now setting O = @(2O + 1), one can express i 5 PW([ @2> @2]) as i (w) =

O X

s=O

³ p´ ³ p´ i ws + Vs w

O

O

in which Vs (w) = hl(wws )( O ) v O (w ws )

O X

1 2lwt O Hts h =

(13.8)

t=O

Since sinc( w) and the functions Vs (w) can be expressed in terms of one another, there are two corresponding matrices whose eigenvectors are the PNS samples of the functions *q . First, we express v (w) = sinc( w) in terms of the Vs , v (w) =

O X X

s=O p

³ p´ ³ p´ v ws + Vs w =

O

O

Then the reproducing and eigenfunction properties give O X ³ ³ X c ´ p´ *q ws + = q *q wt +

O

O s=O p Z W @2 ³ c ´ ³ p´ v | wt × Vs | g|

O

O W @2

=

O X X

s>u=O p>n

×

Z

³ p´ ³ n c´ *q ws + v wu wt +

O

O

³ n ´ ³ p´ Vu | Vs | g|=

O

O W @2 W @2

On the other hand, by the definition of Vs , the reproducing property and the fact that O

O X 2lwn O h v O (w)> v (w) =

n=O

we also have

13 Sampling and Time-Frequency Localization

285

Z W @2 X O ³ c ´ q *q wt + h2ln O (wt +c@ O |) = O

O W @2 n=O ´ ³ c ×v O wt + | *q (|) g|

O Z W @2 O ´ ³ X c 2ln O wt h h2ln O | v O wt + | = O

O W @2 n=O

×

O X X

s=O p

= O

³ p´ ³ p´ *q ws + Vs | g|

O

O

O X X

s=O p

Z

O O ³ X p´ X 1 *q ws + Htn hlws ( O ) Hus

O n=O

u=O

´ ´ ³ ³ c p × h2l O (tnO)| v O wt + | v O ws + | g|=

O

O W @2 W @2

These analogues of (13.5) can be summarized as follows. Proposition 13.8. Let O = @(2O + 1) and, for s = O> = = = > O let Vs be defined as in (13.8). Let *q be the eigenfunction of S[ @2> @2] TW with eigenvalue q . Then the sample vector {*q (wt + p@ O )}t>p is a q -eigenvector for each of the following matrices: Et>c;s>p =

O X X

u=O n

Z ³ n c ´ W @2 ³ n ´ ³ p´ v wu wt + Vu | Vs | g|

O

O

O W @2

and Ft>c;s>p = O ×

Z

O X

n=O

Htn hlws ( O )

O X

1 Hus

u=O

´ ´ ³ ³ c p h2l O (tnO)| v O wt + | v O ws + | g|=

O

O W @2 W @2

13.5 Sampling of Multiband Signals Insofar as uniform sampling is available for PW ([ @2> @2]), Theorem 13.3 and Proposition 13.8 are matters more of flexibility than necessity. In contrast, when is as simple as a finite union of pairwise disjoint intervals, reconstruction from samples taken at the critical rate of | | samples per unit time can rarely be accomplished with uniform sampling. Nonetheless, there are applications to analysis of multiband signals in which interpola-

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Jegrey A. Hogan and Joseph D. Lakey

tion from samples is needed and for which quantification of time-frequency localization properties is desired. Here we want to review briefly a method due to Bresler and Venkataramani [20] for interpolating multiband signals from PNS samples. To maintain consistency with the notation in [20], and since the time localization operator TW is not mentioned again until the very end of this section, the symbol W will be used here to denote the sampling period. One can think of 1@W as a “large” integer sampling rate. Thus one can recover any i in the class PW([0> 1@W ]) from samples along W Z. When S i is assumed to have energy only in a small subset = Q l=1 [dl > el ] where 0 = d1 ? e1 ? d2 ? · · · ? eQ = 1@W , then i should be recoverable from samples along a concomitantly small S 1 subset of W Z. The approach is to express the sampling lattice W Z as S c=0 S W Z+cW as a union of cosets indexed by Z@S Z and to come up with a criterion for deciding which of the cosets are redundant, and can thus be omitted, ST1when it comes to interpolating i . Thus one seeks a subunion of cosets s=0 S W Z + ws W with ws 5 Z@S Z for s = 0> = = = > T 1 along which i can still be interpolated from its PNS samples. Suppose for simplicity that | | 5 N. As Landau [15] first showed, an average sampling rate of at least | | is required to recover any i 5 PW( ) from its samples. Therefore (T@S )(1@W ) | | or T@S W | |. The complexity of the PNS sampling pattern, reflected in the magnitudes of S and T, depends on the spatial layout of the Q intervals comprising . Overcoming this complexity then boils down to finding an ecient way of subdividing the spectrum. This subdivision depends on the interplay of continuous and discrete Fourier transforms, as can be illustrated through a simple example. Suppose that i = i1 + i2 = i1 + h3l· i0 where ib0 > ib1 are supported in [0> 1@2] so ib2 lives in [3@2> 2]. Then i0 and i1 can be recovered from i by “predicting” the values of i on 2Z + 1 from those along 2Z based on the “null hypothesis” that i 5 PW([0> 1@2]), then subtracting the actual values. The same is not true if j = j1 + j2 = j1 + h2l· j0 where jb0 and jb1 are supported in [0> 1@2]. For example, all integer samples of (1 h2l· )j0 vanish. One remedy in this case is to sample, instead, along 2Z ^ (2Z + 1@2). In [20], the subdivision of is expressed in terms of a finite set of remainders (dl ) = dl bS W dl c@S W and corresponding (el ) terms. Some of these 2Q remainders might be redundant, so will have P 2Q elements. We can order as 1 ? · · · ? P with 0 = d1 = 1 and define P+1 to be 1@(S W ). One can “pull back” [dl > el ] continuously to [0> 1@(S W )] via 7$ () = bS W c@S W then define an integer-valued function q() on [0> P 1@(S W )) by q() = #{c : 11 ( + c@(S W )) = 1}. In [20], it is proved that c 11 ( + c@(S W )) is constant on Jp = [p > p+1 )–a consequence of the fact that the pullback of [dl > el ] either contains or is disjoint from Jp . If Jp>c = c@S W +Jp is contained in , then one calls Jp>c a spectral cell. Define spectral index f = P \ Kp where P = {0> = = = > S 1}. sets Kp = {c : Jp>c } and let Kp Now we come to a basic observation. Suppose that for each p = 1> = = = > P , there is at most one c such that Jp>c . Then is eectively obtained by cutting a finite number of pairwise disjoint subintervals from [0> 1@(S W )] and

13 Sampling and Time-Frequency Localization

287

shifting them by multiples of 1@(S W ). Then S Z ( ) is isomorphic to a closed subspace of S Z ([0> 1@(S W )]) in an evident way and any i 5 PW( ) can be interpolated from its samples along S W Z. Generally, when the spectral index sets Kp all satisfy #Kp T, it will be possible to recover i 5 PW( ) from sampling along T of the Z@S Z cosets of W Z. Denote by cp (n) and cfp (n) the f , respectively, arranged in increasing order, and n-th elements of Kp and Kp define a matrix ³ 2lw c (n) ´ 1 s p [Dp ]sn = s exp = (13.9) S S Here, Dp is a T × #Kp submatrix of the S × S DFT matrix obtained by choosing T sample osets ws 5 {0> 1> = = = > S 1}. One then defines, for each such s, an interpolating function Vs by defining its Fourier transform as ( s (n) 1 2l(ws cp (n))@S if 5 cp S W + Jp , cs () = W sS [Dp ]ns h f V (13.10) f c (n) W S [Fp ]ns h2l(ws cp (n))@S if 5 p S W + Jp .

In (13.10), the matrix D1 p actually refers to any #Kp ×T left inverse of Dp , and Fp refers to any (S #Kp ) × T matrix that satisfies Fp Dp = 0. These joint conditions require pseudoinvertibility of Dp –a condition that depends in turn on the choice of the sampling osets ws . Pseudoinvertibility can always be guaranteed by choosing ws = s for s = 0> = = = > T 1 in Z@S Z since then Dp is a Vandermonde matrix; however, this choice does not necessarily give the best conditioned matrices Dp for all p = 1> = = = > P . The following interpolation theorem for PW( ) was proved by Bresler and Venkataramani [20]. S Theorem 13.9. For = Q l=1 [dl > el ], let the spectral cells Jp>c and spectral index sets Kp be defined as above. Suppose that #Kp T for each p and suppose that sample osets ws > s = 0> = = = > T 1 are chosen such that Dp in (13.9) has full rank for each p. Then, with Vs defined by (13.10), any i 5 PW( ) can be recovered from its samples along the coset union ^T1 s=0 S W Z + ws W by means of the interpolation formula i (w) =

T1 XX

i ((ws + qS )W )Vs (w (ws + qS )W )=

s=0 q5Z

There are several implementation issues relevant to this approach. First, there is the issue of subdividing the spectrum in order to minimize complexity. Choosing a larger value of S corresponds to a finer slicing of the spectrum and, at least in a probabilistic sense, a more uniform distribution of sizes of spectral index sets and more flexible choices for a sampling lattice. However, it also raises the problem of determining good choices of sampling osets. Here, a good choice would be one for which all of the matrices Dp are wellconditioned. Some of these issues are closely related to recent work of Candes, Romberg, and Tao [5] addressing the problem of estimating a finite signal

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Jegrey A. Hogan and Joseph D. Lakey

from a few of its measurements, based on the assumption that its (finite) frequency support is small. Other related issues of aliasing and noise errors are also addressed in [4]. The following problem proposes formulating a connection between sampling and time-frequency localization in the multiband case. Problem 13.10. Formulate and prove an analogue of Proposition 13.8 with = [ @2> @2] replaced by a finite, pairwise disjoint union of compact intervals. Here, the sinc function would be replaced by v = 11b . While samples of eigenfunctions * of S TW might be enough for computing inner products hi> P *i, all values of * are still needed to compute orthogonal expansions q hi> *q i *q . This brings us to an even more fundamental problem.

Problem 13.11. Provide a systematic means of determining the eigenvalues q and constructing the corresponding eigenfunctions *q of S TW when is a finite, pairwise disjoint union of compact intervals.

In the following section, we consider one possible method for addressing this problem by building eigenfunctions for S 1 ^ 2 TW from those of S 1 TW and of S 2 TW .

13.6 Sums of Time-Frequency Localization Operators Here we consider a possible method for determining the eigenvalues and eigenfunctions of the operator S TW under the hypothesis that is the disjoint union of two compact sets 1 and 2 . As before, S i = (iˆ · 11 )b , and we return to fixing W so that TW i = i · 11[W @2>W @2] . Let 1 > 2 be two 2 1 compact, disjoint frequency support sets, and let {* q }, {*q } be complete sets of eigenfunctions for S 1 TW and S 2 TW , respectively. We want to find complete sets of eigenfunctions for S 1 ^ 2 TW . Suppose, then, that # is an eigenfunction for the union operator with eigenvalue . Write # = # 1 + # 2 =

4 X

2 1 (q * q + q *q )

q=0 1 2 2 2 2 1 where S 1 TW * q = q *q and S 2 TW *q = q *q . We make no as 1 2 sumption that q and q are related. We have

# = (S 1 + S 2 )TW # 1 + (S 1 + S 2 )TW # 2 4 X 2 2 1 2 1 1 = (q q *q + q q *q + q S 2 TW *q + q S 1 TW *q )= q=0

Now use self-adjointness of S 2 to define the transition matrix by

13 Sampling and Time-Frequency Localization 1 S 2 TW * q =

4 X

289

2 2 1 hTW * q > *p i*p =

p=0

4 X

2 qp * p =

p=0

Then by the self-adjointness of S 1 and TW , we have 2 S 1 TW * q =

4 X

1 1 2 hTW * q > *p i*p =

p=0

4 X

1 pq * p =

p=0

If 1 , 2 are disjoint, we get orthogonality of S 1 > S 2 . In this case # 1 =

4 X

1 1 q q *q +

q=0

and # 2 =

4 X

4 X

q

4 X

q

q=0

2 2 q q *q +

q=0

q=0

which gives the joint eigenvector problem 1 q = q q +

4 X

qp p ;

p=0

4 X

1 pq * p

p=0 4 X

2 qp * p

p=0

2 q = q q +

4 X

pq p >

p=0

which can be summarized as follows. Proposition 13.12. Suppose that 1 and 2 are disjoint, compact sets and l that the eigenvectors {* q } of S l TW , as operators on PW( l ), have correl sponding nondegenerate eigenvalues listed in decreasing order as q , l = 1> 2. l Let l denote the diagonal matrix with q-th diagonal entry q and let 2 1 > * i. Then any eigenvector— be the matrix with entries qp = hTW * q p eigenvalue pair # and 2 TW , as an operator on PW( 1 ^ 2 ), can P4for S 1 ^ 2 be expressed as # = q=0 (q *q 1 + q * q ) where the vectors = {q } and = {q } together form a discrete eigenvector for the block matrix eigenvalue problem ¶µ ¶ µ ¶ µ 1 = = W 2 In summary, finding the discrete eigenvectors (and eigenvalues) of the matrix on the right is tantamount to finding the eigenvectors (and eigenvalues) of S 1 ^ 2 TW . As noted before, when 1 > 2 are intervals, the blocks are essentially of order Q × Q where Q W × max{| 1 |> | 2 |}. In some cases, the matrix is somewhat simpler. For example, if 1 = L and 2 = M are two intervals of the same length, then any function in PW(M) has the form H* for some * 5 PW(L), where H denotes the operator of multiplication by h2l(M L )w where L denotes the center of L. In this R W @2 case, qp = W @2 h2l(L +M )w *q (w)*p (w) gw, where *q is the q-th prolate spheroidal wavefunction, that is, the q-th eigenfunction of S[ @2> @2] TW .

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Jegrey A. Hogan and Joseph D. Lakey

As noted earlier, Landau, Slepian, and Pollack showed that when L is an interval, there are essentially W |L| eigenvalues of SL TW that are close to one. In fact, Landau [16] showed that with D = W |L|, the ranked eigenvalues of SL TW satisfy bDc1 1@2 and bDc+1 1@2. The method of proof in [16] makes use of the fact that the frequency support is an interval. Nevertheless, it is reasonable to conjecture that this area formula remains true when L is replaced by a finite union of p intervals (for comparison, the method in [17] only yields bDc2p 1@2 bDc+2p ). Proposition 13.12 lends some credence to this area conjecture, since if |L| = |M|, then one expects the large eigenfunction/eigenvalue pairs of the separate operators S l TW to beget at least 2bDc correspondingly large pairs for S 1 ^ 2 TW , while decay away from principal axes suggests that the matrix in the proposition has numerical rank on the order of 2D. Acknowledgments It was a great pleasure for both authors to participate in the 2006 Boulder conference commemorating Larry Baggett’s distinguished career. Both authors would like to thank the anonymous referee for helpful comments. Hogan would like to acknowledge support from a University of Arkansas Fulbright College Research Incentive Grant. This work was undertaken while Hogan was visiting the research centers NUHAG, EUCETIFA, and the Erwin Schr¨ odinger Institute at the University of Vienna. Lakey would like to acknowledge support from Los Alamos National Labs through the LANL-NMSUMOU.

References 1. A. Aldroubi and K. Gr¨ ochenig, Nonuniform sampling and reconstruction in shiftinvariant spaces, SIAM Review, 43 (2001), 585—620. 2. J. J. Benedetto, “Harmonic Analysis and its Applications,” CRC Press, Boca Raton, 1997. 3. R. Bracewell, “The Fourier Transform and its Applications,” McGraw-Hill, New York, 1965. 4. Y. Bresler and R. Venkataramani, Sampling theorems for uniform and periodic nonuniform MIMO sampling of multiband signals, IEEE Trans. Signal Proc., 51 (2003), 3152—3163. 5. E.J. Cand´ es, J. Romberg and T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory, 52 (2006), no. 2, 489—509. 6. I. Daubechies, “Ten Lectures on Wavelets,” CBMS-NSF Regional Conference Series in Applied Mathematics, 61. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. 7. Y.C. Eldar and A.V. Oppenheim, Filterbank reconstruction of bandlimited signals from nonuniform and generalized samples, IEEE Trans. Signal Proc., 48 (2000), no. 10, 2864—2875. 8. C. Herley and P.-W. Wong, Minimum rate sampling and reconstruction of signals with arbitrary frequency support, IEEE Trans. Inform. Theory, 45 (1999), 1555—1564. 9. J.A. Hogan and J.D. Lakey, “Time—Frequency and Time—Scale Methods,” Birkh¨ auser, Boston, 2005.

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10. J.A. Hogan and J.D. Lakey, Sampling and oversampling in shift-invariant and multiresolution spaces I: validation of sampling schemes, Int. J. Wavelets Multiresolut. Inf. Process., 3 (2005), no. 2, 257—281. 11. J.A. Hogan and J.D. Lakey, Periodic nonuniform sampling in shift-invariant spaces, in “Harmonic Analysis and Applications,” 253—287, Appl. Numer. Harmon. Anal., Birkh¨ auser Boston, Boston, MA, 2006. 12. A.J.E.M. Janssen, The Zak transform: a signal transform for sampled time-continuous signals, Philips J. Res., 43 (1998), 23—69. 13. A.J.E.M. Janssen, The Zak transform and sampling theorems for wavelet subspaces, IEEE Trans. Signal Proc., 41 (1993), 3360—3364. 14. K. Khare and N. George, Sampling theory approach to prolate spheroidal wavefunctions, J. Phys. A, 36 (2003), no. 39, 10011—10021. 15. H.J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math., 117 (1967), 37—52. 16. H.J. Landau, On the density of phase-space expansions, IEEE Trans. Inform. Theory, 39 (1993), 1152—1156. 17. H.J. Landau, The eigenvalue behavior of certain convolution equations, Trans. Amer. Math. Soc., 115 (1965), 242—256. 18. D. Slepian, On bandwidth, Proc. IEEE, 64 (1976), 292—300. 19. D. Slepian, Some comments on Fourier analysis, uncertainty and modeling, SIAM Rev., 25 (1983), no. 3, 379—393. 20. R. Venkataramani and Y. Bresler, Perfect reconstruction formulas and bounds on aliasing error in sub-Nyquist nonuniform sampling of multiband signals, IEEE Trans. Inform. Theory, 46 (2000), no. 6, 2173—2183. 21. G.G. Walter and X.A. Shen, Sampling with prolate spheroidal wave functions, Sampl. Theory Signal Image Process., 2 (2003), 25—52. 22. G. Walter, A sampling theorem for wavelet subspaces, IEEE Trans. Inform. Theory, 38 (1992), 881—884. 23. J. L. Yen, On the nonuniform sampling of bandwidth-limited signals, IRE Trans. Circuit Theory, 3 (1956), 251—257.

Chapter 14

Entropy Encoding in Wavelet Image Compression Myung-Sin Song This paper is dedicated with thanks to Larry: For his generosity, his encouragement, and his strength in overcoming adversity

Abstract Entropy encoding is a method of lossless compression that is performed on an image after the quantization stage. It enables one to represent an image in a more ecient way with less memory needed for storage or transmission. In this chapter, we will explore various schemes of entropy encoding and how they work mathematically where applicable.

14.1 Introduction In the process of wavelet image compression, there are three major steps that make the compression possible, namely, decomposition, quantization, and entropy encoding steps. While quantization may be a lossy step where some quantity of data may be lost and may not be recovered, entropy encoding enables a lossless compression that further compresses the data [14], [19], [5]. Unlike thresholding, which results in lossy compression, entropy encoding is completely reversible without losing any data and yet results in compression, i.e., less memory is used to store the data. In this chapter we discuss entropy encoding schemes that are used by engineers (in various applications), how they are implemented in an engineering sense, and how they were “derived” in a mathematical sense. There are a number of dierent entropy encoding schemes and some are better suited for a specific application. It is not trivial to make a decision on which one to use. The discussion on the Karhunen—Lo`eve transform, Kolmogorov entropy, Shannon— Fano entropy, Human coding, and arithmetic coding will give the readers an idea about how entropy encoding is done, what is involved, what is being considered during the process, thus how to choose based upon what is needed. Myung-Sin Song Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026; e-mail: [email protected]

293

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Myung-Sin Song

14.1.1 Wavelet Image Compression In wavelet image compression, after the quantization step (see Figure 14.1), entropy encoding, which is a lossless form of compression, is performed on a particular image for more ecient storage. Either 8 bits or 16 bits are required to store a pixel on a digital image. With ecient entropy encoding, we can use a smaller number of bits to represent a pixel in an image; this results in less memory usage to store or even transmit an image. The Karhunen— Lo`eve theorem enables us to pick the best basis thus to minimize the entropy and error, to better represent an image for optimal storage or transmission. Also, Shannon—Fano entropy (see Section 14.3.3), Human coding (see Section 14.3.4), Kolmogorov entropy (see Section 14.3.2), and arithmetic coding (see Section 14.3.5) are ones that are used by engineers. Here, optimal means it uses least memory space to represent the data, i.e., instead of using 16 bits, it uses 11 bits. Thus, the best basis found would make it possible to represent the digital image with less storage memory. In addition, the choices made for entropy encoding varies; one might take into account the eectiveness of the coding and the degree of diculty of implementation step into programming codes. We will also discuss how those preferences are made in Section 14.3.

14.1.2 Geometry in Hilbert Space While finite or infinite families of nested subspaces are ubiquitous in mathematics, and have been popular in Hilbert space theory for generations (at least since the 1930s), this idea was revived in a dierent guise in 1986 by St´ephane Mallat, then an engineering graduate student. In its adaptation to wavelets, the idea is now referred to as the multiresolution method. What made the idea especially popular in the wavelet community was that it oered a skeleton on which various discrete algorithms in applied mathematics could be attached and turned into wavelet constructions in harmonic analysis. In fact, what we now call multiresolutions have come to signify a cru-

z p

GGm G{

x¡

GGl Gl

j GGGGp

zG { y GGGGGp

Gi G{

GGGGp x¡

GGl Gk

Fig. 14.1 Outline of the wavelet image compression process [14].

j GGGGp

14 Entropy Encoding in Wavelet Image Compression

295

cial link between the world of discrete wavelet algorithms, which are popular in computational mathematics and in engineering (signal/image processing, data mining, etc.) on the one side, and on the other side continuous wavelet bases in function spaces, especially in O2 (Rg ). Further, the multiresolution idea closely mimics how fractals are analyzed with the use of finite function systems. But in mathematics, or more precisely in operator theory, the underlying idea dates back to the work of John von Neumann, Norbert Wiener, and Herman Wold, where nested and closed subspaces in Hilbert space were used extensively in an axiomatic approach to stationary processes, especially for time series. Wold proved that any (stationary) time series can be decomposed into two dierent parts: The first (deterministic, i.e., predictable) part can be exactly described by a linear combination of its own past, while the second part is the opposite extreme; it is unitary (chaos, i.e., unpredictable), in the language of von Neumann. von Neumann’s version of the same theorem is a pillar in operator theory. It states that every isometry in a Hilbert space H is the unique sum of a shift isometry and a unitary operator, i.e., the initial Hilbert space H splits canonically as an orthogonal sum of two subspaces Hv and Hx in H, one that carries the shift operator, and the other Hx the unitary part. The shift isometry is defined from a nested scale of closed spaces Yq , such that the intersection of these spaces is Hx . Specifically, · · · Y1 Y0 Y1 Y2 · · · Yq Yq+1 · · · ^ _ Yq = Hx > and Yq = H= q

q

An important fact about the wavelet application is that then Hx = {0}. However, St´ephane Mallat was motivated instead by the notion of scales of resolutions in the sense of optics. This in turn is based on a certain “artificialintelligence” approach to vision and optics, developed earlier by David Marr at MIT, an approach that imitates the mechanism of vision in the human eye. The connection from these developments in the 1980s back to von Neumann is this: Each of the closed subspaces Yq corresponds to a level of resolution in such a way that a larger subspace represents a finer resolution. Resolutions are relative, not absolute! In this view, the relative complement of the smaller (or coarser) subspace in larger space then represents the visual detail that is added in passing from a blurred image to a finer one, i.e., to a finer visual resolution. This view became an instant hit in the wavelet community, as it oered a repository for the fundamental father and mother functions, also called the scaling function * and the wavelet function #. Via a system of translation and scaling operators, these functions then generate nested subspaces, and we recover the scaling identities that initialize the appropriate algorithms. What

296

Myung-Sin Song

Scaling Operator

ϕ

ψ

W0

... V-1 V0 V1 V2 V3 ... Fig. 14.2 Multiresolution. O2 (Rg )-version (continuous); * M Y0 , # M Z0 .

results is now called the family of pyramid algorithms in wavelet analysis. The approach itself is called the multiresolution analysis (MRA) approach to wavelets. And in the meantime, various generalizations (GMRAs) have emerged. Haar’s work in 1909—1910 had implicitly the key idea that got wavelet mathematics started on a roll 75 years later with Yves Meyer, Ingrid Daubechies, St´ephane Mallat, and others–namely the idea of a multiresolution. In that respect, Haar was ahead of his time. See Figures 14.2 and 14.3 for details. Multiresolutions are represented by strings of subspaces: doubly infinite nested spaces. They are initialized with two spaces as follows: · · · Y1 Y0 Y1 · · · , Y0 + Z0 = Y1 . The word “multiresolution” suggests a connection to optics from physics. So that should have been a hint to mathematicians to take a closer look at trends in signal and image processing! Moreover, even staying within mathematics, it turns out that as a general notion, this same idea of a “multiresolution” has long roots in mathematics, even in such modern and pure areas as operator theory and Hilbert space geometry. Looking even closer at these interconnections, we can now recognize scales of subspaces (so-called multiresolutions) in classical algorithmic construction of orthogonal bases in inner-product spaces, now taught in lots of mathematics courses under the name of the Gram—Schmidt algorithm. Indeed, a closer look at good old Gram—Schmidt reveals that it is a matrix algorithm, hence new mathematical tools involving noncommutativity!

...

2

S0S1

... Fig. 14.3 Multiresolution. o2 (Z)-version (discrete).

S0 S 1

S1

S0

S0

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If the signal to be analyzed is an image, then why not select a fixed but suitable resolution (or a subspace of signals corresponding to a selected resolution), and then do the computations there? The selection of a fixed “resolution” is dictated by practical concerns. That idea was key in turning the computation of wavelet coecients into iterated matrix algorithms. As the matrix operations get large, the computation is carried out in a variety of paths arising from big matrix products. The dichotomy, continuous vs. discrete, is quite familiar to engineers. In the formulas, we have the following two indexed number systems a := (kl ) and d := (jl ), a is for averages, and d is for local dierences. They are really the input for the discrete wavelet transform (DWT). But they also are the key link between the two transforms, the discrete and continuous. The link is made up of the following scaling identities: X kl *(2{ l); *({) = 2 l5Z

#({) = 2

X

jl *(2{ l);

l5Z

P and (low-pass normalization) l5Z kl = 1. The scalars (kl ) may be real or complex; they may be finite or infinite in number. If there are four of them, it is called the “four tap,” and so forth. The finite case is best for computations since it corresponds to compactly supported functions. This means that the two functions * and # will vanish outside some finite interval on a real line. The two number systems are further subjected to orthgonality relations, of which X ¯ l kl+2n = 1 0>n (14.1) k 2 l5Z

is the best known. Our next section outlines how the whole wavelet image compression process works step by step. In our next section, we give the general context and definitions from operators in Hilbert space that we shall need: We discuss the particular orthonomal bases (ONBs) and frames that we use, and we recall the operator-theoretic context of the Karhunen—Lo`eve theorem [1]. In approximation problems involving a stochastic component (for example, noise removal in time-series or data resulting from image processing), one typically ends up with correlation kernels; in some cases as frame kernels; see [11]. In some cases, they arise from systems of vectors in Hilbert space that form frames (see [11]). In some cases, parts of the frame vectors fuse (fusion-frames) onto closed subspaces, and we will be working with the corresponding family of (orthogonal) projections. Either way, we arrive at a family of self-adjoint positive semidefinite operators in Hilbert space. The particular Hilbert space depends on the application at hand. While the Spectral Theorem does allow us to diagonalize

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these operators, the direct application of the Spectral Theorem may lead to continuous spectrum, which is not directly useful in computations, or it may not be computable by recursive algorithms. The questions we address are optimality of approximation in a variety of ONBs, and the choice of the “best” ONB. Here “best” is given two precise meanings: (1) In the computation of a sequence of approximations to the frame vectors, the error terms must be the smallest possible; and similarly (2) we wish to minimize the corresponding sequence of entropy numbers (referring to von Neumann’s entropy). In two theorems, we make precise an operator-theoretic Karhunen—Lo`eve basis, which we show is optimal both in regard to criteria (1) and (2). But before we prove our theorems, we give the two problems an operator-theoretic formulation; and in fact our theorems are stated in this operator-theoretic context. See [11].

14.2 How It Works In wavelet image compression, wavelet decomposition is performed on a digital image. Here, an image is treated as a matrix of functions where the entries are pixels. The following is an example of a representation for a digitized image function: 3 4 i (0> 0) i (0> 1) · · · i (0> Q 1) E i (1> 0) i (1> 1) · · · i (1> Q 1) F E F f (x> y) = E (14.2) F= .. .. .. .. C D . . . . i (P 1> 0) i (P 1> 1) · · · i (P 1> Q 1) After the decomposition, quantization is performed on the image. The quantization may be lossy (meaning some information is being lost) or lossless. Then a lossless means of compression, entropy encoding, is performed on the image to minimize the memory space for storage or transmission. Here the mechanism of entropy will be discussed.

14.2.1 Entropy Encoding In most images, their neighboring pixels are correlated and thus contain redundant information. Our task is to to find less correlated representation of an image, then perform redundancy reduction and irrelevancy reduction. Redundancy reduction removes duplication from the signal source (for instance, a digital image). Irrelevancy reduction omits parts of the signal that will not be noticed by the Human Visual System (HVS).

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Entropy encoding further compresses the quantized values in a lossless manner, which gives better compression overall. It uses a model to accurately determine the probabilities for each quantized value and produces an appropriate code based on these probabilities so that the resultant output code stream will be smaller than the input stream.

14.2.1.1 Some Terminology (i) Spatial Redundancy: correlation between neighboring pixel values. (ii) Spectral Redundancy: correlation between dierent color planes or spectral bands. When a digital image is 1-level wavelet decomposed from the matrix representation in Section 14.2, in this chapter we use (*l ) and (#l ) to denote generic ONBs. However, in wavelet theory [4], there is a tradition for reserving * for the father function and # for the mother function. A 1-level wavelet transform of an Q × P image can be represented as 4 3 1 a | h1 (14.3) f 7$ C D v1 | d1 where the subimages h1 , d1 , a1 , and v1 each have the dimension of Q@2 by P@2. P P a1 = Yp1 Yq1 : *D ({> |) = *({)*(|) = l m kl km *(2{ l)*(2| m) P P h1 = Yp1 Zq1 : # K ({> |) = #({)*(|) = l m jl km *(2{ l)*(2| m) P P 1 v1 = Zp

Yq1 : # Y ({> |) = *({)#(|) = l m kl jm *(2{ l)*(2| m) P P 1 d1 = Zp

Zq1 : # G ({> |) = #({)#(|) = l m jl jm *(2{ l)*(2| m) (14.4) where * is the father function and # is the mother function in sense of wavelet, Y space denotes the average space, and the Z spaces are the dierence space from multiresolution analysis (MRA) [4]. k and j are low-pass and high-pass filter coecients. a1 : the first averaged image, which consists of average intensity values of the original image. Note that only * function, Y space, and k coecients are used here. h1 : the first detail image of horizontal components, which consists of intensity dierence along the vertical axis of the original image. Note that * function is used on | and # function on {, Z space for { values, and Y space for | values; and both k and j coecients are used accordingly. v1 : first detail image of vertical components, which consists of intensity difference along the horizontal axis of the original image. Note that * function

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is used on { and # function on |, Z space for | values, and Y space for { values; and both k and j coecients are used accordingly. d1 : the first detail image of diagonal components, which consists of intensity dierence along the diagonal axis of the original image. The original image is reconstructed from the decomposed image by taking the sum of the averaged image and the detail images and scaling by a scaling factor. It could be noted that only # function, Z space, and j coecients are used here. See [20], [18]. This decomposition is not limited to one step, but it can be done again and again on the averaged detail depending on the size of the image. Once it stops at a certain level, quantization (see [16], [14], [19]) is performed on the image. This quantization step may be lossy or lossless. Then the lossless entropy encoding is performed on the decomposed and quantized image as Figure 14.6. Figure 14.4 illustrates how mathematically wavelet image decomposition is done. An example would illustrate how average, horizontal, vertical, and diagonal details are obtained through the wavelet decomposition of a digital image of an octagon as in Figure 14.5. There are various means of quantization and one commonly used is called thresholding. Thresholding is a method of data reduction where it puts 0 for the pixel values below the thresholding value or some other “appropriate” value. Soft thresholding is defined as follows:

S0SH

S

H

S 0 SD

S0SV

S

V

Fig. 14.4 How the subdivision works.

S

D

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; A if |{| ?0 Wvri w ({) = { if { A A = { + if { ?

(14.5)

and hard thresholding as follows: ( 0 if |{| Wkdug ({) = { if |{| A

(14.6)

where 5 R+ and { is a pixel value. It can be observed by looking at the definitions that the dierence between them is related to how the coecients larger than a threshold value in absolute values are handled. In hard thresholding, these coecient values are left alone. Whereas in soft thresholding, the coecient values are decreased by if positive and increased by if negative [21]. Also, see [20], [9], [18]. Another way of quantization is as follows: Definition 14.1. Let [ be a set, and N be a discrete set. Let T and G be mappings T : [ $ N and G : N $ [. T and G are such that k{ G(T({))k k{ G(g)k>

for all

g5N

Applying T to some { 5 [ is called quantization, and T({) is the quantized valued of {. Likewise, applying G to some n 5 N is called dequantization, and G(n) is the dequantized value of n [16].

Fig. 14.5 Original octagon image.

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Fig. 14.6 Octagon after 2-level decomposition.

During the quantization process, the number of bits needed to store the wavelet-transformed coecients is reduced by reducing the precision of the values. This is a many-to-one mapping, meaning that it is a lossy process resulting in lossy compression. Entropy encoding further compresses the quantized values in a lossless manner, which gives better compression overall. It uses a model to accurately determine the probabilities for each quantized value and produces an appropriate code based on these probabilities so that the resultant output code stream will be smaller than the input stream.

14.2.2 Benefits of Entropy Encoding One might think that the quantization step suces for compression. It is true that the quantization does compress the data tremendously. After the quantization step, many of the pixel values are either eliminated or replaced with other suitable values. However, those pixel values are still represented with either 8 or 16 bits. See Section 14.1.1. So we aim to minimize the number of bits used by means of entropy encoding. The Karhunen—Lo`eve transform makes it possible to represent each pixel on the digital image with the least bit representation according to their probability thus yielding the lossless optimized representation using the least amount of memory.

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14.3 Various Entropy Encoding Schemes In this section, we discuss various entropy encoding schemes with regard to how they work and the mathematics behind them.

14.3.1 The Karhunen—Lo` eve Transform The Karhunen—Lo`eve transform, also known as Principal Components Analysis (PCA), allows us to better represent each pixel on the image matrix using the smallest number of bits. It makes it possible to assign the smallest number of bits for the pixel that has the highest probability, then the next number to the pixel value that has second highest probabilty, and so forth; thus the pixel that has smallest probability gets assigned the highest value among all the other pixel values. An example with letters in the text would better depict how the mechanism works. Suppose we have a text with letters a, e, f, q, in order of probabilty. That is, ‘a’ shows up most frequently and ‘q’ shows up least frequently. Then we would assign 00 to ‘a’, then 01 to ‘e’, 100 to ‘f’, and 101 to ‘q’. In general, one refers to a Karhunen—Lo`eve transform as an expansion in Hilbert space with respect to an ONB resulting from an application of the Spectral Theorem.

14.3.1.1 The Algorithm Our aim is to reduce the number of bits needed to represent an image by removing redundancies as much as possible. The algorithm for entropy encoding using the Karhunen—Lo`eve expansion can be described as follows: 1. Perform the wavelet transform for the whole image (i.e., wavelet decomposition). 2. Quantize all coecients in the image matrix, except the average detail. 3. Subtract the mean from each of the data dimensions. This produces a data set whose mean is zero. 4. Compute the covariance matrix Pq ¯ l \¯ ) ([l [)(\ = fry([> \ ) = l=1 q 5. Compute the eigenvectors and eigenvalues of the covariance matrix. 6. Choose components and form a feature vector (matrix of vectors), (hlj1 > ===> hljq )=

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Eigenvectors are listed in decreasing order of the magnitude of their eigenvalues. Eigenvalues found in step 5 are dierent in values. The eigenvector with highest eigenvalue is the principal component of the data set. 7. Derive the new data set Final Data = Row Feature Matrix × Row Data Adjust= Row Feature Matrix is the matrix that has the eigenvectors in its rows with the most significant eigenvector (i.e., with the greatest eigenvalue) at the top row of the matrix. Row Data Adjust is the matrix with mean-adjusted data transposed. That is, the matrix contains the data items in each column with each row having a separate dimension [16]. Starting with a matrix representation for a particular image, we then compute the covariance matrix using the steps 3 and 4 in the algorithm above. We then compute the Karhunen—Lo`eve eigenvalues. Next, the eigenvalues are arranged in decreasing order. The corresponding eigenvectors are arranged to match the eigenvalues with multiplicity. The eigenvalues mentioned here are the same eigenvalues l in this section, thus yielding the smallest error and smallest entropy in the computation. In computing probabilities and entropy, Hilbert space serves as a helpful tool. For example, take a unit vector i in some fixed Hilbert space H, and an orthonormal basis (ONB) #l with l running over an index set L. We now introduce two families of probability measures, one family Si (·) indexed by i 5 H, and a second family SW indexed by a class of operators W : H $ H. Definition 14.2. Let H be a Hilbert space. Let (#l ) and (!l ) be orthonormal bases (ONB), with index set L. Usually L = N = {1> 2> ===}=

(14.7)

If (#l )l5L is an ONB, we set Tq := the orthogonal projection onto vsdq {#1 > ===> #q }= We now introduce a few facts about operators that will be needed. In particular, we recall Dirac’s terminology [6] for rank-one operators in Hilbert space. While there are alternative notations available, Dirac’s bra-ket terminology is especially ecient for our present considerations. Definition 14.3. Let vectors x, y 5 H. Then hx|yi = inner product 5 C> |xihy| = rank-one operator> H $ H>

(14.8) (14.9)

where the operator |xihy| acts as follows |xihy|z = |xihy|zi = hy|zix>

for all z 5 H=

(14.10)

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Consider an ensemble of a large number Q of similar objects, of which Q z , = 1> 2> ===> where the relative frequency z satisfies the probability axioms: X z 0> z = 1= =1

Assume that each type specified by a value of the index is represented by i () in a real domain [d> e], which we normalize by Z

e

d

|i ()|2 g = 1=

Let {#l ()}, l = 1> 2> ===> be a complete set of orthonomal base functions defined on [d> e] Then any function i () can be expanded as i () =

4 X

()

(14.11)

#l ()i ()g=

(14.12)

{l #l ()

l=1

with { l

=

Z

e

d

Here, { l is the component of i in #l coordinate system. With the normal ization of i , we have 4 X 2 |{ (14.13) l | = 1= l=1

Then substituting (14.12) in (14.11) gives i () =

Z

e

i ()[

d

4 X

#l ()#l ()]g =

l=1

X h#l ()|i i#l

(14.14)

l=1

by definition of ONB. Let H = O2 (d> e). #l : H $ l 2 (Z) and X : l 2 (Z) $ l 2 (Z) where X is a unitary operator. Note that the distance is invariant under a unitary transformation. Thus, using another coordinate system {!m } in place of {#l } would not change the distance. Let {!m }, m = 1> 2> = = = , be another set of ONB functions instead of {#l ()}, l = 1> 2> = = = . Let |m be the component of i in {!m } where it can be expressed in terms of { l by a linear relation |m

4 4 X X = h!m > #l i{l = Xl>m { l l=1

l=1

where X : l 2 (Z) $ l 2 (Z), X is a unitary operator matrix

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Xl>m = h!m > #l i =

Z

e d

!m ()#l ()g=

Also, { l can be written in terms of |m under the following relation:

{ l =

4 4 X X 1 h#l > !m i|m = Xl>m |m m=1

m=1

1 where Xl>m = Xl>m and Xl>m = Xm>l

i () =

4 X

{ l ()#l () =

l=1

X

|l ()!l ()=

Thus X ({l ) = (|l ) and 4 X

{ l #l () =

l=1

{ l = h#l > i i =

4 X

|m !m ()

m=1

Z

e

#l ()i () ()g=

d

()

The squared magnitude |{l |2 of the coecient for #l in the expansion of can be considered as a good measure of the average in the ensemble i ()

Tl =

q X

=1

()

z() |{l |2

and thus can be considered as the measure of importance of {#l }. Note that X Tl 0> Tl = 1= l

See [22]. Then the entropy function in terms of the Tl ’s is defined as X V({#l }) = Tl log Tl = l

We are interested in minimizing the entropy, that is, if {m } is one such optimal coordinate system, we shall have V({m }) = min{#m } V({#l })= P matrix and Let J(> 0 ) =P z i ()i ( 0 ). Then J is a Hermitian P z { { where the normalization T = 1 gives us Tl = J(l> l) = l l l trace J = 1 where the trace means the diagonal sum.

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Then define a special function system {n ()} as the set of eigenfunctions of J, i.e., Z e J(> 0 )n ()g 0 = n n ()= (14.15) d

Thus Jn () = n n (). When the data are not functions but vectors y s whose components are () {l in the #l coordinate system, we have X J(l> l0 )wnl0 = n wnl (14.16) l0

where wnl is the l-th component of the vector n in the coordinate system {#l }. So we get # : H $ ({l ) and also : H $ (wl ). The two ONBs result in X X n { f wn l = n wl for all l> fn = l {l l

n

which is the Karhunen—Lo`eve expansion of i () or vector y . Then {n ()} is the K-L coordinate system dependent on {z } and {i ()}. Then we arrange the corresponding functions or vectors in the order of eigenvalues 1 2 = = = n1 n = = =. P Now, Tl = Jl>l = h#l J#l i = n Dln n where Dln = wnl wn l , which is a doubly stochastic matrix. Then 4 3 1 · · · 0 F E J = X C 0 . . . 0 D X 1 = 0 · · · n

14.3.2 Kolmogorov Entropy This is an example of hard implementation into coding. Thus, it is not very commonly used in industry compared to other methods mentioned.

14.3.2.1 Implementation Let [ be a metric space with distance function . If i 5 [ and u A 0, let B(i> u) := B(i> u){ := {j 5 [ : (i> j) ? u} be the open ball with radius u centered at i . For N [ compact, there is a finite collection of balls B(il > ), l = 1> = = = > q, for each A 0, which cover

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Sq

N: N l=1 B(il > )= Then the covering number Q (N) := Q (N> [) is the smallest integer q for which there is such an -covering of N. Definition 14.4. The Kolmogorov -entropy of N is defined as K (N) := K (N> [) := log Q (N)> A 0 where the log is the logarithm to the base two [3].

14.3.3 Shannon—Fano Entropy For each on an image, i.e., pixel, a set of probabilities sl is computed, Pdatum q where l=1 sl = 1. The entropy of this set gives the measure of how much choice is involved, in the selection of the pixel value of average. Definition 14.5. Shannon’s entropy H(s1 > s2 > = = = > sq ), which satisfy the following: • H is a continuous function of sl . • H should be steadily increasing function of q. • If the choice is made in n successive stages, then H = sum of the entropies of choices at each stage, with weights corresponding to the probabilities of the stages. Pq H = n l=1 sl log sl . n controls the units of the entropy, which is “bits.” Logs are taken base 2 [2, 15]. Shannon—Fano entropy encoding is done according to the probabilities of data, and the method is as follows: • The data are listed with their probabilities in decreasing order of their probabilities. • The list is divided into two parts that have roughly equal probability. • Start the code for those data in the first part with a 0 bit and for those in the second part with a 1. • Continue recursively until each subdivision contains just one datum [2, 15]. An example with letters in the text would better depict how the mechanism works. Suppose we have a text with letters a, e, f, q, r with the following probability distribution: Letter Probability a 0.3 e 0.2 f 0.2 q 0.2 r 0.1

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Then applying the Shannon—Fano entropy encoding scheme on the above table gives us the following assignment. Letter Probability Code a 0.3 00 e 0.2 01 f 0.2 100 q 0.2 101 r 0.1 110 Note that instead of using 8 bits to represent a letter, 2 or 3 bits are being used to represent the letters in this case.

14.3.4 Human Coding This was developed by Human shortly after Shannon’s work. This gives a greater compression compared to Shannon entropy encoding. Human coding is performed as follows: • The data are listed with their probabilities. • The two data with the smallest probabilities are located. • The two data are replaced by a single set containing both, whose probability is the sum of the individual probabilities. • These steps are repeated until the list is left with only one member. See [2].

14.3.5 Arithmetic Coding This is a recent and popular encoding scheme. In arithmetic coding, symbols are restricted in such a way that translation is done into an integral number of bits, thus making the coding more ecient. In this coding, the data are represented by an interval of real numbers between 0 and 1. As the data become larger, the interval required for representation becomes smaller, and the number of bits required to specify that interval increases. Successive symbols of the data reduce the size of the interval according to the probabilities of the symbol generated by the model. The data that are more likely have more reduced ranged compared to the unlikely data, thus fewer bits are used [2, 23]. The above-mentioned entropy encoding schemes are chosen in application in wavelet image compression with the preference of coding simplicity, eectiveness in minimization of entropy, and the lossless compression ratio.

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14.4 Conclusion Among the various entropy encoding schemes some, are relatively easy to implement for applications. The Karhunen—Lo`eve transform method is straightforward to code compared to Kolmogorov’s. In contrast, mathematically it is less transparent compared to Shannon—Fano and Human coding. On the other extreme, there is Kolmogorov encoding, which is a monstrosity for implementation purposes. In this chapter, we have tried to work out the mathematics with an emphasis in entropy encoding. Acknowledgments The author would like to thank Professor Palle Jorgensen, the members of WashU Wavelet Seminar, Professors David Larson, Gestur Olafsson, Peter Massopust, Dorin Dutkay, and Simon Alexander for helpful discussions, Professor Victor Wickerhauser for suggesting [1, 8], and Professor Brody Johnson for suggesting [22].

References 1. Ash RB (1990) Information Theory. Corrected reprint of the 1965 original. Dover Publications, Inc., New York 2. Bell TC, Cleary JG, Witten IH (1990) Text Compression. Prentice Hall, Englewood Cligs, NJ 3. Cohen A, Dahmen W, Daubechies I, DeVore R (2001) Tree approximation and optimal encoding. Applied Computational Harmonic Analysis 11:192—226 4. Daubechies I (1992) Ten Lectures on Wavelets. SIAM, Philadelphia 5. Donoho DL, Vetterli M, DeVore RA, Daubechies I (1998) Data compression and harmonic analysis. IEEE Trans. Inf. Theory, 44 (6):2435—2476 6. Dirac PAM (1947) The Principles of Quantum Mechanics. 3rd ed. Clarendon Press, Oxford 7. Egros M, Feng H, Zeger K (2004) Suboptimality of the Karhunen—Lo` eve transform for transform coding. IEEE Trans. Inf. Theory, 50 (8):1605—1619 8. Field DJ (1999) Wavelets, vision and the statistics of natural scenes. Phil. Trans. R. Soc. Lond. A 357:2527—2542 9. Gonzalez RC, Woods RE, Eddins SL (2004) Digital Image Processing Using MATLAB. Prentice Hall, Englewood Cligs, NJ 10. Jorgensen PET (2006) Analysis and Probability Wavelets, Signals, Fractals. Springer, Berlin, Heidelberg, New York 11. Jorgensen PET, Song M-S (2007) Entropy encoding, Hilbert space, and Karhunen— Lo` eve transforms. J. Math. Phys. 48:103503, 22 pp 12. Pierce JR (1980) An Introduction to Information Theory: Symbols, Signals and Noise. 2nd Edition. Dover Publications, Inc., New York 13. Schwab C, Todor RA (2006) Karhunen—Lo` eve approximation of random fields by generalized fast multipole methods. J. Comput. Phys. 217:100—122 14. Skodras A, Christopoulos C, Ebrahimi T (2001) JPEG 2000 Still Image Compression Standard. IEEE Signal Processing Magazine 18:36—58 15. Shannon CE, Weaver W (1998) The Mathematical Theory of Communication. University of Illinois Press, Urbana and Chicago 16. Smith LI (2002) A Tutorial on Principal Components Analysis. Available at http://csnet.otago.ac.nz/cosc453/student tutorials/principal components.pdf

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17. Song M-S (2005) Wavelet image compression. Ph.D. thesis, The University of Iowa, Iowa City, IA 18. Song M-S (2006) Wavelet image compression. Operator theory, operator algebras, and applications. Contemp. Math. 414:41—73, Amer. Math. Soc., Providence, RI 19. Usevitch BE (2001) A tutorial on modern lossy wavelet image compression: Foundations of JPEG 2000. IEEE Signal Processing Magazine 18:22—35 20. Walker JS (1999) A Primer on Wavelets and Their Scientific Applications. Chapman & Hall, CRC, Boca Raton 21. Walnut DF (2002) An Introduction to Wavelet Analysis. Birkh¨ auser, Boston 22. Watanabe S (1965) Karhunen—Lo` eve expansion and factor analysis: Theoretical remarks and applications. Transactions of the Fourth Prague Conference on Information Theory, Statistical Decision Functions, Random Processes. Academia, Prague 635—660 23. Witten IH, Neal RM, Cleary JG, (1987) Arithmetic coding for data compression. Communications of the ACM 30(6):520—540

Symbols

The number systems C, R, Z, Rq , Zq , T, Tq The spaces O2 (R), O2 (Rq ), O2 (Tq ), c2 AR atomic regular (group) FMRA frame multiresolution analysis GMRA generalized multiresolution analysis IFS iterated function system(s) MRA multiresolution analysis MSF minimally supported frequency wavelets (wavelets arising from wavelet sets) OEP oblique extension principle PCA principal components analysis UEP unitary extension principle

313

Index

A adjoint lattice 76 admissible group representation see representation, admissible algebra 13, 19, 28, 35, 37, 42, 74, 79, 194—196, see also group algebra F W - xvi, 36, 74, 243, 251, 267, 269 Feichtinger 84 Fourier xiii, 34, 37 Fourier—Stieltjes 34, 37 – isomorphism 21 measure see measure algebra – of operators 195 operator xv, 195—197, 219 analysis xvii, 13, 18, 116, 132, 134, 219, 285 Fourier xvii, 78, 139 harmonic x, xii, xiv, xvi, xvii, 34, 47, 84, 218, 236, 294 multiresolution xiii—xvii, 49, 50, 59, 104, 107, 136, 155, 178, 241—243, 245, 249, 251, 258, 269, 270, 296, 299, see also multiresolution frame xiii, 59, 60, 137, 138, 178 generalized xiii, xiv, 47, 59, 60, 138, 153—155, 178, 181, 249, 250 projective xvi, 239—251, 254, 257—259, 262, 266, 268—270 – operator 85 principal components 303 time-frequency 71, 84, 218 wavelet 116, 218, 296 approximation xiv, 105, 109, 119, 120, 122—124, 126—128, 133, 166, 234, 235, 297, 298 homogeneous – property 71, 81, 82, 92

Ramanathan—Steger strong 94 Ramanathan—Steger weak 93 strong 94 weak 93 – of integrals xii, 3, 4 atom 38, 41, 72, 222 atomic regular representation xii, 33, see also group, [AR] B E-spline 107—109, 117, 127 Baggett, Lawrence W. ix—xvii, xxiii, xxv, xxix, 33, 34, 39—41, 47, 54, 59, 61, 65, 71, 74, 103, 106, 131, 136, 150, 153, 154, 158, 159, 177, 179, 193, 207, 217, 218, 236, 239, 249, 270, 275, 290, 293 Baggett consistency equation of see equation, consistency –’s problem 162, 163 – for frame wavelets xiv, 153, 154 – for Parseval wavelets 159, 161 band-limited function see function, band-limited basis xiii, 23, 28, 29, 71, 85, 98, 169, 182, 219, 258, 294, 298 dual 98 Fourier 218, 219 incomplete xiv, 71, 73, 75, 77, 79, 80, 100 orthogonal 25, 282, 288, 296, see also basis, orthonormal orthonormal xiii, xvi, 20, 29, 48, 50, 51, 56, 61, 62, 64, 65, 72, 73, 78, 85, 100, 117, 131, 136, 137, 139, 140, 153, 155, 168, 169, 177—179, 181, 193, 197, 218,

315

316 220, 222, 224, 233, 235, 242, 280, 282, 283, 297, 304 redundant 74 Riesz 71—80, 85, 86, 94, 97, 100, 114 symplectic 29 wavelet 51, 62, 64, 65, 114, 194, 295 Bessel – ane system 170 – bound 85 – sequence 49, 58, 61, 85, 87, 121, 134, 165, 168, 170 – wavelet 121, 195 Beurling density 79, 80, 86 bimodule 244, 268 biorthogonal 75, 76, 85, 86, 98 – wavelet 107, 108, 114 biorthogonality relation 107 Wexler—Raz 75, 79 C F W -algebra see algebra, F W F W -module see module, F W canonical dual frame see frame, dual, canonical Cantor dust 64, 65 coecient criterion 194, 196 complete sequence 85 consistency equation see equation, consistency continuous wavelet transform see transform, continuous wavelet convergence 4, 6, 7, 34, 73, 88, 113, 134, 143, 224, 269 almost everywhere 35, 148 O2 - 75 norm 165 unconditional 72, 85, 86, 131, 134, 141 weak 82, 87, 89, 165, 166 convolution 5, 13, 18, 19, 28, 35, 36, 105, 120, 121, 167, 219, 221, see also measure, Bernoulli convolution D I. Daubechies xiii, xiv, 48, 73, 74, 78, 81, 160, 296, see also wavelet, Daubechies decomposition 21, 22, 30, 54, 171, 218, 266, 293 frame 121, 122 – of a representation x, 61 wavelet 298—300, 302, 303 density theorem xiv, 71—82, 93, 99, 100

Index dilation xiii—xv, 40, 49, 60, 64, 65, 73, 117, 131, 134—136, 145, 154—156, 177—185, 187—191, 196—198, 200, 201, 203—205, 208—212, 240, 242—249, 251, 252, 254, 255, 257—261, 267—269 – matrix see matrix, dilation – operator xiii, 195 dimension function see function, dimension discrete wavelet transform see transform, discrete wavelet domain Fourier 167 frequency 115, 120, 131, 195, 196, 243, 244, 247, 252 time 115, 245 duality 60, 157, 218, 235 – principle 76, 78, 79 dyadic – wavelet 197, 199, 207 E eigenfunction 276, 282—285, 288—290, 307 eigenspace 27 entropy 293, 294, 298, 304, 306—309 – encoding xvii, 293, 294, 298—300, 302, 303, 308—310 equation characteristic wavelet xiv, 131, 133, 136, 144, 147, 149 consistency 145, 158, 159, 181, 182 refinement 137 scaling see scaling identity exact sequence 85 expansion 6, 223, 228, 229 Fourier 219, 280 nonharmonic 72 frame 71, 72, 75, 76, 85 Karhunen—Lo` eve 303, 306, 307 lacunary 219 nonorthogonal see painless nonorthogonal expansions orthogonal 288 exponential xvi, 80, 81, 220, 222, 224, 225, 229, 233, 240, 244, 250 complex 219 orthogonal xvi, 217—220, 224—230, 232, 233 F father function

see function, father

Index filter

xiv, 106, 113, 114, 117, 137, 138, 146, 147, 162, 179, 248, 251, 258, 260 finite-impulse-response xiii, 105, 121 high-pass 127, 137, 146, 299 low-pass 65, 137, 146, 162, 242, 243, 248, 251, 252, 258—261, 266—268, 299 FMRA see analysis, multiresolution, frame J. Fourier xvi Fourier – algebra see algebra, Fourier – analysis see analysis, Fourier – basis see basis, Fourier – expansion see expansion, Fourier nonharmonic see expansion, Fourier, nonharmonic – series xii, 4, 7, 8, 141, 277, 280 lacunary 219 random 219 – transform see transform, Fourier Fourier—Stieltjes – algebra see algebra, Fourier— Stieltjes – transform see transform, Fourier— Stieltjes Fr´ echet distance 87 fractal xv, xvi, 179, 218, 219, 295 – measure 219 measure on 217 frame xiii, xiv, xvi, xvii, 48, 49, 53, 57—63, 71—82, 85, 86, 89, 93—97, 99, 100, 105, 117, 119, 120, 122, 124, 126—128, 131, 137, 153, 155, 165, 236, 245, 267, 268, 297, 298 – bounds 85, 89, 95, 138, 164, 170, 171, see also Bessel bound – coecients 72 – decomposition see decomposition, frame dual 75, 86, 97, 133, 135, 154, 164, 171 canonical 75, 76, 85, 86, 93, 95, 160 – expansion see expansion, frame Gabor xiii, xiv, 71, 73—75, 78—82, 86, 89, 92—94, 97, 100 localized 80—82 module see module frame – multiresolution analysis see analysis, multiresolution, frame multiwavelet 105 dual 117—120, 123—126, 128 – operator 75, 85 Parseval see Parseval frame quasiane 135 – sequence 86

317 tight xiii, 104, 131, 137, 153, 155—157, 160, 179, 267, 269, 270 – transform see transform, frame wavelet xiii, xiv, 58—62, 103, 104, 106, 109, 110, 113, 114, 116, 117, 127 dual xiv, 103—112, 114—117, 119—121, 123, 124, 126—128 tight 108, 109, 117, 121, 127 – wavelet 60, 62, 153—157, 162, 164, 167, 170, 171, 194, 196, see also Baggett’s problem for frame wavelets Weyl—Heisenberg xiii framelet 107, 153, 154, 156, 160, 164, 170, 171, see also frame wavelet frequency – domain see domain, frequency –-localized 73, 74, see also wavelet set function 283 E-spline see E-spline band-limited xvi, xvii, 80, 81, 275, 276, 278, 279, 282, 283 dimension xiv, 138, 140, 154, 157, 158, 163, 187 eigen- see eigenfunction exponential see exponential father 295, 299, see also scaling function mother 295, 299, see also wavelet function multiplicity see multiplicity function refinable 103—105, 107—109, 111, 113, 116—120, 125—128 scaling see scaling function – space see space, function spectral xiv, 131, 136, 139, 140, 147, 156, 157 unitary see unitary function wavelet see wavelet function window see window function G Gabor – frame see frame, Gabor – system xiv, 71—75, 77—82, 93, 135, 168 Gelfand pair xii, 13, 14, 18—20, 22, 24—27, 29—31 generalized multiresolution analysis see analysis, multiresolution, generalized GMRA see analysis, multiresolution, generalized

318 group 13, 33, 37—42, 47, 48, 51, 56, 57, 60, 62, 64, 133, 145, 194, 198, 217, 218, 242, 251, 266 d{ + e 48, 218 Abelian 41, 145 admissible 47 admissible – representation see representation, admissible ane xii, xiii, 38, 47, 52 – algebra 18, 33, 36 amenable 36 [AR] xii, xiii, 33, 34, 39—44, see also atomic regular representation Baumslag—Solitar 64, 65 Borel sub– 27 compact xii, xiii, 13, 33, 38, 39, 41, 43 continuous 218 countable 56 discrete 41, 57, 74, 218 finite 13 general linear 27, 40 Heisenberg xii, 13, 14, 18, 30, 40, 48, 52, 57, 74, 77, 218, 251 hyperoctahedral 13 Klein four 201 Lie xii, 13 locally compact ix, x, xii, xiii, 33, 34, 36, 37, 39, 41, 47—49, 51, 84 locally compact Abelian 33, 34, 39, 41, 44 multiplicative 30 nilpotent ix non-Abelian xii, 27, 34 quotient 256, 263 – representation see representation, group second countable 34, 39, 41 semidirect product 218 symmetric 13 symplectic 16, 17, 19—24, 26, 27, 29 unimodular xiii, 35, 40 unitary 13, 28—30, 194, 196 H harmonic analysis see analysis, harmonic Hilbert space see space, Hilbert I image processing see signal/image processing inner product 7, 14, 15, 20, 23, 29, 30, 35, 104, 106, 109, 110, 198, 224, 240, 244,

Index 246, 255—257, 259, 262, 264, 265, 267, 288, 296, 304 interpolation xv, 80, 179, 193—203, 205, 208, 209, 213, 276, 277, 279, 281, 285, 287 J Janssen representation

79

K Kirillov

xii

L lattice xiv, xvi, 71—82, 86, 134, 135, 161, 177, 183, 187, 193, 207, 219, 239, 241, 244, 252, 257—259, 279, 286, 287, see also adjoint lattice, rectangular lattice, separable lattice dual 78 M matrix xvi, 17, 33, 35, 51, 77, 79, 82, 105, 110—120, 124—126, 161, 194, 197, 207, 210, 219, 251, 254, 257—259, 266, 268, 270, 276, 279, 281, 283—285, 287, 289, 290, 296—299, 303—307 dilation xv, xvi, 105, 106, 133—136, 154, 165, 177—182, 190, 191, 193, 210, 213, 239—242, 245—252, 254, 257, 258, 266, 268, 270 symplectic 77 transition 288 measure absolutely continuous 47, 48, 61 – algebra 37, 198 atomic 42 Bernoulli convolution 217, 218, 220—222 Borel 34, 138, 145 continuous 42 counting 56, 145 Haar 34, 39, 41, 43, 44, 47, 51, 56, 60, 139, 145, 217, 218 Hausdorg 64 Hutchinson 217 Lebesgue 155, 197, 200, 203, 219, 221 probability 41, 60, 217, 222, 304 projection-valued 60 rotation invariant 34 spectral see spectral measure minimal sequence 85

Index minimally supported frequency wavelet see wavelet, minimally supported frequency modulation 72 – operator 83 module 239, 241, 245, 248, 249, 251, 255, 268, 269 F W - 242, 249, 267, 269 F(T2 )- 239, 241, 243—249, 251, 254, 255, 257—259, 262, 266, 268, 270 F(Tq )- 239, 240, 249, 270 – frame 267—270 free 244, 249, 262 Hilbert 243, 249, 269 initial 241, 244—249, 251, 254, 259, 267—269 projective 243, 244, 246—251, 255, 257, 258, 262, 266—268 right 240 wavelet xvi, 239, 241, 245, 249, 251, 258, 262, 270 mother function see function, mother MRA see analysis, multiresolution MSF wavelet see wavelet, minimally supported frequency multiplicity function xiv, 47, 60, 139, 141, 142, 145, 147, 149, 154, 157, 181, 187, 249, 250 multiresolution 47, 59, 136, 145, 294—296 – analysis (frame, generalized, projective) see analysis, multiresolution N number theory xii Nyquist density 80 O operator – algebra see algebra, operator dilation see dilation integral x, 179, 218 modulation see modulation operator positive 38, 75, 76, 85, 297 scaling see scaling operator subdivision see subdivision translation see translation operator unitary see unitary operator optimality criteria 298 orthogonal – basis see basis, orthogonal – exponential see exponential, orthogonal

319 – wavelets 196, see also biorthogonal, semiorthogonal, orthonormal orthonormal – basis see basis, orthonormal – wavelet see wavelet, orthonormal P painless nonorthogonal expansions 73 Parseval – frame xiii, 49, 56, 106, 131, 135, 138—145, 147, 148, 194, 267, 268, 270 –’s identity 8, 53, 233 – wavelet xiv, 131, 132, 137, 138, 141, 142, 144—147, 149, 153—155, 157—164, 167, 170, 179, 194, see also Baggett’s problem for Parseval wavelets partition 202 measurable 198, 200—202, 204, 205, 209 periodization xiii, 244 R Radon—Nikodym derivative 44, 139 reconstruction xvii, 121—123, 134, 267, 285 rectangular lattice 73 redundancy 245, 268, 286, 298, 299, 303, see also basis, redundant refinable – distribution 114, 118 – function see function, refinable representation ix, 14—17, 20, 21, 34—39, 41—44, 48, 51—63, 65, 71, 250, 275, 278, 298, 299, 302, 304, 309 admissible xiii, 47, 49, 52—56, 58, 62 atomic regular see atomic regular representation frame 53, 54, 56, 58, 60, 122 group xii, xiv, 17, 33, 38, 41, 43, 47, 51, 52, 56, 64, 74, 77, 218 Janssen see Janssen representation oscillator 14, 16, 17, 20—22, 26 square-integrable xii, xiii, 34, 37, 38, 43, 44, 47—49, 51, 52, 55, 57 unitary see unitary representation Walnut see Walnut representation wavelet 107, 111, 123, 179 resolution see also multiresolution coarse 122 fine 122 Riesz basis see basis, Riesz Riesz sequence 86 Ron—Shen Duality Principle 76

320 S sampling xiii, xvi, xvii, 47, 80, 81, 178, 275—279, 282—288 scaling 55, 64, 300 – function xiv, 47—51, 64, 65, 136, 137, 139, 141, 178, 190, 241, 242, 244, 245, 252, 258, 261, 270, 295 generalized 139, 141, 142, 145—147 pseudo- 137—139 – identity 295, 297 – operator 295 – set 177, 181, 183—186, 189 generalized 181, 182, 187 – vector 60, 62 semiorthogonal 136, 138, 141, 142, 144—147, 149, 155—160, 162—164 separable lattice 78 set wavelet see wavelet set wedding cake xv, 207, 211 signal/image processing xiii, xiv, xvi, xvii, 103, 123, 273, 275—277, 283, 285—287, 295—297, see also wavelet image compression smoothness xiv, 113, 116, 153, 154, 164, 165, 178, 270 space see also subspace, eigenspace amalgam 83, 84 Banach 34, 41, 42, 84, 110 Bargmann—Fock 80 Besov 106 core 47, 48, 59, 156—158 dual 36, 42 fractal xvi function 27, 34, 84, 103, 104, 106, 109, 246, 282, 295 Hardy 171 Hilbert xiii, xiv, xvi, 35, 48, 49, 71, 83, 85, 109, 110, 131, 134—136, 144—147, 149, 165, 217—220, 222, 224, 229, 242, 249, 250, 267, 269, 270, 294—297, 303, 304 measure 41, 42, 194 modulation 83, 84 – of negative dilates 153, 154, 156—164, 167, 168, 170, 171 Paley—Wiener xvii, 276, 277, 283 probability 221 resolution 138, 141 Riemannian symmetric 13 sequence 120 shift-invariant 120, 136, 139, 140, 156, 157, 159, 160, 163, 277, 283

Index Sobolev xiv, 103, 104, 106, 109—114 tiling 219 wavelet 141, 194 spectral – band 299 – cell 286, 287 – function see function, spectral – index set 286, 287 – measure 218 – multiplicity 138 – set 222 – theorem 297, 298, 303 spectrum 286, 287 continuous 48, 298 purely 48, 51 Stone—von Neumann theorem 16, 17 Stone—Weierstrass theorem 233 Stone’s theorem 60, 138, 249 subdivision 116, 118, 120, 286, 300, 308 subspace xiii, 21, 25—27, 35, 37, 38, 43, 47, 52, 55, 56, 59, 83, 125, 133, 136, 138, 145, 147, 155, 178, 181, 240, 242, 243, 245, 249, 250, 254, 259, 269, 270, 276, 277, 287, 294—297 core xiv, 138, 145, 146, 249 shift-invariant 51, 139, 140, 145, 276 wavelet 243 support xv, xvi, 8, 34, 43, 73, 105, 107— 109, 111, 113—116, 118, 120, 121, 124—128, 134, 135, 139, 155, 167, 168, 177, 194, 196—198, 217, 219, 222, 248, 286, 288, 290, see also wavelet, compactly supported symplectic – basis see basis, symplectic – group see group, symplectic – matrix see matrix, symplectic synthesis operator 85 T theorem density see density theorem spectral see spectral theorem Stone—von Neumann see Stone—von Neumann theorem Stone—Weierstrass see Stone— Weierstrass theorem Stone’s see Stone’s theorem thresholding xvii, 293, 300, 301 time – domain see domain, time –-frequency

Index – analysis see analysis, timefrequency – localization xvi, xvii, 74, 275, 276, 282, 286, 288 – plane 72—74 – shift operator 72, 84 transform Bargmann 84 compact 121 Fourier xiii, xv, 34, 37, 42, 43, 50, 52, 53, 82, 83, 104, 115, 131, 135, 139, 147, 155, 167, 177—179, 181, 193, 195, 198, 220—222, 224, 240—245, 250, 252, 254, 259, 261, 270, 275—277, 287 discrete 18, 286 operator-valued 20, 23 Fourier—Stieltjes 34, 37 frame 104, 105, 120, 121 fast 104, 105, 117, 118, 120—124, 126—128 – reconstruction 121—123 Karhunen—Lo` eve 293, 302, 303, 310 metaplectic 77 multiwavelet 105, 123, 124 Plancherel 37 wavelet xiii, 47, 48, 57—61, 63, 65, 120, 123, 299, 302, 303 continuous xiii, 33, 44, 47, 48, 51, 52, 57, 297 discrete xiii, 47—49, 56—58, 297 Zak 74, 79, 277, 278 translation xiii, 36, 47, 48, 64, 72, 83, 133—136, 177, 180—182, 184, 186, 189, 198, 201, 203—205, 207, 208, 210—212, 242, 243, 249, 250, 253, 270 – operator xiii, xiv, 48, 49, 56, 83, 131, 155, 195, 219, 295 U UEP see unitary extension principle uniformly separated 87 unitary 295 – dual 15, 16 – extension principle xiv, 103, 108, 109, 117 – function xv – group see group, unitary interpolation xv, 194, 196 – operator xv, 16, 21, 23, 35, 36, 48—51, 54, 56—58, 60, 63, 64, 73, 77, 131, 138, 145—147, 149, 194—196, 242, 250, 257, 295, 305

321 – representation x, 15, 16, 21, 22, 33, 35, 42, 56, 138, 249 W Walnut representation 79 wavelet ix, xii—xvii, 47, 48, 50, 57, 61, 63—65, 81, 103, 104, 108—111, 117, 119, 122, 123, 131, 132, 135—137, 141, 153—156, 158, 160, 163, 178, 179, 181, 190, 193—196, 219, 236, 241—244, 246, 248, 266, 270, 294—297, 299, 310 – algorithm 295 – analysis see analysis, wavelet Baggett’s problem for frame –s see Baggett’s problem for frame wavelets Baggett’s problem for Parseval –s see Baggett’s problem for Parseval wavelets – basis see basis, wavelet Bessel see Bessel wavelet biorthogonal see biorthogonal wavelet characteristic – equation see equation, characteristic wavelet Cohen 132, 144 compactly supported 104, 105, 108, 109, 126, 127, 133, 154, 163, 164, 178, 179, 195, 196, 252, 270, 297 Daubechies 178 – decomposition see decomposition, wavelet discrete 295 discrete – transform see transform, discrete wavelet dual 107, 114, 153, 170 dyadic see dyadic wavelet – expansion see expansion, wavelet frame see frame wavelet – frame see frame, wavelet frequency localized see wavelet set – function 107, 127, 295 Haar 132, 178, 252, 258 – image compression xvii, 293, 294, 297, 298, 309 Littlewood—Paley 178 localized 178, 179 Meyer 252 minimally supported frequency xv, 134, 177—179, 194—196 – module see module, wavelet MSF see wavelet, minimally supported frequency

322 orthonormal xiii, xiv, 47, 48, 59, 62, 65, 104, 117, 131, 132, 135, 137, 154, 177, 193—198, 242 Parseval see Parseval wavelet Riesz 114, 194, 196 – set xiv, xv, 177—191, 193—199, 201—203, 205, 207—213 Journ´ e xv, 48, 178, 179, 185, 188 Shannon 60, 178 – space see space, wavelet

Index – subspace see subspace, wavelet – transform see transform, wavelet weak convergence see convergence, weak Wexler—Raz biorthogonality relation see biorthogonality relation, Wexler—Raz window function 72, 73, 78—81, 83, 84 Z Zak transform

see transform, Zak