ISBN 978-90-78677-26-0
I
9 789078 677260
ATLANTIS STUDIES IN MATHEMATICS FOR ENGINEERING AND SCIENCE VOLUME 4 SERIES EDITOR: C.K. CHUI
Atlantis Studies in Mathematics for Engineering and Science Series Editor: C. K. Chui, Stanford University, USA
(ISSN: 1875-7642)
Aims and scope of the series The series 'Atlantis Studies in Mathematics for Engineering and Science (AMES) publishes high quality monographs in applied mathematics, computational mathematics, and statistics that have the potential to make a significant impact on the advancement of engineering and science on the one hand, and economics and commerce on the other. We welcome submission of book proposals and manuscripts from mathematical scientists worldwide who share our vision of mathematics as the engine of progress in the disciplines mentioned above. All books in this series are co-published with World Scientific. For more information on this series and our other book series, please visit our website at: www.atlantis-press.com/publications/books
. , ATLANTIS ., PRESS AMSTERDAM -PARIS
w-
World Scientific
@ ATLANTIS PRESS I WORLD SCIENTIFIC
The Wavelet Transform
Ram Shankar PATHAK Department of Mathematics Banaras Hindu University Varanasi, India
~ATLANTIS
.,
PRESS
AMSTERDAM - PARIS
11»
World Scientific
Atlantis Press
29, avenue Laumiere 75019 Paris, France For information on all Atlantis Press publications, visit our website at: www.atlantis-press.com Copyright
This book, or any parts thereof, may not be reproduced for commercial purposes in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system known or to be invented, without prior permission from the Publisher.
Atlantis Studies in Mathematics for Engineering and Science
Volume I: Continued Fractions: Volume I: Convergence Theory- L. Lorentzen, H. Waadeland Volume 2: Mean Field Theories and Dual Variation - T. Suzuki Volume 3: The Hybrid Grand Unified Theory- V. Lakshmikantham, E. Escultura, S. Leela
ISBN: 978-90-78677-26-0 ISSN: 1875-7642
@ 2009 ATLANTIS PRESS I WORLD SCIENTIFIC
To my wife Indrawati Pathak
Preface
The wavelet transform has emerged as one of the most promising function transforms with great potential in applications during the last four decades. The present monograph is an outcome of the recent researches by the author and his co-workers, most of which are not available in a book form. Nevertheless, it also contains the results of many other celebrated workers of the field. The aim of the book is to enrich the theory of the wavelet transform and to provide new directions for further research in theory and applications of the wavelet transform. The book does not contain any sophisticated Mathematics. It is intended for graduate students of Mathematics, Physics and Engineering sciences, as well as interested researchers from other fields. The Fourier transform has wide applications in Pure and Applied Mathematics, Physics and Engineering sciences; but sometimes one has to make compromise with the results obtained by the Fourier transform with the physical intuitions. The reason is that the Fourier transform does not reflect the evolution over time of the (physical) spectrum and thus it contains no local information. The continuous wavelet transform (W111 f)(b,a), involving wavelet lfl, translation parameter b and dilation parameter a, overcomes these drawbacks of the Fourier transform by representing signals (time dependent functions) in the phase space (time/frequency) plane with a local frequency resolution. The Fourier transform is restricted to the domain LP (JRn) with I defined for I
~
~ p ~
2, whereas the wavelet transform can be
p < oo with appropriate kernel lfl. The frequency resolution is controlled by
dilation parameter a, and for small
Ia I, (W111 j)(b,a) represents high frequency components
of the signal f. Hence, it is desirable to know asymptotic expansions of (W111 f)(b,a) for small and large values of the parameters. The convolution operation associated with an integral transform increases considerably the applicability and theory of the transform. Motivated from the work of 1.1. Hirschman vii
viii
The Wavelet Transform
Jr. on Hankel convolution, a theory of wavelet convolution is developed. This provides the interesting property that the product of two wavelet transforms could be a wavelet transform. The book assumes that the reader has a background in the elements of analysis. Chapter I essentially deals with the perquisite material for the theory of distributions and certain integral transforms and related topics. It gives a brief idea about wavelets and wavelet transforms. In Chapter 2 using properties of Fourier transform certain approximation properties of the wavelet transform are obtained. Certain relations between wavelet transform, Hilbert transform, generalized Hilbert transform and Riesz fractional integrals, and also between their generalizations are obtained. These relations are used to derive new inversion formulae for the wavelet transform of functions belonging to LP -space under different conditions. Inversion formulae, Parseval formulae, boundedness and approximation results are also obtained in certain weighted LP -spaces. Some of the results are extended to distributions. In Chapter 3 it is shown that the wavelet transform is a continuous linear map of the Schwartz space Y' (ffi.n) into a similar space Y' (ffi.n x ffi.+) when the wavelet belongs to
Y'(ffi.n). The composition of two wavelet transforms is defined and the continuity of the composition operator is investigated. A reconstruction formula for the composition of the wavelet transforms is obtained. Distributional extensions of these transforms are given. Boundedness results regarding these transforms on certain weighted Sobolev spaces are obtained. Abelian theorems are of considerable importance in solving boundary value problems of Mathematical Physics. Abelian theorems for various integral transformations are available in the literature. In this chapter we establish Abelian theorems for the wavelet transform of functions and afterwards derive certain distributional results. Chapter 4 is devoted to the study of the continuous wavelet transform on certain Gel'fandShilov spaces of typeS. It is shown that, for wavelets belonging to the one type of S-space defined on ffi., the wavelet transform is a continuous linear map of the other type of the Sspace into a space of the same type (latter type) defined on ffi. x ffi.+. The wavelet transforms of certain ultradifferentiable functions are also investigated. Chapter 5 contains a study of the continuous wavelet transform on certain Gel'fand-Shilov spaces of type W. The continuity and boundedness results for continuous wavelet transform are obtained on some suitably designed spaces of type W defined on ffi. x ffi.+, C x ffi.+ and
CxC.
Preface
ix
In Chapter 6 the continuous wavelet transform is studied on the generalized Sobolev space B~k·
Roundedness results in this Sobolev space are obtained. Local generalized Sobolev
space is defined and some of its important properties are discussed. Wavelet transform with compactly supported wavelet is also studied. Generalized translation and convolution operators for a general integral transform are defined in order to develop a unified theory of convolutions for all commonly used integral transforms. Thereby, definitions of translation and convolution for continuous wavelet transforms are given and their properties are investigated. A brief account of translation and convolution for discrete wavelet transform is also presented. A basic function D(x,y, z) associated with general wavelet transform is defined and its properties are investigated. Using D(x,y, z), translation and convolution associated with the wavelet transform are defined and certain existence theorems are proved. An approximation theorem involving wavelet convolution is also proved in Chapter 7. Chapter 8 contains derivation of a relation between the convolution associated with the wavelet transform and convolution associated with the Fourier transform. This relation is used to define the wavelet convolution transform and study its properties. Existence theorems are proved. We show that the product of two wavelet transforms could be a wavelet transform. Application of the wavelet convolution in approximation of functions is given. In Chapter 9 asymptotic expansion of the wavelet transform of a function f with respect to the wavelet II' is derived when the dilation parameter belongs to an open subset of (O.oo) and the translation parameter goes to infinity. It is assumed that both
f and II' possess
power series representations in descending powers oft. Asymptotic expansion of Mexican hat wavelet transform is obtained. Following Wong's technique asymptotic expansion of the wavelet transform is derived when translation parameter is fixed and dilation parameter goes to infinity and also to zero. Asymptotic expansions for Morlet wavelet transform, Mexican hat wavelet transform and Haar wavelet transform are obtained as special cases in Chapter 10. The author received inspiration and guidance from Professor C. K. Chui in writing this monograph. His sincere thanks are due to Professor Chui, the Editor of AMES series and Dr. Z. Karssen, Publisher Atlantis Press. The author is also thankful to Professor K. K. Azad (University of Allahabad) for his assistance in so many ways. The author's coworkers G. Pandey, A. Pathak and S. Verma were of great help in the preparation of the
X
The Wavelet Transform
monograph. In preparation of the manuscript the author received assistance from Mr. P. K. Sinha, Ms. Shikha Gaur and staff of Pushpa Publishing House, Allahabad (India). The research work was supported by the U.G.C.-Emeritus Fellowship (New Delhi). R. S. Pathak
Contents
vii
Preface 1.
1.1 1.2 1.3 1.4 1.5 1.6 1.7
1.8
2.
3.
1
An Overview Introduction . Introduction to distribution theory The Fourier transform . . . . . . . The Fourier transformation in Y'(!Rn) 1.4.1 Operation-transform formulae . The Hilbert transform . . . . . . . Wavelets . . . . . . . . . . . . . . . . . The continuous wavelet transform . . . 1.7.1 Relationship with Fourier transform 1.7.2 Parseval relation for the wavelet transform 1.7.3 Reconstruction formula . . . . 1.7.4 The discrete wavelet transform . Asymptotic expansion . . . . . . . . . . 1.8.1 The Mellin transform technique . The distributional approach 1.8.2
I 5 6 7 8 9 12 14 14 15 15 16 18
19
The Wavelet Transform on lJ'
21
2.1 2.2 2.3
21 26
2.4 2.5
Introduction . . . . . . . Approximation properties . . . . . . . . . . . . . . . . . . . . Wavelet transform, Hilbert transform and Fractional integrals . 2.3.1 Inversion formulae . . . . . . . . . . . . . . . . . . . 2.3.2 Approximation . . . . . . . . . . . . . . . . . . . . . 2.3.3 lnvariance under rational transformations of translation parameter . Wavelet transform on weighted LP- spaces . . . . . . . . . . . . . . Boundedness of the wavelet transform on LP with different weights
2.6 2.7 2.8
Wavelet transform and operators The Wavelet transform on F~ . . . The wavelet transform on lJ' (!Rn)
41 44 47
H(v), Htk and Itk
Composition of Wavelet Transforms 3.1 3.2 3.3 3.4
29 32 34 35 37 39
49
Introduction . . . . . . . . . . . The wavelet transform of tempered distributions . Composition of wavelet transforms . Weighted Sobolev Spaces . . . . . . xi
49 50 52 56
The Wavelet Transform
xii
3.5 3.6 3.7 3.8
4.
5.
6.
. . . .
59 61 65 67
4.1
Introduction . . . . . . . . . . . . . . .
67
4.2 4.3 4.4 4.5 4.6
Wavelet Transform on Spaces of Type S The Wavelet Transform of Generalized Functions The Wavelet Transform of Tempered Ultradistributions Wavelet Transform of Gevrey Functions of Compact Support . Band Limited Wavelets . . . . . . . . . . . . . . . . . . . . .
69 74 75
78 80
The Wavelet Transform on Spaces of Type W
83
5.1 5.2 5.3 5.4
83 83
Introduction . . . . . . . . . . . . . . . . The spaces WM,a, wn.,B and w:i:~ ... . The wavelet transformation on W -spaces . Examples . . . . . . . . . . . . . . . . .
86 91
93
The Wavelet Transform on a Generalized Sobolev Space
6.3
6.4 6.5 6.6
Introduction . . . . . . . . . . . . . The generalized Sobolev space Bmk . . . . . . . . . p, 6.2.1 Examples . . . . . . . . . . . . . . . . . . The wavelet transform on generalized Sobolev space B~k . 6.3.1 Adjoint of L'l' . . . . . . . . . . . . . . . 6.3.2 Bm k -approximation of wavelet transforms Asymptotfc' behaviour for small dilation parameters Local convergence Example . . . . . .
93
94 95 96 97 99 100 103 106
A Class of Convolutions: Convolution for the Wavelet Transform
109
7.1 7.2 7.3 7.4 7.5 7.6 7.7
109 109 112 118 120 122 127
Introduction . . . . . . . . . . . . . . . . . The generalized translation and convolution Special cases . . . . . . . . . . . . . . . . Convolution for the wavelet transform .. . Convolution for the discrete wavelet transform . Existence theorems . . . . . An Approximation Theorem
129
8. The Wavelet Convolution Product 8.1 8.2 8.3 8.4 8.5
9.
57
The Wavelet Transform on Spaces of Type S
6.1 6.2
7.
Wavelet transforms on weighted Sobolev spaces . . . . Abelian theorems for wavelet transform of functions . Abelian theorems for wavelet transform of distributions An Application . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . The Wavelet Convolution Product Existence Theorems . . . . . . . . Wavelet Convolution in a Generalized Sobolev Space Approximations in U' and B p,k Spaces . . . . . . . .
lbl is Large
137
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Asymptotic Expansions of the Generalized Stieltjes Transform .
137 142
Asymptotic Expansions of the Wavelet Transform when 9.1 9.2
129 130 132 135 135
Contents
9.3 9.4 9.5
10.
xiii
Asymptotic expansions for Tz and T3 . . . . . . . . . . . . Asymptotic Expansions of f*Syg and W(x,y) . . . . . . . Asymptotic Expansion of Mexican Hat Wavelet Transform
Asymptotic Expansions of the Wavelet Transform for Large and Small Values of a 10.1 10.2 10.3 10.4 10.5 I 0.6 10.7
Introduction . . . . . . . . . . . . . . . . . . . . . Asymptotic expansion for large a . . . . . . . . . . Asymptotic expansion of Morlet wavelet transform Asymptotic Expansion of Mexican Hat Wavelet Transform Asymptotic expansion of Haar wavelet transform Asymptotic expansion for small a . . . . . Asymptotic expansion for small a continued 10.7.1 The case It= 1 . . . . . . . . . .
143 145 150
155
155 156 158 161 162 164 168 169
Bibliography
171
Subject Index
175
Chapter 1
An Overview
1.1
Introduction
In this chapter we present some elements of theory of classical and distributional Fourier and Hilbert transforms that are to be exploited in the development of the theory of the wavelet transform in U -spaces ( 1 ::::;; p < oo) and in certain distribution spaces. For this purpose an elementary theory of distributions is given that is essential for proper understanding of various developments in the theory of wavelet transform presented in this book. The relationship between the wavelet transform and Fourier transform is well known. We shall find relations of the wavelet transform with Hilbert transform and Riesz fractional integral operator. The application of Fourier transform restricts itself to LP-spaces with I ::::;; p::::;; 2 but Hilbert transform approach gives results valid in U-spaces with (1::::;; p < oo). Hilbert transform has applications in signal processing, aerofoil problems, dispersion relations, high-energy physics, potential theory problems and others [53]. Therefore, using aforesaid relation the wavelet transform can also be applied to tackle all such problems. A brief idea about convolution operator, and asymptotic expansion of a general integral transform is also given, that will form a basis for development of corresponding results for the wavelet transform.
1.2
Introduction to distribution theory
In this section we give a brief account of the theory of distributions (or generalized functions) that we shall need in subsequent chapters. For more thorough discussion of these ideas we may refer to Friedman [20], Zemanian [97, 98].
Definition 1.2.1. Let K be a compact subset of JR.n. Let ~K be the set of all complex-valued infinitely differentiable functions on JR.n that vanish at all points outside K.
~K
is a linear
2
The Wavelet Transform
space under addition and multiplication by a complex number; the zero element in
~K
is
the identically zero function. The union of all by
~-
~K,
The space
where K varies through all possible compact subsets of IR.n, is denoted
~
is also a linear space under the usual definitions of addition and
multiplication by a complex number. Identically zero function is the zero element of also.
~
~
is called a test function space.
Example 1.2.2. An element in
~(IR.n)
is given by
cfJ(x) = { exp( lxiLI ), lxl < I 0, lxl ~ l. For each non-negative integer k E N 0 define a seminorm Yk on the linear space
(1.2.1) ~K,
as
follows:
Yk(C/J) :=sup ID*cp(x)l < oo, cfJ
E ~K,
(1.2.2)
xE~"
where D* = (a I axl )k 1 (a I ax2)k 2 ... (a I axn )kn. The topology of ~K is generated by the sequence of semi norms { Yk}, k E N0. The space
~K
is a complete countably multinormed
space. Definition 1.2.3. The space of all continuous linear functionals on ~K is denoted by ~~.
also called dual of ~K- ~~ is the dual of the space~- Its elements are called Schwartz distributions. The space of all those distributions in ~' which have compact support is denoted by lff'. Example 1.2.4. Dirac delta function D concentrated at xo, defined by
(D(x-xo),cp(x)) := cp(xo), cfJ E
~(l~n),
( 1.2.3)
is an example of a distribution. Example 1.2.5. Let K be a compact subset of IR.n, and let f(x) be a locally integrable func-
tion on IR.n. Define the functional f on
~K
(f,cp) :=
by
r f(x)cp(x)dx,
}~n
(1.2.4)
where cfJ E ~K.Then f E ~~Differentiation of a distribution is a basic property which has significant applications. Definition 1.2.6. Distributional derivative~ ox off E ~'(IR.n) is defined by 1
I at ) I aq,) . n \ax;'q, =-\f'axj ,J=l,2, ... ,n, C/JE~(IR. ).
(1.2.5)
An Overview
3
· I ""(JX:• of J·- 1, 2 , ... ,n, ts . a Iso a d.tstn"b utwn. . F unctwna 1
Example 1.2.7. The Heaviside unit function His defined by H(x)={1,x;:,O O,x
( 1.2.6)
Then H'(x)
= DH(x) = 8(x).
( 1.2.7)
Example 1.2.8. Define -a
X+
=
{ 0,
x::::;; 0
x-a, x
( 1.2.8)
> 0.
For 0 < a < 1, this function is locally integrable and defines a regular distribution on IR by
(xf_a,q,) =
fo'" [alf>(x)dx.
Note that x-a is absolutely continuous on (O,oo), so Dxf_a
( 1.2.9)
= -ax-a-l on (O,oo).
Hence we define xf_a-l on IR by ( 1.2.10)
In general, we write (x-a-s l/>) = 1- lo~ x-aq,(s)(x)dx + ' (a)s o '
(1.2.11)
where(a)s=a(a+1) ... (a+s-1),s= 1,2, ....
Next, we define log+x
={
x::;;; 0 logx, x > 0. 0,
( 1.2.12)
Then, as in the above case, we have (1.2.13) where s = 0, 1,2 .... In order to study distributional Fourier transform we define the Schwartz space .9'(1Rn) of rapidly decreasing infinitely differentiable functions on IRn.
4
The Wavelet Transfonn
c= - functions
lf> on JRn such that for each
Ya,fJ(l/>) :=sup lxavf3q,(x)l < oo.
(1.2.14)
Definition 1.2.9. Y is the vector space of all
a,f3 E N0, xEIR"
The space Y(lRn) is equipped with the topology generated by the collection of seminorms
{ Ya.{J }; it is a Fre'chet space. The dual of .Y is denoted by Y'; its elements are called tempered distributions. Generalizations of these spaces are studied in chapters 4 and 5.
Example 1.2.10. lf>(x) = e-lxl 2 E Y(lRn). Proposition 1.2.11.
(i)
~ C
(ii)
~
Y with continuous injection
is dense in Y
(iii)YCLP,l (iv) LP
c Y' c
~p
~'. I~ p
< oo.
Definition 1.2.12. The set of all elements lf> E C"(JRn) such that for every
a
0
E N there is
a polynomial Pa(x) for which (1.2.15) is denoted by
eM
eM.
is the space of multipliers in Y' in the sense that iff E Y' and g E
eM.
then the
product gf defined by
(gf,l/>) := (f,glf>),l/> EY, is an element of Y', and the map f
->
(1.2.16)
gf is continuous in Y'.
Let f E Y' (JRn ), then the convolution f
* g is defined by
(f*g,lf>) := (f(x), lfl(x)) ,l/> E Y(JRn), where lfl(x) = (g(y), lf>(x+ y)) E Y(lRn). It can be proved that f
( 1.2.17)
* g E Y'(Rn).
Another space, which is used in the development of theory of Hilbert transform, is ~LP(JRn).
It consists of all those COO -functions lf> on JRn such that
Yk(lf>) :=
(k." ioklf>(x)IPdxYIP<
oo, k E N0, 1 < p < oo.
The topology on ~LP is generated by the sequence { Yk}. For 1 < p
< oo,
(1.2.I8) ~LP is a locally
convex, Haussdorff and sequentially complete topological vector space. We denote by ~I;, the dual of ~LP', where p' = pj(p -I); i.e., ~I;, = (~LP' )'. Clearly,~
c
~LP and~};, C ~'.I< p
< oo.
5
An Overview
1.3
The Fourier transform
In this section we give an introduction to the theory of the Fourier transform. A detailed account can be found in the books of Bochner and Chandrasekharan [6], Lieb and loss [47], Sneddon [85], Zemanian [97] etc. Iff E L 1 (IR.n), then its Fourier transform lis defined by
(ff!)(m) =f(m) := where (x, m)
= XJ ro1 + Xzl.02 + ... + Xnmn.
r
}'J{n
e-i(x,w)f(x)dx,
Some fundamental properties of
(1.3.1)
j
are given
below.
Theorem 1.3.1. Let f E L 1 (JR.n). Then (iJ llilloo ~ ll!ll1 ( ii) j( (J)) is uniformly continuous on JR.n
(iii) limlwl_,oof(ro) = 0 (iv) f( m)
= 0 =? f(x) = 0 almost everywhere.
Proof. See [6, pp. 57-59]. Next, we recall the inversion theorem [6, p. 65].
Theorem 1.3.2. If f(x) E L 1 (JR.n) and j( m) E L 1(JR.n), then
r
1ei(x.w)j(m)dm a.e. ( ff- 1i)(x)=f(x)=(2n)n }F.n
( 1.3.2)
Example 1.3.3. ( 1.3.3)
Definition 1.3.4. Let f and g be locally integrable functions on JR.n. The convolution f
*g
off and g is defined by
(f*g)(x)=
r f(x-y)g(y)dy= }'J{nr f(y)g(x-y)dy=(g*f)(x),
}'J{n
( 1.3.4)
provided that the integrals exist. For different conditions under which the convolution exists we may refer to Chapter 8.
Theorem 1.3.5. If f,g E L 1 (JR.n), then (1.3.5)
The Wavelet Transform
6
Proof. An application of Fubini's theorem gives
(f*g)/\(ro) = =
=
r r e-i(x,w)f(x-y)g(y)dydx r r e-i(x-y,w)f(x-y)e-i(y,w)g(y)dydx }JRn }JRn j(w) r e-i(y,w)g(y)dy }JRn }JRn }JRn
= j(w)g(ro).
Let ga(x) := (4na)- 112 exp( 4~ ), a> O+,x E JR. Then it can be proved that
§[ga](ro)
=
2
e-aw ::::::: 1 as a--+ 0+.
Using the above approximate identity property of Gaussian function ga one can prove the following [12, Theorem 2.8, p. 29].
Theorem 1.3.6. Let f E L 1 (IR). Then lim (f*ga)(x) =f(x)
a-->0+
( 1.3.6)
at every point x where f is continuous.
Indeed, following general result can be proved using n-dimensional Gaussian [47, p. 118].
Theorem 1.3.7. Let f, g E L 2 (!Rn). Then
)n (!,g) ~~
(1.3.7)
r
( 1.3.8)
1 (!,g)= ( 2n
and
r
}JRn
f(x)g(x)dx
=
}JRn
j(x)g(x)dx.
The following generalization of Theorem 1.3.5 is of importance [47, p. 122].
Theorem 1.3.8. Let f E LP(!Rn) and g E U(!Rn),where 1+ 1/r = 1/ p+ Ijq. Suppose that
I::::;; p,q,r::::;; 2. Then f*g E U(!Rn), and (1.3.5) holds in L"1 (!Rn), where 1/r+ 1/r' = l. 1.4
The Fourier transformation in Y'(!Rn)
The distributional theory of the Fourier transformation ( 1.3.1) can be found in [20,21,22,57,97]. Since Y(!Rn) C L 1(!Rn), the Fourier transformation of lf> E Y(!Rn) is well-defined.
Theorem 1.4.1. The Fourier transformation is an isomorphism of Y(!Rn) onto itself
7
An Overview
Proof. For lfJ E .9'(~11 ), from (1.3.1), we have
4i(ro) =
r e-i(w.x)cp(x)dx.
( 1.4.1)
}JRn
Differentiating under the integral sign we get
of3wa4i(w) = (-t)lf31jlf31-1al f e-i(w.x)xf3Dalf>(x)dx, }JRn where
a,f3
EN
1/31 = /3 1 + /32 + ... + f3n; so that of3 wa4'( w) is bounded for all a, f3.
0,
It then follows
by the product rule for derivatives and induction on f3 that waof34' is bounded for all Hence
a,f3.
4' E .9'(~11 ).
Next, applying same arguments to (1.3.2), we can show that if 4' E .9'(~11 ) then l/J E Y(~11 ). By uniqueness of the Fourier transform it follows that the Fourier transformation is an isomorphism on .9'(~ 11 ). Now, the Fourier transform §[f] of tempered distribution f E YJI is defined by (§[fJ,cp) := (f,§[cp]) ,l/J E Y.
( 1.4.2)
It specifies an automorphism of the tempered distribution spaceY'. For functions f(t) E
L 1 (~11 ), §[f] coincides with (1.4.I).
Let us define for f E Y' the operator §- 1 [f(t)] = (2n)-n§[f(-t)];
( 1.4.3)
it is also an automorphism of Y' and defines the inverse transformation of§(!): .~-I[§(!)]= §[§- 1(!)] =
f, fEY'.
For f(t) E L 1 (~ 11 ) the formula( 1.4.3) has the form §-I [f] = (2n)-n
r
}!Rn
ei(x,t) f(t)dt.
The following relation holds : §[§[f(t)]] = (2nt f( -t). Example 1.4.2. (i) §[<5] =I 1.4.1
(1.4.4)
(ii) §[I]= (27r) 11 D.
Operation-transform formulae
Suppose thatf(t),g(t) E Y'(~ 11 ),k = (kJ,kz, ... ,k11 ) E N0and cis a complex number. Then (i) <~[( -it)kf(t)](x) (ii) §[D~f(t)](x)
= D~§[f](x),
= (ix)k§[f](x),
(iii) §[f(t -to)](x) = e-i(x,to)§[f](x),
The Wavelet Transform
8
(iv) §[e-i(t.xo) f(t)](x) = §[f](x+xo), (v) §[f(ct)](x)
=
~§[!]
(vi)The convolution f
0) ,c =1- 0.
* g of two distributions f E .Y' and g E
of compact support, is defined by (1.4.5) The right-hand side makes sense, since (g( 1:), l/J (t + 1:)) E .Y.
Theorem 1.4.3. Let f E .Y' and g E
Proof. See [97]. 1.5
The Hilbert transform
The Hilbert transform of a suitably restricted function f(t) is defined by (Hf)(x)
=
I -p.v. 1r
joo -(f(t) -)dt -oo
] hm . --
7r
E->0+
t -x
(fx-e + -oo
1x+e 00
)
-J(t) -dt
(t -x)
'
(1.5.1)
where x is real [85, p. 233]. The classical theory of this transformation is given in [53], [85], [87], [96] and its distributional theory can be found in [53], [57]. Some of the important properties of the classical Hilbert transformation are stated below [53, pp. 92-98]. (i) Iff E U(IR.), I < p <
oo,
then (Hf)(x) exists a.e. on JR., and it belongs to U(IR.). The
limit in ( 1.5.1) also exists in U -sense. There exists Cp > 0 independent off such that (1.5.2) (ii) Let H be the operator defined by (1.5.1 ). Then (H 2 f)(x)
= -f(x),
and also in U sense for all f E U(IR.), 1 < p <
a.e.
( 1.5.3)
oo.
(iii) If f(t) E U(IR.) and g(t) E U(!R.), 1/ p +I /q
= 1, then
j_~ (Hf)(x)g(x)dx = - j_~ f(x)(Hg)(x)dx.
(1.5.4)
9
An Overview
(iv) ( 1.5.5) (v)
( 1.5.6) (vi) For I < p < oo, His a linear homeomorphism from q&LP (IR) onto itself and H- 1 = -H. (vii) Generalizing the Parseval relation(l.5.4), the Hilbert transform off E q&Jy is defined by (1.5.7) (viii) The generalized Hilbert transform defined by (1.5.7) is an isomorphism from 9&~ onto itself and ( 1.5.8) (ix) ( 1.5.9) (x)
Ho =
I
I
n
t
--p.v.-.
( 1.5.10)
(xi) A relation between Fourier transform and Hilbert transform is given by
(.9?[Hf])(~)
= insgn(~)($f)(~), f E L2 (1R),
(1.5.11)
where
I, { sgn(~) = 0,
~ >0 ~ =0
-1, ~
1.6
(1.5.12)
< 0.
Wavelets
The word wavelet stands for small wave. Wavelets have been introduced for representation of functions in a more efficient manner than Fourier series. Wavelets are generated from one single function, called mother wavelet, by translation and dilation. Wavelets have found applications in data compression, noise removal, pattern recognition, fast computation etc.
The Wavelet Transform
10
Wavelet theory provides a unified framework and coherent theory for a number of different ideas and techniques that have been independently developed in several fields. Example 1.6.1. Haar wavelet V' is defined by 1, O~t
0,
otherwise.
Clearly, {I dt j oo l/f(t)dt = lot/ dt- l1;2 2
= 0.
-00
Definition 1.6.2. A function V' which satisfies
I:
l!'(t)dt =
o
(1.6.1)
represents a wavelet because this condition implies that V' changes sign in ( -oo, oo ), and it vanishes at-oo and +oo. Haar wavelet is localized in time, i.e., it vanishes outside a finite interval; but it is not continuous. Its Fourier transform:
= {1/2 e-iwtdt- {I e-iwtdt lo l1;2 . 2
(J)
_ · -iro/2sm 4 -
( 1.6.2)
ro/4 .
le
Then
The requirement
o < c"' := joo I i/1( ro) 12 dro < oo -00
lrol
( 1.6.3)
is called "admissibility condition" for a wavelet. The existence of integral (1.6.3) yields that i/1(0) = 0. Then for V' E L 1 (~),
I: I:
i/1( ro) = and i/1(0) = 0 implies that
e -iwt l!'(t )dt,
l!'(t)dt =
o;
so that V' changes sign, hence it is a wavelet. From ( 1.6.2) it follows that as
Irol
->
i/1( ro)
is not localized in frequency domain and it tends to zero
oo, the decrease is as fast as 1 ~ 1 ; not very fast.
An Overview
t
~(t)
II
t
1.5
1
w(t)
1
0
05
-1
0
0
05
-2
15
0
05
~~
t
Fig. 1.1: Haar scaling function cp (t)
______...
15
Fig. 1.2: Haar wavelet function lfl(t)
Example 1.6.3 (Shanon wavelet). It is defined by means of the Fourier transform
~ lfl(ro)
= Xr(ro) :=
{l,roE/ 0, roE lR \I,
where I= [-2n, -n] U [n, 2n]. Then by inversion of Fourier transform,
lfl(t)
=
l("'
2 7r 1-oo eiwrXI(ro)dro
= __!__ [ r-R: ei(J)t dt
27r
1-2R:
sin ~t
3
2
2
+
{2R: ei(J)t dt]
1R:
= -----;c;- cos - 7rt It decays as fast as t- 1, hence it has poor time localization. Its Fourier transform has good frequency localization and the Fourier transform has compact support. This possesses properties contrast to Haar wavelet.
Example 1.6.4. Mexican hat wavelet
lfl(t)
=
fi/( ro) =
2
(I-t2)e-r /2
d2
2
= - dt2e-r /2,
V2iiro2e-w2 /2.
It is a C"" -function and well localized in time and frequency domains.
Definition 1.6.5. If 1f1 E L 2 (JR) and satisfies the admissibility condition (1.6.3) (or sometimes (1.6.1)), then it is called mother(a basic) wavelet, or just wavelet.
12
The Wavelet Transform
1.0
0.
Fig. 1.3: Shanon wavelet 0.4
0.3
0.2
0.1
l.
- 10
- 5
5
10
Fig. 1.4: Fourier transform of Shanon wavelet 1.7 The continuous wavelet transform Wavelet transforms were recognized in one form or another much earlier by different scientists, cf. Bochner and Chandrasekharan [6]. The wavelet transform acts as a time and frequency localization operator.
Definition 1.7.1. The continuous wavelet transform of a function f with respect to the wavelet 1f1 is defined by
(W11,J)(b,a) =
j_~J(t)lf!b;a(t)dt,b E ~,a E ~ \ {0} := ~o,
(1.7.1)
An Overview
13
-4
4
Fig. 1.5: Mexican hat wavelet
-4
4
Fig. 1.6: Fourier transform of Mexican hat wavelet
where li'b;a (t) = ~ 11' e~b)' provided the integral exists. Using HOlder's inequality one can show that the integral exists when f E U(IR), 11' E U(!R), 1I p + 1I q = 1. In particular, for p
= q = 2, we have
The classical framework of the continuous wavelet transform is the space L2 (IR), since in this space, the transform defines an isometry; see Section 1.7 .3. In most of applications, the mother wavelet is assumed to be continuous and to vanish at ±oo; therefore, the continuous
14
The Wavelet Transform
wavelet transform exists for a larger class of functions than L2 (~). If the mother wavelet is bounded, then the wavelet transform exists for f E L 1 (~).
Relationship with Fourier transform
1.7.1
We have ~[lf/b;a(x)]
= =
1 JT::T
V lal
joo e-lXYlfl .
(x-b) -
a
-oo
dx
r= e-i(b+az)ylfi(Z)-a-dz; vTaT
l-oo
so that
~[lf/h;a(x)] = .Jrare-ibyi/f(ay).
(1.7.2)
Using Parseval formula (1.3.7) for n = 1, we obtain 1
~ ~
(Wvd)(b,a) = (!, lf/b;a) = 2 7r (!, lf/b;a)
= =
r=
alal-1/2 j(ro)eibwi/f(aro)dro 2Jr l-oo alal- 1/ 2~-I [j(ro)o/(aro)](b,a).
Hence
~[(Wvd) (b, a)] ( ro)
= alal- 1/ 2f( ro) 1/f(aro ).
(1.7.3)
In view of Theorem 1.3.8 the above relation holds in general, for fEU(~) and 1f1 E U(~), where 1+1/r= 1/p+1/qand 1 ~p,q,r~2. 1.7.2 Parseval relation for the wavelet transform Let us assume that a E ~ \ {0} = ~o- Define the space
W=
L2 (~ x ~o,dbdafa 2 ) := { F(b,a): ~ ~0 IF(b,a)i 2db ~~ < oo}.
(1.7.4)
This is a Hilbert space with inner product
(F,G)w where dJ.L(b,a)
=
f f F(b,a)G(b,a)dJ.L(b,a),
lJR.lJR.0
= dbdaja 2.
Let f,g E L2 (~), and 1f1 E L2 (~) satisfy admissibility condition (1.6.3). Then, using (1.7.3), we have
(Wv,f, Wll'g)w =
f f da~b (WII'f)(b,a) aiarl/2 f
}JR. }JR.0 a
7r
}JR.
g{ro)e-ihwi/f(aro)dro
(1.7.5)
15
An Overview
=
f d~ f alal-l/2 g(ro)fji(aro)dro f (W11d)(b,a)e-ib(J)db JIR a lJR.0 27r JIR
l
da alal- 112 - ~ -g(ro)fji(aro)alal- 112 f(ro)fji(aro)dro JR. a Ro 2n
= 1m 2
=
rf(ro)g(ro)dro_!__ r lfii(aw)IZ da 27r lJRo lal
JIR 1
~
(1.7.6)
= 2nelfl(f,g) =elfl(f,g),
where elf/ is given by (1.6.3), is the desired Parseval relation. In particular for g = f, we have the Plancherel formula:
IIW
= v'c;llfll2·
(1.7.7)
1.7.3 Reconstructionformula For analyzing functions (signals), we often need to transform the functions from time domain to frequency domain and reconstruct them from frequency domain to time domain. Therefore, we need a formula for inverse wavelet transform. Now, from (I. 7 .6) we have
(f,g)z = el 1fl
1 -e
=
f f (Wlflf)(b,a)(Wlflg)(b,a)db d~ JJR 1JR0 a
hh
(Wlflf)(b,a)
1f!IRIR.o
I =-
JR
da g(t)lJfb;a(t)dtdb-z a
frn frn 1m (Wlflf)(b,a)lJib;a(t)db----zg(t)dt da-
elf/ JR
=
h -
_l ( elf/
JR.
Ro
a
f f (Wlflf)(b,a)lJib;a(t)db d~ ,g(t)). lJR lJRo a
Since this holds for all g E L2 (1~). we have
f
1.7.4
= el f f (Wlflf)(b,a)lJib;a(t)db d~ lf/~~o
( 1.7.8)
a
The discrete wavelet transform
Sometimes it is more convenient to deal with discrete wavelet transform. We discretize the parameters a and b as follows: Set
a= a0 and b = a0nbo ao > I and bo > 0. Definition 1.7.2. The discrete wavelet transform off is defined by
J(m,n)
=
(W f)(m,n)
=I:
f(x)lJim.n(x)dx,
( 1.7.9)
The Wavelet Transform
16
where
lflm.n ( x ) = a0-m/2 If! ( a0-m x- nbo) .
(1.7.10)
The series I:,';,n=-ooj(m,n)lflm,n(x) is called the wavelet series off and the functions
{ lf/m,n (x)} are called wavelets. We note that a function
f E L2 (IR) is completely determined by its discrete wavelet trans-
form (i.e. wavelet coefficients) if the wavelets form an orthonormal basis in L2 (IR). Thus 00
f =
L
(1.7.11)
J(m,n)lflm,n·
m,n=-oo
It is well-known that wavelets can form a basis not only for L2 (IR), but also for a large
class of functions [96]. It is interesting to note that many currently used function spaces can be characterized by the behaviour of the wavelet coefficients
J( m, n).
This is surpris-
ing, if we compare the "wavelet transform" with the Fourier transform, for which such characterizations are not true in general.
1.8
Asymptotic expansion
Asymptotic series are of great value in many computations. We define an asymptotic power series representation of a function f(x) as x-> xo.
Definition 1.8.1. Let f(x) be defined and continuous on domain JR. The formal power series [;;'= 0 an (x- xo)n is said to be an asymptotic power series expansion off, as x-> xo in IR, if the conditions lim {(x-xo)-m[f(x)- Ean(x-xo)n]} =0,m=0,1,2, ...
(1.8.1)
n=O
X->Xo
are satisfied. It can be readily seen that conditions ( 1.8.1) are equivalent to m
f(x)= L,an(x-xot+O(x-xo)m+l,x ->Xo iniR, m=0,1,2, ....
(1.8.2)
n=O
Definition 1.8.2. The formal power series E;;'=oanx-n is said to be an asymptotic power series expansion off, as x
-> oo
if the following equivalent set of conditions are satisfied
lim {?[f(x)x----+oo
Eanx-n]} =
n=O
0, m = 0, 1,2, ... ,
( 1.8.3)
m
f(x) = L,anx-n+O(x-m-l),x->oo,m=0,1,2, .... n=O
( 1.8.4)
An Overview
17
If (1.8.3) or (1.8.4) holds, then we write
r. anX-n ,X---> oo. n=O 00
f(x) rv
( 1.8.5)
In this book we are concerned with asymptotic expansion of integrals. As an example, we consider Laplace transform:
I(x)
= fooo e-xtf(t)dt,
X> 0.
( 1.8.6)
We assume that f(t) is locally absolutely integrable on (0, oo) and that, as t
-->
oo, ( 1.8.7)
for some real number
a.
In order to study l(x) for large x, let us further suppose that, as t
-->
0+,
00
f(t) rv
L
a5 ts+a-l, a> O,t __, 0+. (1.8.8) s=O Then the integral (1.8.6) is absolutely convergent, if x > a and a > 0. Replacing f(t) in (1.8.6) by the expansion (1.8.8) and then integrating term by term we get the asymptotic expansion
=
E
as f'(s
s=O
XS
:a
a) ,x--> oo.
( 1.8.9)
This is the conclusion of the following [92]:
Lemma 1.8.3 (Watson's lemma). In ( 1.8.6) let f(t) be a locally integrable function on
(O,oo ), bounded for finite t and let (1.8.7) and (1.8.8) hold. Then (1.8.9) holds. If we set n-1
f(t)
=
L asts+a-l + fn(t)
(1.8.10)
s=O then ( 1.8.9) can be written as
I(x)
=
n-1
~ a5 L..
s=O
f'(s+ a) xs+a
+ Dn(x),
(1.8.11)
where (1.8.12) is called the remainder term.
18
The Wavelet Transform
We can find an estimate to the remainder term [92]. The growth conditions ( 1.8.7) and (1.8.8) suggest that (1.8.13) for some number M11
> 0. The best value of Mn is given by Mn
=
sup lfn(t)tl-n-ae-atl·
(1.8.14)
(O,oo)
The error term in (1.8.12) is bounded by 18n(x)l :(; Mn fooo tn+a-le-(x-a)tdt
= Mn r(n+a) = O(x-a-n). (x- a)n+a
1.8.1
(1.8.15)
The Mellin transform technique
We often use Mellin transform in deriving asymptotic expansion of integrals. We define the Mellin transform of a locally integrable function on (0, oo) by
M[f;z] = fooo tz-lf(t)dt,
(1.8.16)
provided the integral converges. Its domain of analyticity is usually an infinite strip a < Re z < b. The inversion formula for this transform is given by
f(t)
= -2
1 1c+ioo
.
'!Cl
c-ioo
rzM[f;z]dz,
( 1.8.17)
where a < c < b. In asymptotic analysis sometimes we use Abe/limit of a function defined by lim
r= f(t)e-E dt,
E->O+}o
1
(1.8.18)
if this limit exists, in which case we say that fis Abel summable. Iff is absolutely integrable on (O,oo), then Abel limit is simply the integral of f. The improper Riemann integral of
f on (0, oo) is also equal to the Abel limit off [93, p. 197]. However, in the following lemma, we show that the Abel limit may exist even when improper Riemann integral does not converge.
(1.8.19)
An Overview
Proof.
19
We know that
{"" tA.-Ie-ztdt =
lo Replacing z by £
-
ix with
£
f'(~) ,Re z > O,Re z
A> 0.
( 1.8.20)
> 0, gives
lim {"" tA.-Ie-(e-ix)tdt E--+O+}o
=
lim f'(A.) e--+0+ (t:- ix)A.
=
f'(A.) . ( -ix)A.
This proves the lemma.
D
Using Mellin transform and Abel limit, asymptotic expansion of the Fourier transform has been obtained in [91]. A similar technique will be used to obtain asymptotic expansions of wavelet transforms in Chapter 10. 1.8.2
The distributional approach
Distributional technique developed by Wong [93] is often used in deriving asymptotic expansions of certain integral transforms. Assume that
I(x)
= looo f(t)h(xt)dt,
(1.8.21)
and f(t) possesses asymptotic expansion
r. a rs-a ,r-> co, s=O 00
f(t)'"" where 0 <
a
~
I. For n
~
(1.8.22)
5
I, set n-1
f(t)
= Last-s-a+ fn(t).
(1.8.23)
s=O Also, assume that 00
h(t) rv eictP L bst-s-{3 as t-> +co, s=O where c is real, p
> 0 and 0 < f3
~
( 1.8.24)
1.
Let f be a locally integrable function on (0, co) and let it define a tempered distribution by
(f,l/J) := fooo f(t)l/J(t)dt,l/J E Y(JR).
( 1.8.25)
Let us define
fn,o(t) fn,j+l(t) = -
1
oo
t
fn,j('r)d'r=
= fn(t) (-1 )j+ I .1 1·
1
oo
t
.
(-r-t) 1 fn('r)d-r,
20
for j
The Wavelet Transform
= 0, I, 2, ... , n -
O(t-a) as t---->
co.
I. For 0
<
< 1, fn,n (t) is bounded on [O,R] for any positive R and
a
For a= I, fn,n(t)
= O(t- 1) as t----> co and fn,n(t) = O(logt) as t----> 0+.
Therefore, by integration by parts, we find that (1.8.26) The following lemmas give basic identities for derivation of asymptotic expansion off [93, pp. 296-297].
Lemma 1.8.5. ForO< a< I, n;;:: I and q, E Y(O,oo), n-1
n
L
(f,l/') =Las (t.:;:s-a,q,)s=O
Cs (
D(s-1) ,q,)
+ (fn,l/'),
( 1.8.27)
s-1
where
(-1)'
Cs
(1.8.28)
= (s -I)!M[f;s],
M[f;z] is the Mellin transform off(t), or its analytic continuation.
Lemma 1.8.6. If a= I in (1.8.22) then for each integer n;;:: I and for any q, E Y(O,oo), we have
(f,q,) =
n-1
n
s=O
s-1
L as(t.:;:s-l ,l/')- Lds ( D(s-l),q,) + (fn,l/'),
(1.8.29)
where
.
ds = hm
t__,O+
[
fs,s(t)
+ ((-1)'-1 ) s- I .1
]
as- I logt .
(1.8.30)
These lemmas were used by Li and Wong [46] for obtaining asymptotic expansion of convolution ( 1.3.4 ). We shall use these results for deriving asymptotic expansion of the wavelet transform for large values of lbl in Chapter 9.
Chapter 2
The Wavelet Transform on LP
2.1
Introduction
The Fourier transform has been the most useful technique for the frequency analysis of a signal for a long time, but to deal with signals which are not localized in frequency but also in space one needs wavelet transform. The wavelet transform off with respect to the wavelet 1f1 in n dimensions is defined by
t-b (W f)(b,a) =a-n I2 1m f(t)lfl(-)dt, a> 0, bE JR.n a
JRn
=
where Xa(x)
U* Xa)(b),
= a-n/ 21f/( -xfa), provided the integral exists [45, p.
(2.l.l)
(2.1.2) 28].
If f(x) and lfl(x) are both bounded functions of class L 1 then by [6, Theorem 13, p. 24], for
n = I, we can write (2.1.1) in the form:
(Wf)(b,a)
=
al/21oo ~ . = 2 7C -oof(ro)e'brolfl(aw)dw,a > O,b E JR.,
(2.1.3)
where f denotes Fourier transform off defined by ( 1.3.1 ). Iff, 1f1 E L 2 (JR.), then also by Parseval formula, (2.1.3) follows from (2.1.1) [ 12, p. 61]. For more general situations refer to (8.1.9). The form (2.1.3) of the wavelet transform was used by Bochner and Chandrasekharan [6, p.l3] as early as 1949 as a summability integral in the development of the theory of Fourier transform. According to [6, p.l4] we write
S~ (b)=_!_ 1 2n
00
j(ro)eirobK(rojR)dro, R > 0.
(2.1.4)
-00
From (2.1.3) and (2.1.4) it follows that s~ (b)= a- 112(W f)(b,a), 21
(2.1.5)
22
The Wavelet Transform
for R = I fa and K( ro) = lif( ro ). Iff E L 1 (IR) and lfl E L 1 (IR), then from (2.1.1) we get (2.1.6)
(cf. [6, lemma 2, p. 15]).
The modified Hilbert transform Ha(f) is defined by (Haf)(x)
= n- 112 r(l -
a)cos(naj2)
1:
which reduces to Hilbert's transform (1.5.1) when
It -xla-l sgn(t- x)f(t)dt
(2.1.7)
a= 0.
If 1 < p < a- 1 and f E U(JR.), then Haf E L', r- 1 = p- 1 - a and there exists a constant M = M(p, a) such that
IIHafllr ~ MIIJIIP
(2.1.8)
[43].
Iff E U(JR.), and g = Haf, then f(x) = - lim n- 1r(l 8---.0+
+ a)cos(naj2)T8(g)(x)
(2.1.9)
for almost all x E Rand also in LP- norm [33], where T8(g)(x) =
r
Jlt-xi-;:,8
It -xl-l-asgn(t -x)g(t)dt.
(2.1.10)
The range of Ha was characterized by Herson and Heywood [33] as follows: A function g E L'(IR) is also in Ha(LP) if and only if IIT8giiP is bounded for all positive 8.
A variant of Ha is Riesz fractional integral operator Ia. 0
I, 0 L' is continuous
and there exists a constant K = K(p, a) such that (2.1.12) If g(x)
= (Iaf)(x),then
f(x) = (I;; 1g)(x) =-lim n- 1r(l- a)sin(na/2) 8--+0
X
r
Jlt-xi-;:,8
It -xl-l-a[g(t)- g(x)]dt
pointwise almost everywhere and also in U-norm [33].
(2.1.13)
23
The Wavelet Transform on LP
For Parseval relations involving the operators Ha and Ia we may refer to Theorem 1.1, given below. Let f be defined on ( -oo, oo) and x E ( -oo, oo); then the extended Weyl integral is defined by (2.1.14) and the extended Riemann-Liouville integral is given by
(Raf)(x) = (W; f)(x) := (r(a))- 1 For f
E
J:,., (x-t)a- /(t)dt. 1
(2.1.15)
U(JR.), the following properties ofthe above operators obtained by Kober [44] will
be used in the sequel:
2sin(naj2)Haf = (Wa- w;)J,
(2.1.16)
2cos(naj2)1af = (Wa + W;)J,
(2.1.17)
lxl-aWaf-> f, lxl-aw; f-> f, lxl-aHaf-> Hf in U- norm.
(2.1.18)
Many properties of these operators can be found in [54]. The following generalizations of the operators H, Ha and Ia were given by Okikiolu [52]. For J.l, v E JR. and 0
H(v)(f)(x) := n- 1 lxl-v p.v.
Ht~ (f)(x)
i:
ltnt -x)- 1/(t)dt,
:= n- 1r(1 -a) cos( anj2)lx1JL-v-a
and
It~ (f)(x)
:= n- 1r(1- a) sin( anj2)lxiJL-v-a
i: i:
(2.1.19)
ltlvlt -xla-l sgn(t -x)f(t)dt (2.1.20)
Itnt -xla-l f(t)dt.
(2.1.21)
Clearly, H 0 (!) = H(f), H~~~(f) = Ha(f) and~~~~(!)= Ia(!). We write H(v) a for H(v) a,o and /v) a for /(v) a,o· Properties of these operators are summarized in the following theorems [52]. Theorem 2.1.1. Let p > 1, 0 < J.l ~a< 1 (or 0 < a < 1 if J.l = 0), 1/p-1 < v < 1/ p- a, 1/r = 1/p- J.l > 0. Then there exists a constant K = K(p, a, V,J.l)
such that
IIH~~~ (f)llr ~ Kllfllp, IIIt~(f)llr ~ tan(na/2)KIIfllp·
(2.1.22)
Also,for g E L1 (JR.), 1/r+ 1/r' = 1, we have
i: g(t)Ht~(f)(t)dt i: =-
/_: g(t)ItMt)(t)dt where v'
= J.l - v - a.
f(t)Ht)(g)(t)dt,
= /_: f(t)ItJ
(g)(t)dt,
(2.1.23) (2.1.24)
24
The Wavelet Transform
Theorem 2.1.2. Let f E LP(JR), p
(i) (ii)
> 1, 1I p- 1 < v < 1I p. Then IIH(v)(f)IIP ~ Ap,vllfllp H(v) { H(v)(f)} = - f
(2.1.25)
a.e.
(2.1.26)
(iii)forgELP'(IR), llp+llp'= 1,
L~ g(t)H(v)(f)(t)dt = - L~ f(t)H(-v)(g)(t)dt
(2.1.27) (2.1.28) (2.1.29)
Now we recall certain results on weighted LP- spaces which are to be used in Section 2.4. Suppose that 0
a- 1 and r- 1 =
p- 1 -
a.
(2.1.30)
Let a non-negative weight function w(x) satisfy
(III-I j(w(x) )' dx) I/'(III- 1 j(w(x))-p' dx) l/p' ~ c,
(2.1.31)
for some constant c > 0 and all interval I
c JR.., where III denotes the length of I. Here as usual, if 1 < p < oo, p' denotes the conjugate exponent p' = p(p - I) -I. Andersen [ 1] has
shown that if w(x) satisfies ( 1.1.36) then ( w(x) )P( 1 + lxl) -p is integrable over JR.., (w(x))- 1 (1 and (w(x))-p' (1
(2.1.32)
+ lxl)-r' is integrable over IR,
+ lxl)(a-l+e)p' is integrable over JR..,
for some e
(2.1.33)
> 0.
(2.1.34)
For a non-negative weight function U(x) define L'(U) =
{f: llfllr,U =
(L~ lf(x)I'V(x)dx) 11r < oo}.
(2.1.35)
According to Andersen [I] if w(x) satisfies (2.1.31) then Ia and Ha are bounded operators from LP(wP) into L'(w'). Moreover, we have the following inversion theorems. Theorem 2.1.3. Suppose that (2.1.30) and (2.1.31) hold. Let g = Haf with f E LP(wP). Then (2.1.9) holds pointwise almost everywhere and also in the norm of LP(wP).
25
The Wavelet Transform on LP
Theorem 2.1.4. Suppose that (2.1.30) and (2.1.31) hold. Let g
=
1af with f E U(wP).
Then (2. 1.13) holds pointwise almost everywhere and also in the norm of U(wP). Theorem 2.1.5. Suppose that (2.1.30) and (2.1.31) hold and g E U(wr). Then g = Haf (resp. g = 1af) for some f E U (wP)
if and only if there is a constant c > 0 independent of
o > 0 such that (2.1.36) The following Parseva1 relations hold.
Theorem 2.1.6. Suppose that (2.1.30) and (2.1.31) are satisfied. Let f E U(wP) and g E U' (w-q'), then
i:
f(x)(Hag)(x)dx
=-
i:
(2.1.37)
g(x)(Haf)(x)dx
and
i:
f(x)(Jag)(x)dx
=
i:
(2.1.38)
g(x)(Jaf)(x)dx.
A boundedness result for Ia on LP spaces with different weights was obtained by Harboure et. al [30, p. 640], and the inequality (2.1.12) was generalized. Let us recall it here for our future use in Section 2.5. To a given pair (a, p ), such that 0 :::;; a < 1 and I :::;; p < co, associate a real interval as follows:
Q(a,p)
=
{q : 1 :::;; q < 1I (1 - a)} , { {q: I:::;; q:::;; p/(1- ap)},
{q:1:s;;q
if p
=1 (2.1.39)
if 1 < p < 1/a
if1ja:s;;p
Theorem 2.1.7. Let w be a non-negative weight function on a set of positive measure. Let 0
(a) There exists a non-negative function v, different from zero on a set of positive measure, and a finite constant C such that
(! 00 IIaf(xWv(x)dx)
1/q
-oo
:::;; C
(!00 lf(x)IPw(x)dx) -oo
1/p
.
(2.1.40)
(2.1.41)
(ii) w(x) ~ C(1
+ Jxl)a- 1
if p = 1.
(2.1.42)
26
The Wavelet Transform
For a characterization of the weight v for which there exists a nontrivial weight w such that (2.1.40) holds we may refer to [30, Theorem 6, p.643]. The following form of Hardy-Littlewood-Sobolev inequality involving operator Ia [47, p. 98] will also be used in the sequel. Theorem 2.1.8. Let p,r > 1 and 0
1[~ la(f)(y)v(y)dyl:;::; C(a,p)llfllpllvllr·
(2.1.43)
In this chapter using summability theorems and other results of Bochner and Chandrasekharan on Fourier transforms and relation (2.1.5), certain approximation properties of the wavelet transform are investigated in Section 2.2. Certain relations between wavelet transform W(f), Hilbert transform H(f), modified Hilbert transform Ha(f) and Riesz fractional integral Ia(!) are obtained which are used to prove inversion theorems and Parseval relations for the wavelet transform in U- space in Section 2.3, and in weighted U -space in Section 2.4. Using the relation of wavelet transform with the operator Ia. certain boundedness results for the wavelet transform in LP and in weighted U- spaces with different weights are obtained in Section 2.5. Relations between W(ltlv f), H(v)(f), Ht~U) and ~~~~(f) are obtained and approximation results and general Parseval formulae for the wavelet transform are established in Section 2.6. Some of the results are extended to distributions in Section 2.7.Section 2.8 contains extensions to
U(~n)
of certain results of
Section 2.3.
2.2
Approximation properties
In this section we exploit approximation properties of S~ [6] to get approximation results for the wavelet transform (2.1.3 ). Theorem 2.2.1. Assume that f(x) E L 1 (~),and (i)
lf!(x) E L 1 (~),
(ii) lfl(x) is continuous at x
=0
(iii) lfi(O) = 1 = 217f:
1 00
-oo
o/(t)dt
(iv)
lfl(x) is even
(v)
fii( ro) is monotonically decreasing in 0 :;::; ro < oo, and hence
(2.2.1)
The Wavelet Transform on 1..1'
27
wfii( ro) -t 0 as ro-t
(vi)
oo,
Let gb(t)- f(b+t) + f(b-t)- f(b) . 2 -
(2.2.2)
Then at a point b, the condition
Iloh gb(t)dt
-
h
=
0
(2.2.3)
o(I), ash -t 0
implies I
a-'1. (W f)(b,a)- f(b)
= o(I) as a-t 0.
(2.2.4)
Proof. Apply relation (2.1.5) to the proof of Theorem 6 given in [6, pp.l7-18]. Theorem 2.2.2. Let f(x) E L 1 (JR.), and let lfl(x), fii(ro) satisfy assumptions of the Theorem
2.2.1. Then I
(2.2.5)
lla- 2 (Wf)(b,a)-f(b)llt-t0 asa-tO. Proof. See [6, p. 25].
= e-x 2 , fii(w) = e-w2 f4nl/2 i/i(w) = H~wz lfl(x) = e-lxl, lfl(x) = (I-x 2)e-!x2, fii(w) = J2iw 2e-iw2 (Mexican hat wavelet).
Example 2.2.3. (i) (ii) (iii)
lfl(x)
The following LP- approximation results and pointwise approximations of the wavelet transform can be proved using [6, Theorems 57 and 58, pp. 114-115].
Theorem 2.2.4. Let j, 1f1 E L2 (1R.) and let fii E L 1(JR.) nL2 (JR.) satisfy (2.2.1).
Then lim ll(2n/a) 112 (W f)(b,a)- f(b)ll2 = 0.
(2.2.6)
a-->0
Theorem 2.2.5. Let J, If/ E L 2 (JR.) and iji(ro)
that -21 n
1
00
= OC,)+< ), e > 0, as ro-t oo.
Assume further
i/i(ro)dro = 1.
-00
Then (2nja) 112 (W f)(b,a) -t f(b) as a-t 0, for almost all b, provided that
Iloh gb(t)dt -t 0 as h -t 0.
-
h
0
(2.2.7)
28
The Wavelet Transform
Several other approximation results can be derived using other theorems of Bochner and Chandrasekharan [6]. Theorem 2.2.6 (Degree of approximation). Let 1/f(x) and iji( ro) satisfy assumptions of Theorem 2.2.1. Let iji( ro) now satisfy:
loco lj/( (J)) (J)q d (J) < oo,
q > 0 instead 0 f
loco lj)( ro )dw < oo.
Then (i)
Jt gb(t)dt = o(hl+q)
implies
(2.2.8) ( ii)
if Iij/( ro) I ~ ij/o (ro ),
where ij/o ( ro) satisfies aforesaid assumptions of iji( ro ), then
conclusion (i) holds.
Proof. See [6, p. 31 ]. Theorem 2.2.7. Let f(x) E U'(~n), I < p < oo, and ij/(ro) E L 1 (~n). Assume that ) f iji(ro)dw=l. -1( 2n n }JRn
Then I
a-:zn(w f)(b,a) f(b)ll 0 II 2n-I (r(n/2))n/2P-->
0 b
as a--> '
lll>n
E~ .
Proof. From [6, p. 100], for R = I/ a, and definition (2.1.1 ), we have I I f t-b SR(b) := 2n-l(r(n/2))n/2 an }JRnf(t)l!'(-a-)dt
I I 2n-l(r(n/2))n/2 an/2 (Wf)(b,a).
Now, the proof follows by an application of Theorem 54 in [6, p. 101 ]. The next theorem gives a result on pointwise approximation in Theorem 2.2.8. Assume that (i) (ii) (iii)
f(x) E
U'(~n),
p
>I
lj/(ro) is radial, and ij/(w) ~ 0
(Z~)n fJRn lj/(ro)dro =I
~n.
(2.2.9)
The Wavelet Transform on U
(iv)
29
ijl(co) = 0 Ccol~+e), e > 0, as Icol--+ oo,
then
l
gb(t)dt = o(t), as t---> 0
implies
as a---> 0. Proof. See [6, p. 103]. 2.3
Wavelet transform, Hilbert transform and Fractional integrals
In this section we consider the general form of the wavelet transform studied by Kaiser [41]:
1
t- b dt (Wf)(b,a)= -oof(t)l!'(-a-)aP' p>O, 00
(2.3.1)
and obtain its relations with the Hilbert transform H, fractional integral operators Ha and
Ia, 0
E
< oo.
Multiplying both sides of (2.3.1) by a-r, oo,
f
r E JR, and integrating with respect to a from 0 to
we get
{""(Wf)(b,a)a-Yda= {"" a-P-Ydajoo f(b+u)l!'('!.)du
lo
lo
a
-00
(2.3.2)
(2.3.3) Iff E U(JR) and 1/f E U' (JR), 1/p + 1/p' = 1, then the u-integral in (2.3.2) is absolutely
convergent and a-integral is absolutely convergent if (2.3.4) Therefore, under above conditions the change in order of integration in (2.3.2) is justified by Fubini's theorem. We can rewrite (2.3.3) as follows:
foo(Wf)(b,a)a-Yda= foo f(b+u)du {"" l!'('!.)a-P-Yda
lo
lo
lo
a
+j0 f(b+u)du {"" l!'('!.)a-P-Yda -00
lo
a
30
=
The Wavelet Transform
r f(b+u)ul-p-ydu lor 1/f(t)tP+Y-2dt+ lor f(b-u)ul-p-ydu lor 1!'(-t)tP+Y-2dt.
lo
(2.3.5)
Putting 2- p - y =
a and using definitions (2.1.14 )-(2.1.15), we get
lo"' (W f)(b,a)a-Yda = r(a)(Waf)(b) lo"' l!'(t)radt +r(a)(W~ f)(b) lo"' 1!'( -t)t-adt.
(2.3.6)
Let us assume that (2.3.7) Then, by (2.1.16) and (2.1.17),
lo"' (W f)(b,a)a-Yda
2Aar(a)sin(naj2)(Haf)(b), if 1/f is odd,
= 2Aar(a)cos(naj2)(1af)(b),
if 1/f is even.
(2.3.8) (2.3.9)
These are the desired relations between the wavelet transform and the operators Ha and Ia, where 0 <
a < 1. In case 11' is neither odd nor even function, then the desired inversion
formulae may not be obtained but certain boudedness results can be obtained using (2.3.6) and boundedness properties of Wa, W~ [57, p. 204]. From (2.3.8)-(2.3.9) we may conclude that the Mellin transform of (Wf)(b,•) is a fractional integral. Hence (Wf)(b,a) can be expressed as inverse Mellin transform of Ha(f) (resp. Ia(!)) according as 1/f is odd or even.
Therefore, theory of Mellin transform can also be applied for investigating properties of the wavelet transform. Let us consider the most interesting case
a
= 0, not covered by the above analysis. If we
assume that 1/f is a basic wavelet, then condition( 1.6.1) yields: 0 =F Ao := fo"' 1/f(t)dt = -fo"' 1!'( -t)dt.
(2.3.10)
Then from (2.3.5) we have
=Ao p.v.
! "'
f(t) . . -bdt (p.v. = prmczpal value). -"' t
(2.3.11)
Thus
lo"' (W !)(b,a)aP- 2da = Aon(Hf)(b).
(2.3.12)
The Wavelet Transform on LP
31
This gives a relation between the wavelet transform and Hilbert transform. Example 2.3.1. Let us find a relation between Haar wavelet transform and Hilbert transform. A translated version of the Haar wavelet 1/f is given by 1, -1/2 ~ X < 0 { 1/f(x)= -1, O~x<1/2 0,
otherwise.
This satisfies
~~ 1/f(x)dx = 0 and
Ao
r
= lo 1/f(x)dx = - 21 .
Hence <~>2-p(b) = -
Now, if we choose f(x)
I
2n(Hf)(b).
= eir:x-ir:fx, -r > 0, then from [26], we get (Hf)(b)
=
ieir:b-ir:fb.
lo (Wf)(b,a)aP-2da
=
-lnie'r:(b-1/b), -r>O.
Therefore, by relation (2.3.12),
r
1
.
Example 2.3.2. Consider the Mexican hat wavelet defined by
which is an even wavelet. Then for 0 <
Aa
a < I,
= {"" (1- t2)e-(l/2)tz t-adt = r(a+t)/2ar( I -a). 2
lo Hence
r = 2(t-a)/2ar( 1 ~a )r(a) cos(na/2)(/af)(b).
Example 2.3.3. Choosing 1/f(t) as in Example 2.3.2 and f(t)
Ao
=
(IL
= -J(n/2) and
1
00
o
(W f)(b a)aP- 2da = -n312 r
'
b
112 - - . I +b 2
2),
we can show that
The Wavelet Transform
32
Example 2.3.4. Newland's odd harmonic wavelet [14, p. 476] is defined by
lflo(t)
= -2
1
m
(cos 2m- cos4m)
I sin 3m sin m 1r
Then, by L18, p. 321 (40)],
Aa =
!o
=
1
t-alflo(t)dt = r( -a) cos -7ra(27r)a-l [I- 2a].
2
0
Hence
= ~(27r)a-l [2a- I](Haf)(b). a
Example 2.3.5. Newland's even harmonic wavelet [14, p. 476] is defined by
lfle(t)
1-(sin4m- sin 2m) 2m
= -
1 cos 3m sin m 1r
so that by [18, p. 321 (41)],
A a=
!o=t-a lfle(t)dt 0
=
1
r( -a) sin -na(27r)a-t [1- 2a]. 2
Therefore,
= ~(27r)a-l [2a -1](Iaf)(b). a
lnversionfonnulae
In order to study mapping properties of the wavelet transform we define the space W{ (JR( x JR(+), y E JR(, 1 :::;; p <
W{(JR( x JR(+)
= {
oo, as follows: g(b,a):
llgll~ =(~~I ~n= g(b,a)a-YdaiPdb) 11P < oo}.
Since H: U(JR()---> U(JR(), 1 < p <
oo, and H(Hf) = - f,
(2.3.13)
from (2.3.12) it follows that
W maps U(JR() onto Wf_p(JR( x JR(+), and from (1.5.3) and (1.5.4), we have the following inversion and Parseval formulae.
Theorem 2.3.6. Assume that f E U(JR(), I < p < ""· Let 1f1 E u' (JR() be a basic wavelet satisfying (2.3.10). Then 1 p.v. f(t) =--A 1r 0
/=
-=
-db b ~o= (W f)(b,a)aP- 2 da, - t
0
pointwise almost everywhere and in the LP- norm.
(2.3.14)
The Wavelet Transform on U'
33
1.0
0.5
-4
4
-0 5
-1.
Fig. 2.1: Graph of l!fo(t)
05
-4
4
-0.5
-1.0
Fig. 2.2: Graph of l!fe(t) Further, if g E U' (~).then
I:.fo"" = Next,assume that 0
g(b)(Wf)(b,a)aP- 2dadb
-[:fo""
f(b)(Wg)(b,a)aP- 2 dadb.
1. Then, for 1 < p
p.44 Theorem l(a)], Haf E U(~), r- 1 = p- 1 -
a
(2.3.15)
< a- 1 and fEU(~). by [43,
and (2.3.16)
34
The Wavelet Transform
for some K > 0. Now, using (2.3.8) and (2.1.9), we get the following inversion theorem for the wavelet transform (2.3.1 ). Theorem 2.3.7. Let f E LP(~), 1 < p < a- 1, where 0
Let 1/f E u' (~)be an odd function which satisfies (2.3.7) and let
L"' ill'(t)lt-adt <
(2.3.17)
oo.
Then, W: LP(~)--> Wy(~ x ~+), y = 2- p- a, r- 1 = p- 1 - a, and lim acot(naj2) f sgn(b-t) db {""' W(b,a)a-Yda s~o+ 2nAa Jlb-tl~o lb- t!Y+I lo pointwise everywhere and also in LP- norm.
f(t)
=-
(2.3.18)
Theorem 2.3.8. Let p, a, f and 1/f satisfy all conditions of Theorem 2.3. 7, except one prop-
erty, viz. 1/f is odd, which is now assumed to be even. Then
w: U(~)--> w;(~ x ~+), r= 2- p- a, r- 1 = and f(t)
=-
. atan(naj2) hm 2 nAa
s~o+
h
lb-tl~o
lb
p- 1 -
a,
1 ~o= !Y+I db [(W f)(b,a)- (W f)(t,a)]a-Yda
- t
o
(2.3.19)
pointwise almost everywhere and also in LP-norm. The range of the wavelet transform can be characterized using the characterization of the range of Ha (and also of Ia) [33]. Theorem 2.3.9. Let p, a, f and 1/f satisfy conditions of Theorem 2.3. 7 when 1/f is odd and
that of Theorem 2.3.8 when 1/f is even. Let F(b,a) be any function in Wy(~ x ~+), r- 1 =
= (W f)(b,a)for some f E LP(~) if and only ifiiTs
p- 1 - a. Then F(b,a) for all positive 8,
2.3.2 Approximation Applying Parseval formulae (2.1.23), (2.1.24) with Jl =a and v
= 0 to the relations (2.3.8)
and (2.3.9) respectively we get Parseval formulae for the wavelet transform. Theorem 2.3.10. Let p, a, f and 1/f satisfy conditions of Theorem 2.3.7 when 1/f is odd
and that of Theorem 2.3.8 when 1/f is even. Let f E LP(~) and gEL''(~), r- 1 = p- 1 - a, 1/r+ 1/r' =I, then
l~looo g(b)(W f)(b,a)a-Ydadb = =f ~~looo f(b)(Wg)(b,a)a-Ydadb where - or + sign is taken according as 1/f is odd or even.
(2.3.20)
35
The Wavelet Transform on LP
Next, we give an approximation property of the wavelet transform. Theorem 2.3.11. Let fEU(~), I < p < a- 1, where 0
u' (~) be an odd (resp. even) function which satisfies (2.3. 7) and (2.3.17).
Then, for~( a)= 2Aar( a), we have (i)
(ii)
II~(~) lbl-a fooo (W f)(b,a)a-Yda- fliP--> 0 as a--> 0+,
(11' even),
(2.3.21)
ll~(a)si~(naj 2 )ibl-a fooo(Wf)(b,a)a-Yda-Hfllp -->Oasa-->0+,(11' odd). (2.3.22)
Proof. From (2.3.8),(2.3.9) and (2.l.l7) we have
lbl-a
= 2~(a)lbl-a(w~ f(b)
+ Waf(b)), (11' even).
Now,
II~~~~
and
ll~(a)~~~:aj 2 )
a
-->
llibl-a(Haf)(b)-(Hf)(b)IIP' (11' odd).
0+ and invoking (2.I.18), we arrive at the proof of the theo-
rem. 2.3.3
Invariance under rational transformations of translation parameter
Using relation (2.3.I2) and invariance of the Hilbert transform under certain rational transformations of the integration variable [24], we obtain invariant properties of the wavelet transform under certain rational transformations of the translation parameter. Let {at,a2,a3, ... ,an} be a sequence of positive numbers and {bt,b2,b3, ... ,bn} be an increasing sequence of arbitrary real numbers. Glasser [6] has established the following two properties of the Hilbert transform. Theorem 2.3.12. For any F whose Hilbert transform is defined by
H {F(x);y}
I ~oo F(x)
=-
n
-oo
-dx, x-y
(2.3.23)
36
The Wavelet Transform
we have (i) H{F(I/Jt(x));y} =H{F(x);lf>t(y)},
(2.3.24)
where n-1
1/11 (x) = x-
L x-b __!!)__. 1
(2.3.25)
J=l
(ii) H {F(tf>2(x));y} = H {F(x);O} -H {F(x);tf>2(y)},
(2.3.26)
where n-1
th(x)
=
L x:Jb··
)=I
(2.3.27)
J
Theorem 2.3.13. Assume that 11' satisfies (2.3.10). Then for any F whose Hilbert transform
is defined by (2.3.23 ), we have (2.3.28)
loco W(F(tf>2))(b,a)aP- 2 da =loco (WF)(O,a)aP- 2 da
(ii)
-loco (W F)(tf>2(b),a)aP- 2 da. Proof. (i) Using (2.3.12) and (2.3.24) we have
loco W (F( 1/11)) (b, a)aP- 2 da =AonH(F( 1/JI) )(b, a) =AotrH(F(x); 1/11 (b)) =loco (W F)( 1/JI (b),a)aP- 2 da; from which (2.3.28) follows. (ii)
From (2.3.12) and (2.3.26) it follows that
loco (W F)( tf>2(b),a)aP- 2 da =AotrH(F(tf>2(x));b) =
AotrH {F(x);O} -AotrH {F(x); tf>2(b)}
=loco [(WF)(O,a)- (WF)(tf>2(b),a)]aP- 2 da. This yields (2.3.29).
(2.3.29)
37
The Wavelet Transform on LP
2.4
Wavelet transform on weighted U- spaces
Let w be the weight function satisfying (2.1.31) and U(wP), 1 ~ p <=,be the weighted U - space defined by (2.1.35).
Now, we define the weighted analogue of the space W{ given in (2.3.13) to be the space
W{,uO~
X
=
IR.+) = { g(b,a): llgllp,y,U
(1: I
fooo g(b,a)a-YdaiPU(b)db)
l/p
<
oo}. (2.4.1)
a = 2- p - r < 1, then for 1 < p < a- 1 and f E U ( wP) we have g = Haf E U(wr),r- 1 = p- 1 - a. Using Theorem 2.1.3 and relation (2.3.8) we get the
If we assume that 0 <
following inversion theorem.
Theorem 2.4.1. Assume that 1 < p < a- 1, r- 1 = p- 1 - a andy= 2- p- a. Let w satisfy (2.1.31), and Let (W f)(b,a) be defined by (2.1.1) with f E U(wP). Let l/f be an odd function which satisfies (2.3. 7), and Let
Q-r:=supl(1+ltl)-rl!f(t)l <=, -r= 1-a-e,
(2.4.2)
t
for some e > 0. Then
w: U(wP)----> w;,wr(IR. X JR.+) and (2.3.18) holds pointwise almost everywhere and in the norm of U (wP).
Proof. From (2.3.2) we have fooo(WF)(b,a)a-rda= fooo a-r-P
=
fooo a-r-p
(l:f(t)l!fc~b)dt)da
(l:f(t)w(t).l!fc~b)[w(t)t 1 dt)da.
(2.4.3)
(2.4.4)
By HOlder's inequality, fooo lf(t)w(t).l!f
C~b)
[w(t)t 1 1 dt
~ llfllp,wP
lll/lb;a llp'.w-P', 1/ p+ 1/P1 = I, (2.4.5)
where l/fb;a(t) = l/fe~h). Now, for -r = 1 -
a - e, we have
(2.4.6)
38
The Wavelet Transform
Using inequality [95, p. 97];
lul2) s ( I+ I + lvl2
~ 21sl (I+ lu- vl2) lsi "' '
s E JR.,
(2.4.7)
for -r > 0, we can write
I ( where D = 2-rp 12 I+ I~
I2) 'rp1/2 .
Since I I ( I +t2 ) I a2 +t2 = 1 +t2 I+ (tja)2 a2
~max(1,~) ~, a 1+t therefore,
(I+ where
E
=
It~ b I) --rpl ~ Dmax ( 1,a-rpl) (1 +t2r-rp112 ~ E (1 + ltl)--rpl,
(2.4.8)
D2-rp1/ 2 max ( 1, a -rp1). Hence by (2.1.34 ), 00
j_oo [w(t)tP
I
(
1+
It ~ I) b
-'rpl
dt
(2.4.9) Now, if we assume that (2.4.2) holds, then from (2.4.6), using (2.4.9), it follows that (2.4.10) Therefore, t-integral in (2.4.3) is absolutely convergent under the above conditions on f and
l!f. The a- integral is absolutely convergent under the conditions (2.3.4 ). Hence, by Fubini 's theorem, change in order of integration in (2.4.3) is justified.
The Wavelet Transform on LP
39
Now, proceeding as in Section 2.3, we arrive at relation(2.3.8) for f E LP(wP) and lfl satisfying (2.4.2). Then application of Theorem 2.1.3 to this relation proves the theorem. Similarly, relation (2.3.9) and Theorem 2.1.4 yield the following: Theorem 2.4.2. Let p,
r,
a, p, w andfbe the same as in Theorem 2.4.1. Let lfl be an even
function which satisfies (2.3.7) and (2.4.2). Then
w: LP(wP)-+ w;wr (JR. X JR.+) and (2.3.19)holds pointwise almost everywhere and in the norm of LP(wP).
The range of wavelet transform (2.3.1) can be characterized by means of Theorem 2.1.5. Theorem 2.4.3. Let p,
a, p, wand f be the same as in Theorem 2.4.1. Let lfl be an
r,
odd(resp. even) function which satisfies (2.3.7) and (2.4.2). Let FE w;,wr (JR. x JR.+)· Then F(b,a) = (W f)(b,a) for some f E LP(wP) there exists a constant C
if and only if
> 0 independent of o > 0 such that
where y is given by (2.3.8)- (2.3.9).
2.5
Boundedness of the wavelet transform on LP with different weights
In this section, instead of using relation (2.3.9), we derive an inequality between the integral of the wavelet transform and the operator Ia and use (2.1.40) to find a boundedness result for the wavelet transform. We have for y E JR., l!ooo(Wf)(b,a)a-Ydal
~
foooi(Wf)(b,a)la-Yda
C
~ fooo a-P-r (I:if(t) lfl ~b) Idt) da (2.5.1)
(2.5.2) The above change in order of integration can be justified as follows. We see that
I:
jt(t)lfl
C~b) I =I: dt
~
jt(t)[w(t)] 11Plfl
llfllp,w lllflb;allp',w-lfp ·
C~b)
[w(t)t 11PI dt
40
The Wavelet Transform
But, for I
=
(l
< p < oo,
oo
[w(t)]
_'
P fp
(
Jt-bJ)(a-l)p' ( 1 + Jt-bJ)(l-a)p'l (t-b)lp' dt )ljp' 1Jf -
I +-a
. -oo
a
00
a
)
::,:; QE j_oo[w(t)rP'!r (I+ Jtl)(a-I)p' dt (
!jp' <
oo
by(2.I.41) and (2.4.8), where E is a constant and
Q = s~p{ (I+ lti) 1-a llJI(t)l}.
(2.5.3)
If p =I, then
j_~ jt(r) 1Jf C~b) Idt = j_~ jt(t)w(t).lJI C~b) [w(t)r 'I dt ::,:;
s~pllJIC~b) [w(r)r'l/_~1/(t)lw(t)dt.
But, by (2.I.42) and (2.4.8),
::,:; sup I(I+ ltl)'-alJI(t)l E(I + lti)a-I [w(r)r' I
Thus, we see that t-integral in (2.5.1) is absolutely convergent for
f
E
U(w) and 1Jf sat-
isfying (2.5.3). The a-integral is absolutely convergent when (2.3.4) holds. Under these conditions the change in the order of integration in (2.5.1) is justified. Then proceeding as in Section 2.3 we get
fooo I(W f)(b,a)Ja-Yda::,:; r(a)Wa(l/l)(b) fooo ilJI(t)lt-adt +r( a)w; (1/l)(b) fooo llJI( -t) It-adt.
(2.5.4)
41
The Wavelet Transform on LP
If we assume that there exists a constant B
> 0 such that (2.5.5)
and (2.5.6) then by (2.1.17),
l"'
I(W f)(b,a)l a-Yda =
~ B[r(a)Wa(lfl)(b) + r(a)W;(Ifl)(b)J
Br( a)2cos(nai2)Ia(lfl)(b).
(2.5.7)
Then using Theorem 2.1.7 we have the following:
Theorem 2.5.1. Let w be a nonnegative weight function on a set of positive measure satisfying (2.1.40)-(2.1.41). Let 0
~
p
< oo, q E Q(a,p) andy= 2- p- a.
Let f E LP(w) and let IJI satisfy (2.5.3), (2.5.5) and (2.5.6) then there exists a non-negative weight function v, different from zero on a set of positive measure, and a finite constant C( a) > 0 such that
IIWfllq,a,v ~ C(a) llfllp.w·
(2.5.8)
Applying Theorem 2.1.8 to relation (2.3.9) we get the following boundedness result.
Theorem 2.5.2. Let p, r > 1 and 0 < a
< I with 1I p + I I r = I + a.
Let f E LP(~) and v E U(~). Let 1J1 E u' (~)be an even function which satisfies (2.3.7). Then there exists a sharp constant C( a, p) such that
where y = 2 - p - a
2.6
Wavelet transform and operators H(v), Ht~ and It~
In this section we study wavelet transform of ltlv f(t), where v E ~and fEU(~). At first we obtain relations between W(ltlv f), H(v)(f),
Ht~(f) and ~~~~(!).These relations
are used to investigate approximation properties of the wavelet transform and to establish inversion and Parseval formulae. Some of the results of this section are generalizations of those contained in Section 2.3.
42
The Wavelet Transform
Proceeding as in (2.3.2) we find that for any v E JR.,
~a= W(ltlv f)(b,a)a-Yda =~a= a-P-r (l~ f(t)ltlvll' C~ b )dr) da. The t-integral is absolutely convergent iff E U(IR.) and ltlvll'(!~h) E u'(JR.), l / p 1 = l. But, as in (2.4.6) we have, for some r
(2.6.1)
1/p+
> 0,
whereO < v+ 1/p' < r, Q-r
= supi(I + ltlrll'(t)l < oo
(2.6.2)
t
and E' a constant similar to that occuring in (2.4.8). Also, from (2.3.4), the a-integral is absolutely convergent if lJI( ±t)tP+r- 2 E L 1( 0, oo). Then the order of integration in (2.6.1) can be interchanged and for 2 - p - a =
r we can write
~a= W(ltlv f(t))(b,a)a-Yda = r(a)Wa(ltlv f)(b) ~a= lJI(t)t-adt +r( a)W~ (ltlv f)(b)
~a= lJI( -t)t-adt
= 2Aar(a) sin(naj2)Ha(ltlv!)(b),
if lJf is odd
(2.6.3)
= 2Aar(a)cos(naj2)1a(ltlv!)(b),
if lJf is even,
(2.6.4)
where A a is given by (2.3. 7). Therefore, by (2.1.20)-(2.1.21 ),
lblll-v-a ~a= W(ltlv f)(b,a)a-Yda
= 2Aar(a)sin(naj2)HtkU)(b) (II' odd)
(2.6.5)
= 2Aar(a)cos(naj2)1~~1U)(b) (II' even).
(2.6.6)
Next, as in (2.3.12) we find that, for
a = 0,
The Wavelet Transform on U
43
so that lbl-v fo'"" W(ltlv f)(b,a)aP- 2da = AonH(v)(f)(b).
(2.6.7)
Now, applying Theorem 2.1.1 to relations (2.6.5)-(2.6.6) we get the following.
Theorem2.6.l.Letp>1,
0
1I p- 1 < v < 1I p- a, 1I r = 1I p- p.
(or0
> 0 and r = 2- p- a. Assume that f
E U(JR)
and lJf satisfies (2.6.2) with r > max( 1 -a, v + 1I p'). Then there exists a constant K = K(p,a, v,p.) such that
lllbl~-v-a ~o= W(ltlvf)(b,a)a-Ydallr ~KIIfllp ~tan(nai2)KII!IIP
Also,for g E
u' (IR),
(lJI odd)
(2.6.8) (2.6.9)
(lJI even).
1lr+ 1lr' = 1, we have the Parsevalformulae:
L:~o= g(b)W(Itlv f)(b,a)a-rlbl~-v-adadb = =f L:~o= f(b)W(Itlv' g)(b,a)a-rlbl~-v-adadb, where - or + sign is to be taken according as lJI is odd or even and v' = p. -
(2.6.10)
v- a.
Next, applying Theorem 2.1.2 to (2.6. 7) we get the following boundedness, inversion and approximation properties of the wavelet transform.
Theorem 2.6.2. Let f E U(IR), p > 1, 1lp -1
< v < 1lp. Let lJI satisfy
(2.6.2) with
r >max( I, v + 11 p'). Then (i)
lllbl-v
(ii)
f
~o= W(ltlv f)(b,a)aP- 2daL ~ Ap,v II/IlP;
= --1-H(v)[lbl-v nAo
r= W(ltlv f)(b,a)aP-
lo
2 da];
(2.6.11)
(2.6.12)
(iii) for gEU'(!R), 1lp+llp'=1,
L:~o= g(b)W(itlv f)(b,a)aP- 2 ibl-v dadb =-
L:~o= f(b)W(Itl-v g)(b,a)aP- 2 Iblv dadb.
(2.6.13)
44
The Wavelet Transform
Theorem 2.6.3. Assume that conditions of Theorem 2.6.1 hold with 11 =a. Let lfl satisfy (2.6.2) with
II t,(I)\a)
-r >max( I - a, v + 1/ p').Then
lbl-v fo"" W(ltlv f)(b,a)a-Yda- fliP---> 0
as a---> 0+ (lfl even),
(2.6.14)
(2.6.15) where t.(ll(a) = 2Aar(a)cos(anj2) and t.<
2.7
2l(a) =
2Aar(a) sin(an/2).
The Wavelet transform on F~
The wavelet transform has been studied on certain distribution spaces by Holschneider [36], Pathak [62], Pathak and Singh [73]. Hilbert transform was extended to distributions in ~~ by Pandey [53]. Fractional integral operators Ha and Ia were studied in ~~ by Pathak [55]. Using (2.3.12) we extend the wavelet transform to
F~-
distributions studied
by McBride [49]. For this purpose we define at first the test function space Fp, which is the Fo.p-space of McBride [49]. An infinitely differentiable function f/> on JR. belongs to Fp if and only if (2.7.1) Then Fp, 1 ::::;; p < oo, is a locally convex, Hausdorff and sequentially complete topological vector space [49]. Another space useful for the study of wavelet transform is the space G~.v(IR. x JR.+), J1, v E JR.,
1 ::::;; p <
oo,
consisting of all C"'- functions g(b,a) defined on
JR. x JR.+ and satisfying k = 0, 1,2, ...
(2.7.2) Then G~,v (JR. x JR.+) is also a locally convex, Hausdorff and sequentially complete topological vector space. Theorem 2.7.1. Let f/> E Fp(IR.) and lfl E Y(IR.) satisfy (2.3.10). Then,for 1 < p < oo, the wavelet transform Wlf> defined by (2.3.1) is a one-to-one linear continuous map from Fp(IR.) onto
a; l.p -I (JR. x JR.+) and W- 1 is given by (2.3.14 ).
45
The Wavelet Transform on LP
Proof. We note that for a differentiable function lfl,
(adjda+bdjdb+ I) lf!((x-b)/a) = -xdjdx lf!((x-b)ja). a a Invoking this relation to (2.3.1) and using integration by parts, we get (ad/ da + bd / db)k[aP- 1(W If> )(b, a)]
=
1
00
-oo
dt (td / dt/1/> (t) lfl( (t- b) fa)---;;;
so that
1
00
a-P+ 1(adjda+bdjdb/aP- 1(Wq>)(b,a) =
according to (2.3.1), where ek(t)
(2.7.3)
=
-oo
dt fh(t)lf!((t -b)/a) aP
W(Ok)(b,a),
(2.7.4)
(td jJtllf'(t).
=
Since by (2.3.12),
fooo W(Ok(b,a))aP- 2da =Aon(HOk)(b),
(2.7.5)
application of inequality (1.5.2) yields
lliooo W(Ok(b,a))aP- 2daL ~ AonCpiiOkllp· Therefore, by (2. 7.4 ),
lliooo a- 1(adjda+bdjdb)kaP-i(Wq>)(b,a)daiiP ~ AonCp ll(td/dt)klf'(t)IIP. Thus
J3kp,-i,p- 1(W) 1/>
~
AonC,p() P"'k If> , k = 0 , I, 2 , ....
(2.7.6)
Also, since H- 1 = -H, from (2.7.5) we get
r
(2.7.7)
W(Ok)(b,a)aP-2dall = Cp !3{'-l.p-i(Wq>). P Aon
(2.7.8)
W(Ok)(b,a)aP- 2da); Ok(t) = (tdjdt/q>(t) = - -1-H ( lo Aon so that by (1.5.2) and (2.7.5), A.{(q>)
~ ACp II on
kr
In view of (2.7.6) and (2.7.8) we conclude that W is a one-to-one mapping from Fp(lR) onto a;l,p-l(JR X JR.+)· Therefore, w- 1 is defined on a;Lp- 1(1R X JR.+). Since w- 1wq, = 1/> for all If> E Fpc LP, it follows that
w- 1 is given by (2.3.14).
Now, we define the wavelet transform W'T of generalized function T belonging to (G;i,p- 1 ) 1 , the dual of G; 1·p-i (JR. X JR.+) by
(W'T,q>) := (T,Wq>),
(2.7.9)
46
where 1/J
The Wavelet Transform
E
Fp(IR.).
Using duality arguments we can prove Theorem 2. 7 .2. Let T E (
c; 1,p- 1)',
I
< p < oo and lfl E .9' (JR.) satisfy (2.3. 10), then the
generalized wavelet transform defined by (2.7.9) is a one -one continuous linear map of
(c; 1,p- 1)' onto F~. Next, we assume that 0 < 2- p- r
=a<
I and apply relations (2.3.8)-(2.3.9) to get
fo"" W(Ok)(b,a)a-Yda = 2Aar(a) sin(naj2)(Ha0k)(b),
if lfl is odd
= 2Aar(a)cos(naj2)(/a0k)(b),
if lfl is even.
Then applications of inequalities (2.1.8) and (2.1.12) give
lifo"" W(Ok)(b,a)a-Ydallr ~ 2Aar(a)MIIOkiiP,
(2.7.10)
(lfl odd)
~ 2Aar(a)LIIOkiiP, (lfl even),
(2.7.11)
where r- 1 = p- 1 - a, and MandL are certain positive constants. Therefore, using (2.7.4), (2.7.10) and (2.7.11), we get
lifo"" a-Y-P+ (adjda+bdjdb)kap-i(WI/J)(b,a)dallr ~ QIIOkiiP, 1
for some constant Q > 0. Hence (2.7.12)
where J1 = I - y- p and v = p - 1. From this we conclude that for 0 < 2 - p - y =
a < I, W is a continuous, linear map from
Fp into G~'v. Therefore, as in (2.7.9), we can define generalized wavelet transform ( G~,v (JR.
X
w' T
of T E
JR.+) )
1•
Theorem 2.7.3. Let f E (G~,v (JR. x JR.+))', 1 < p
< oo and 0 < 2- p- y =a< 1. Let
lfl E Y'(IR.) satisfy (2.3.7). Then W'f defined by (2.7.9) is a one-one, continuous, linear map from (G~.v)' into F~.
Following the above technique and the distributional analysis contained in [55] the results of Section 2.4 can also be extended to distributions.
47
The Wavelet Transform on LP
2.8 The wavelet transform on LP(I~n) In this section we extend some of the results of Section 2.3 to n-dimensions.
For
this purpose we assume that a= (aJ,az,a3,···,an), a;> 0, i = 1,2, ... ,n and b =
(b h bz, b3, ... , bn) E ~n. Define the wavelet transform by (2.8.1)
(Wy,f)(b,a) = { f(t)lf/b;a(t)dt,
}JRn
where _
lf/b;a(t)-a
(t1-b1 tz-bz tn-bn) If/ - - , - - , ... , - - ' a1 az an
-p
(2.8.2)
and -p a - p-- a -p 1 . .. an .
This transformation with p = 1 has been studied by [74]. Next, we recall the definition and properties of the Hilbert transform inn-dimensions [53, pp. 152-153]. Let f E LP(~n), p > 1, then its Hilbert transform (Hf)(x) defined by
(Hf)(x)=n-nlim~axe;~O+ 1EIIi
{ . On ~(t~ ·)dt Jlr,-x,l>e, l=i t, x,
(2.8.3)
exists a.e and Hf E LP(~n). There exists a constant Cp > 0 such that (2.8.4) Moreover, for all f E LP(~n),
(H 2 f)(x)
=
(-It f(x)
(2.8.5)
in LP(~n) and a.e. Now, multiplying both side of (2.8.1) by a-Y = a~Yl .a~Y2 · · · a;;Yn, with Y1 =
}'2 = · · · Yn =
2 - p and integrating with respect to a = (a 1, az, ... , an) we get
{ (W f)(b,a)aP- 2 da
JJR"+
=
r
lrR"+
f(b+u) du u
r
= { f(b+u)du { lfl(':!_)a- 2da }JRn
lfl(t)dt+
lrR"+
J'iR"+
r
f(b-u) du
lrR"+
u
a
r
lfl(-t)dt.
lrR"+
If we assume that
r
}'!Rn
lfl(t)dt
= o,
(2.8.6)
then for
O=rfAn= { lfl(t)dt=- { lfl(-t)dt,
lrR"+
lrR"+
(2.8.7)
48
The Wavelet Transform
we have ( (Wf)(b,a)aP- 2da=An(f
}~!'+-
}~"+
=An p.v. (
f(b+u)du- ( f(b-u)du) u lJR"+ u f(tb) dt,
lJR"+ t-
where t- b = (t,- b, )(t2- b2) ... (tn- bn) and p.v. is taken as in (2.8.3). This gives the desired relation between n-dimensional wavelet transform and Hilbert transform: (2.8.8) Using (2.8.5) and (2.8.8) and proceeding as in Section 2.3.1, we arrive at the following inversion theorem. Theorem 2.8.1. Assume that f E LP(JR.n), 1 < p
Let II' E
u' (IR.n), 1I p + 1I p' =
< oo.
1, be a basic wavelet which satisfies (2.8.6).
Then
(2.8.9) in LP- norm and a.e.. Remark 2.8.2. It will be worth while to investigate other results of previous sections to n
dimensions and extend to distributions.
Chapter 3
Composition of Wavelet Transforms
3.1
Introduction
In this chapter we consider a variant of (2.1.1) and define the wavelet transform of tf> with respect to the wavelet II' by
(Wtf>)(b,a)
=
f
}JRn
1/>(t)l/f((t-b)/a)dt/an,
(3.1.1)
provided the integral exists, where bE ~n and a> 0. If tf> E L2 (~n) and II' E L2 (~n), then using Parseval formula for Fourier transforms we can write the wavelet transform in Fourier space in the following form ([36], p.9):
W(b,a)
= (Wtf>)(b,a) = (2n)-n f ei(lO,b)fil(aro)~(ro)dro. }JRn
(3.1.2)
This form of the wavelet transform is very similar to that of a pseudo-differential operator with symbol a(a, ro)
= fi/(aro).
Hence a theory of the wavelet transform can be developed
analogous to that of a pseudo-differential operator [90]. Assuming that 1/f E .9"(~n) we investigate continuity of the wavelet transform (3.1.2). We define the composition (product) W1 o W2 of two wavelet transforms W1 and W2, which turns out to be another wavelet transform of three variables. The continuity of W1 o W2 is also investigated and a reconstruction formula is obtained. Certain weighted Lebesgue and Sobolev spaces are defined. Results on the boundedness of the wavelet transform in certain weighted Sobolev norms are obtained. Corresponding results for the composition W1 o W2 are also given. 49
so
The Wavelet Transform
3.2 The wavelet transform of tempered distributions In this section we need the test function space Y(IR.n x JR.+) defined to the space of all functions c/J E C"(IR.n x JR.+) such that for .e, k E No and
Ye,a,k,f3(c!J)
laeba(ajaalD~c/J(b,a)i < oo,
sup
=
a, {3 E N0, (3.2.1)
(h,a)ElR" xlR+ l+lal~k+l/31
Clearly, the Schwartz space Y(IR.n x JR.+) is contained in .9"(1R.n x JR.+). Theorem 3.2.1. Let lfl E .9"(1R.n). Then the wavelet transform (Wc!J)(b,a) is a continuous
linear map of Y(IR.n) into Y(IR.n x JR.+). Proof. For f,k E No and a,{3 E N0, we have after differentiation and integration by parts,
aeba(a jda)kD~Wc/J(b,a) =
(2n)-nbaD~ .~n ei(b.ro)~(ro) { ae(a jJa/iji(aro) }dro
= (2n)-nba { ei(b.rol(iw)f3~(ro)ie+k (w-e { ei(ro,ay)(D )e[(ro,y/lfl(y)]dy) dro }]Rn
= (2n)-nba x
}]Rn
Y
r ei(b,ro)jlf31+f+kwf3-e~(ro)
}]Rn
(lrn ei(ro,ay)D~ [ IA-I=k L (k!jA,!)roA.llfl(y)l dy) dro lR"
=
(2n)-nif+k+lf3+al fnei(b,ro)
JR
x
c;~a
(a) (Dw)a-8(wf3+A.-e~(ro)) 8
(k.n (iay)c;(Dy)e[llfl(y)]ei(ro,ay)dy) dro
= (ln)-nie+k+lf3+a+281
r" ei(b.w) r, (k!JA,!) r,
JJR X
L (k!JA,!) L IA-I=k
IA-I=k
(a) 00 -8v~-8(wf3+A.-e~(w))
Ci~a 8
(k.n (Dy) 8i(Dy)f(/lfl(y))ei(ro,ay)dy) dro
=
(ln)-n/+k+I+I/3+a+2Cil
L (k!JA,!)L, (a) L (a
IA-I=k X
c;
8
P
-8)
P
r ei(b.ro)A({3,.e,;.,,p)wf3+A.-f-p-CiD~-Ci-p~(ro)
}]Rn
x (
L ( 8 ) A'( 8, y) frn"i-rv;+Ci-Y(/lfl(y))ei(ro,ay)dy) dro.
r~c;
r
lR
51
Composition of Wavelet Transforms
1/31 + k ~ Ia I + t', we can write laeba( a I da)k D~ q>(b, a) (W q,) (b, a) I
Therefore, for
(a)o
L. (k!/A,!) L.
~ (2n)-n
8~a
IA.I=k
xA(j3, t',A,, p )A' ( X
X
L.
p~a-8
(a -o) L. (o) P
r~8
L.
( + o-y) t'
Y -r~e+8-y
-r
o, y)A" (A,, -r, If!)
r (1 + lroi)I.BI+k-f-lpl-181+n+IID~-8-p ~( (0) ldro/( 1 + lrol)n+
}!Rn
k..
(1
I
+ lyl)k+l81-1-rl-lrl+n+IIDf+8-y+-rlfi(Y)Idy/(1 + IY!t+l;
so that
Ye.y.k,,B(WI/>)
~ L (k!/A.!) L (a) L 8~a
IA.I=k
X
L
(
p~a-8
(a-o) L (o) P
y~8
Y
~
t'+o-r)
B( a, J3' y, -r, A, o,p, t', n)Y!.BI+k-f-lpl-181+n+l,a-8-p ( q,)
-r
-r~e+8-r
o
x Yk+l81-lrl-l-rl+n+ 1/+8+-r-y( If!)·
Thus Wlf>(b,a) E 9(~n x ~+),and from the above inequality the continuity of WI/) also follows. In view of the above theorem the generalized wavelet transform W'T ofT E 9', the dual of 9 (~n x ~+), can be defined by (3.2.2) Using duality arguments we have
Theorem 3.2.2. The generalized wavelet transform W' : 9'
--->
9' is linear and continu-
ous. We can also analyse the wavelet transform by imposing a condition on the Fourier transform of the wavelet 1f1 as follows: Assume that lj) E Coo(~n) such that (3.2.3) where m E set of all
~.
0
~
p
~
1,
a
E N
0. Let us define the function space 9
1 (~n
x
~+)
to be
c= -functions q, on ~n x ~+ such that fort', k E No and a, J3 E N0, Ye,a,k,,B(I/>) =
sup (b,a)EIR"x!R+ l+lal~k+I.BI l+lal~-m
laeba(a;aa)kD~If>(b,a)l < oo.
(3.2.4)
The Wavelet Transform
52
Theorem 3.2.3. Assume that the wavelet lfl satisfies (3.2.3 ). Then the wavelet transform
.9", (JRn x JR+) and the generalized wavelet transform is a continuous linear map of .9"{ (JRn x JR+) into .9"' (JRn). W is a continuous linear map of Y(JRn) into
Proof. Proceeding as in the proof of Theorem 3.2.1 we can write Jaeba(() jJa)kD~ W !f1(b,a)
J
L (k!/A.!) L (a) j[D~D~i{/(u)]u=aroJalrllwiiA-1
::_;; (2n)-naffm IR"
x
E
8.;;;a-r
::.:;; (2n)-n
IA-I=k
(a-
y.;;;a
r) A(/3,
c5
c5)1rollf3- 8 11D~-r- 8 ;p(ro)ldro
E (k!/A.!) L E IA-I=k
r
y.;;;a8.;;;a-r
(a) Y
(a-r)A(f3,c5)alrl+f c5
r lroJIA-1+1/3-81(1 +aJroJ)m-p(lri+IA-IlJD~-Y- 8 ($(ro)ldro ::.:;; (2n)-n L L L (k!/A.!) (a) (a-r)A(f3,c5) X
./JRn
IA-I=kr.;;;a8.;;;a-r
Y
c5
r (I +aJroi)IYI+f+m-p(lrl+kllwlk+lf31-181-1rHID~-y-8($(ro)Jdro ::.:;; (2n)-n L L L (k!/A.!) (a) (a-r)A(f3,c5) X
./JRn
IA-I=kr.;;;a8.;;;a-r
X
Y
c5
r (l + lwl)k+lf31-181-lrl-f+n+IID~-y-8($(ro)Jdro(I + lrol)-n-1
./JRn
for lrl +t'+m- p(lrl +k) < 0. Therefore, form< -t' -Ia I and t'+ Ia I::.:;; k+ 1/31, we have
Ja'ba(d/da)kD~W!f1(b,a)l::.:;; (2n)-n E E E
(k!/A.!)
IA-I=kr.;;;a8.;;;a-r
(a) (a- r) Y
c5
xA(/3' c5, n)Yk+l/31+181-lrl-f+n+ l.a-y-8( ($). From this we conclude that the wavelet transform is a continuous linear map of Y(JRn) into ,9"1
(JRn
X
JR+).
As in Theorem 3.2.1 we define the generalized wavelet transform W' ofT E .9"{, the dual of .9"1(JRn x JR+ ), by (3.2.2) and get the second part of the theorem.
3.3
Composition of wavelet transforms
Let W1 and W2 be two continuous wavelet transforms defined as follows: (W,q,)(b,a)=(2n)-n { ei(b,ro)i/f,(aw)($(w)dw,
./JRn
bEJRn, aElR+
(3.3.1)
53
Composition of Wavelet Transforms
and (3.3.2) provided the integrals exist. Then their composition(product) Wt o W2 is defined by
(Wlf>)(b,a,c)
= (Wt o W2)l/>(b,a,c) = (2n)-n { ei(b,lO)Vtt(aco)(~d(W2l/>))(co,c)dco }JRn
(3.3.3)
(3.3.4) where
~d
denotes the Fourier transformation with respect to the variable d and
x(a,c,co)
=
(3.3.5)
Vtt(aro) Vt2(cro),
provided the integrals involved are convergent. For instance, W1 o W2 is well defined when if> E L 2 (1R.n),l/fi E L 1(JR.n) and 11'2 E L 2 (JR.n). From (3.3.4) it is obvious that the composition
Wt o W2 is a wavelet transform of three variables mapping from IR.n into IR.n x JR.+ x JR.+. Now, we impose certain conditions on
Vfl, o/2 and if so that (3.3.4) may be meaningful and
of sufficient use. Assume that if> E Y'(IR.n), so that
if E Y'(JR.n) and (3.3.6)
and (3.3.7) where Ca, Ha are constants and m 1 , m 2 are real numbers. As in Section 3.2, let Y'(IR.n x
JR.+ x JR.+) consist of all C" -functions ( b, a, c) such that
11<1>11 =
sup
la'c1 ba(a;aa/(a;ac)PD~(b,a,c)l < oo,
(3.3.8)
(b,a.c)EJRn xiR+ xJR+)
for
Ia I+t'+t::.::; 1/31 +k+ p, t'+ lal +mt::.::; 0
and t + Ia I+m2::.::; 0.
Theorem 3.3.1. The composition operator Wt 0 w2: Y'(IR.n)--> Y'(IR.n
and continuous
X
JR.+
(3.3.9)
X
JR.+) is linear
54
The Wavelet Transform
Proof. Proceeding as in the proof of Theorem 3.2.1, from (3.3.3) for f,t,k,p E No and
a,fj
E
N0, we have
laf d ba(a 1aa)k(a I ac)P D~ (W l/J )(b,a,c) I
= (2n)-n lafc ba 1
~ (2n)-naf c1
k.n (iw)f3ei(b,co) {(a 1aa/ VII (aw)(alac)P Vl2(cw)4}(w) }drol
kn I
(a I aw)a [ {(a I aa)k VII (aw) (a I ac)P Vl2(cw)} wf3 4}( ro)] Idw
~ (2n)-naf c JJRr Iy,;a L (a) (a I aw)Y {(a I aa/ VII (aw) (a lac)P Vl2(cw)} Y 1
n
x l(alaw)Y- 8(a lac)PV12(cw)l
v,;~-r (a~ Y) A(fj, v) lwf3-v(alaw)a-r-v4}(w)l dw.
Now, using estimates (3.3.6) and (3.3.7), we see that the last expression is bounded by
(2n)-n r~
X
(a) y
lwlk+p-f-lrl
(r) ftr
L
o
v,;a-y
f (1+alrol)t+l81(1+clroiY+Irl-l81 C(y,o,p,k) }JRn
(aV
y)
(1 +alrol)-m' (1 +clrol)-mz
A(fj' v) Iwf3-v (a I aw )a-y-v 4' ((J)) Idw
~(2n)-n y,;a L (a) L (y)C(y,o,p,k) v,;a-y L (a-y)A(fj,v) Y 8,;y 0 V ~ (2n)-n y,;a L (a) L (y)C(y,o,p,k) v,;a-y L (a-y)A(fj,v)YA.,Jl(4}(w)) Y 8,;y 0 V (3.3.10)
IPI-Ivl-lrl +k+ p -.e -t +n+ 1, J.l =a- y- v, t+ Ia! +m2 ~ 0 and t+ Ia! +f+t ~ IPI +k+ p.
where It=
From (3.3.1 0) the assertion follows.
f+
lal +m1
~ 0,
55
Composition of Wavelet Transforms
We can define the generalized composition operator (W1 o W2 )' of any T E Y'' (JRn x JR+ x
JR+) by
Again, by duality arguments, we have
Theorem 3.3.2. The generalized composition operator
is linear and continuous. Next, we obtain a reconstruction formula for the product wavelet transform.
Assumption 3.3.3. [Admissibility condition] Let 1/IJ E L2 (JRn), 1112 E L 1(1Rn) be such that there exists a positive constant C'l'1,'lfz
< oo and, for ~ almost everywhere on JRn (3.3.11)
Theorem 3.3.4. Let lflt E L2 (1Rn), lf/2 E L1(1Rn) and c/J E L2 (1Rn) which define the compo-
sition operator CW c/J )(b,a,c) by means of(3.3.3). Then
f
}JRn
Jor= Jor=(Wf)(b,a,c)Wg(b,a,c)dbdadc/(ac)
=C'If1,'1f2 (!,g),
(3.3.12)
for all j,g E L2 (1Rn). Moreover, iff E L2 (1Rn) is continuous at x E JRn, then f(x) = - 1 - { r= r= (W f)(b,a,c)U(x- b,a,c)dbdadc/(ac), c'l'l ,'lfz }JRn lo lo
(3.3.13)
where, (3.3.14)
Proof. Let us write
and
Then, using unitary property of the Fourier transform, we have
-
-
dade ~lo~ (W f)(b,a,c)(W g)(b,a,c)db-llo a c 0 JRn 0
56
The Wavelet Transform
This proves (3.3.12). Now, to prove (3.3.13), we set g(t) = 8a(t- x), x E JR., where 8a(x) is the Gaussian function defined by
8a(x) with 8;;(ro)
=
= T 1(na)-! exp( -x2 /(4a)), a> 0,
e-aro 2 • Then, using the approximation property of the Gaussian func-
tion([12], Theorem 2.8, p. 29), we get lim
- f)(b,a,c) (W 1m looco loco o
a-o+ IR"
(loco e-zbrofi/1 . (aco)ljh(cco)e'roxe-aro . dco 2
o
)
da-de db a
This yields (3.3.13).
3.4
Weighted Sobolev Spaces
In order to study singularities at the origin we need the following function spaces. A measurable function lfl on IR.n is said to belong to the weighted space L~l(JR.n), 1 ~ p
< oo, s E JR. and a E IR.n, if
lllfiiiL~P(JR") = Clearly, Lg'P(JR.n) = U(!R.n). Let us set
(k.n I(I+ !u! )-sl ualf!(u) IP du) 2
2
l/p
< oo.
c
Composition of Wavelet Transforms
57
Then, a tempered distribution f E Y'(JR.n) is said to belong to the weighted Sobolev space H~P(JR.n), I ~ p
< oo, s E JR., a E IR.n, if
d;)
11/IIHt/{IR") = (k.. ILs,af(;)IP
l/p
< 00 •
Clearly, H~·P(JR.n) = W·P(JR.n), the well-known Bessel potential [90]. Using the well-known property of the Fourier transform for a, ann-tuple of non-negative integers, we have
11/IIH~P(JR") = (k.n 1§- 1(1 + 1;1 2 )-s/ 2§(Da f)(;)lp d;) l/p. Therefore, Hd_P C Hs,p for
a
E N
0. For p = 2, Plancherel's theorem gives
11/IIH~{IR")= (k."(I+I;I 2 )-· ·I;aj(;)l 2 1
d;y/
sEIR., aEIR.n;
so that H~' 2 (JR.n) = Hs(JR.n), the usual Sobolev space. When no confusion arises, we write
H~ and H~' 2 and L:X for L~2 • It is easy to see that 1,2,3 ... ,n. 3.5
o E H~
for all s ~ n( 1 + a;), i =
Wavelet transforms on weighted Sobolev spaces
In order to study the wavelet transform W and the composition operator W1 o W2 on the weighted Sobolev space
H~(JR.),
we need the following lemmas which give new integral
representations for the wavelet transform Wand the composition operator W1 o W2 , respectively.
Lemma 3.5.1. For
q, E Y(IR.) and ll' E Y(IR.); we have
(W q, )(b, a) = (2n)-1
L: L: eia~+ih114J(
TJ) ll'(; /TJ )TJ -I d;dTJ'
where b,a E JR.. Proof. We have
(2n)-l I:'~ I:'~ eia~+ih114J( TJ) ll'(; /TJ )TJ -I d;dTJ
= (2n) -1
1:
eib114J( TJ )dTJ
l: eia~
(2n)- 1
1:
eib114J(TJ)dTJ
1:
=
= (2n)- 1 =
1:
(Wcp)(b,a).
l!'(; /TJ )TJ -1 d;
e-ia71ull'(u)du
eib114J( 11) iji(aTJ )dTJ
(3.5.1)
The Wavelet Transform
58
Lemma 3.5.2. For lfJ E ..9"(1R.), l/fl E ..9"(JR.) and
(W, oW2)(lfJ)(b,a,c) =
l/12 E ..9"(JR.), we have
(2n)- 1 l~l~l~ exp(ia~ +ic' +ibT/) (3.5.2)
where b,a,c E R Proof. The proof is similar to that of Lemma 3.5.1. Theorem 3.5.3. Let l/J, l/f E ..9"(1R.). Then for s E JR., J3
A=
= (/3 1,/32) E JR.2 and
/31+ /32- 1/2, we have IIWl/JIIHh(JR2) ~ lll/JIIHl{IR) llli'IILg1 {IR) for s > 0 ~
lll/J IIHf {IR) II li'IILJi 1 {IR) for s ~ 0.
Proof. From (3.5.1 ), we have
(WlfJ)(b,a)
= (2n)-'l~l~ eia~+ib1Jl/I(~/T/){ji(T/)T/-ld~dT/.
Hence,
IIWl/JII~h(JR2)
= (2nf l~l~ (1 +~ 2 +T/ 2)-s lli'(~/T/)I 2 1~f3 1 T/f3z(ji(T/)1 2 d~dT//T/ 2 =
(2n) 2 l~ ITII 2132 - 2Iiii(TI)I 2 X
(l~ (1 + e + T/ 2 )-sl~l 2 f3 1 lli'(~/T/)I 2 d~) dT/.
(3.5.3)
For s > 0, we have
x l~ lul 2f3 1 ll/f(uWdu ~ (2n) 2lll/J ll~sA(IR) II ll'llioflJ {IR) · Since for s ~ 0, (1 + ~ 2 11 2 + 11 2 )-s ~ (1 + ~ 2 )- 5 (1 + 11 2 )-s, from (3.5.3) we derive
IIW lfJ ll~h(JR2) ~ (2n) 2 l~ IT/ l2(f3I +f3z)-l ( 1 + T/ 2) -sl(ii( 1'/) l2dT/ x l~ lul2f3I (1 +u2)-sll!f(u)l2du
~ (2nf lll/JII~"(IR) llll'lli_sfl1 (IR) · A
Composition of Wavelet Transforms
59
Special cases. The following special cases may be noted. For f3 = (0, 1/2),
(i)
IIWC/JIIHp(JR2)::::; (2n) IIC/JIIH'"(IR) llli'IIL2(JR) (s > 0), ::::; (2n) IIC/JIIHs(JR) llli'IILo(IR) (s::::; 0). For f3
(ii)
= (0,0), IIWC/JIIH'"(JR2)::::; (2n) IIC/JIIH~ 112 (JR) llli'IIL2(JR) (s > 0), ::::; (2n) IIC/JIIH~ 112 (JR) llli'IIL0(JR)
Theorem 3.5.4. Let
f3I
q,, lJII, 11'2 E ..9"(~).
Then for s E ~.
(s::::; 0).
f3 = (/31 ,f3z,/33)
E IR3 and 8
=
+ f3z + /33- 1, we have
Proof. The proof is similar to that of Theorem 3.5.3, where Lemma 3.5.2 is to be applied instead of Lemma 3.5.1.
3.6
Abelian theorems for wavelet transform of functions
Initial and final value theorems for wavelet transform (3.1.2) with n=l are given in this section. In what follows we assume that
lj/(m) = O(JroJil), Jml----> 0
(3.6.1)
and set (3.6.2)
+ 2,
> 0. Assume that JroJ 1-17lj/( co) E L 1 (IR), Jlj/( m)J ::::; M, M > 0, and ~ (co) E L 1 ( 8, oo), \f8 > 0. If Theorem 3.6.1. Let 2 < 1] < J1
J1
(3.6.3) then lim a2 -11W(b,a) = aH(T]).
a->=
(3.6.4)
60
The Wavelet Transform
Proof. Using (3.6.1) we can write
ia 2 -11W(b,a)- aH(1J)I
=
la
2 -17 (2n)- 1
l~ eibwfji(am)~(m)dm- a l~ fji(am)(alml) 1- 11 adml
=a \l~ [(2nt 1 eihro~(m)lmi-I+TJ- a J (almi) 1-1J fji(am)dm\
~a
sup lrol<8
1(2n)- 1 eibro~(m)lml- 1 + 17 - al f
(almi) 1-1Jifii(am)ldm
J1ro1<8
(3.6.5) The first of the above two integrals in (3.6.5) is convergent. Therefore, given e > 0, the first tenn can be made less than t::/2 by choosing D small enough. Since 1J > 2, keeping D fixed the second tenn in (3.6.5) can be made less than t::/2 for all sufficiently large a. Theorem 3.6.2. Let 2 < 1J < J1+2, J1 > 0. Imill~( co) E L 1(-X ,X), VX
Assume that lmi 1-1Jlji(m)
E
L1(IR) and
> 0. If
lim (2n)- 1 eibro~(m)lml-l+1J lrol_,oo
=a,
(3.6.6)
then Iima 2 -11W(b,a) = aH(1J).
(3.6.7)
a--->0
Proof. As in the proof of the previous theorem, for X> 0, we have ia 2 -11W(b,a)- aH(1J) I
=a
r
Jlroi<X
1(2n)-leibro~(m)lml-l+1]_ al (alml) 1- 11 1lfi(am)ldm
~ a2 -11
1:
+
1(2n)- 1 eibro~(m)lmi-I+1J- al Joo
sup IWI>X
I (2n)- 1 eibro~(m)lmi-I+1J- allml 1- 17 1lfi(am)ldm
-00
lml 1- 11 llfi(m)ldm,
61
Composition of Wavelet Transforms
e is an arbitrary small positive number. In view of the asymptotic behaviour (3.6.1 ), there exists a constant A > 0 such that Ii/f(am) I ~A (al ml)ll. where
Hence
la 2 -11W(b,a)- aH(1J)I
~ Aa2 -11+Jl /_: I(2n) -I eibwc/)( m)- al ml 1- 17 11mllldro +
sup 1(2n)- 1eibroc/)(m)lml- 1+17 IWI>X
-
alj= lml
1- 17
1i/f(m)ldm.
(3.6.8)
-=
Since both the integrals on the right-hand side of (3.6.8) are convergent, and second term is independent of a, for given e > 0, the second term can be made less than ej2 by choosing X sufficiently large. Then there will exist B > 0 such that when 1J < 2 + J1 the first term is less than ej2 for 0
3.7
Abelian theorems for wavelet transform of distributions
In this section we define wavelet transform of tempered distributions using kernel method and study some of its properties. Assume that ll'(x) E Y(IR), so that eibw i/f(am) is also an element of Y (IR) as a function of m. Let cfJ E .9'' (IR); so that
c/) (m) E .9'' (IR).
Hence, we can define the wavelet transform of c/) E Y'(IR) by
W (b, a) = (W cfJ )(b, a) =
2~ ( c/)(m), eibw i/f(am)).
(3.7.1)
Theorem 3.7.1. The function W(b,a) is differentiable andforeach i E No, j E No, we have
(djda);(djdb)iW(b,a) =
2~ (cl)(m),(im)ieibw(ajaa);i/f(am)).
(3.7.2)
Proof. First we prove the differentiability of W(b,a) with respect to the variable a> 0. For h > 0, we have
k[W(b,a+h)- W(b,a)]- 2~ (cl)(m), -Jaeibw~) =
2~ ( c/)( m),eibw { ~ ( i/f[(a + h)m]- i/f(am)) -
:a i/f(am)}).
We need to show that
eibw{~(i/f[(a+h)mJ-~)- :ai/f(am)}->Oin Now, denoting
(fro
1(t)k
r
Y(JR:) as h--->0.
i/f(am) by i/lr(am ), we have
(a~)
m
[eibw { ~ ( i/f[(a +h)m]- i/f(am)) _ :a i/f(am)} J1
62
The Wavelet Transform
Thus . W(b,a+h)- W(b,a) _ I \~( ) d ibro-=---()) I1m h - t/J m , :~e 1f1 am . 2n aa
h--->0
Similarly, we can prove the differentiability with respect to the variable b, and in general we have (3.7.2). Next we obtain asymptotic orders ofW(b,a).
Theorem 3.7.2. Let W(b,a) be the wavelet transform of~ E S"''(IR) defined by (3.7.1). Then, fork large, we have W(b,a)
= O(a- 2klblk),a ---7 0; = O(ak),a
---7 oo;
=
O(a-k(I +a 2 )k), lbl---7 0;
=
O(a- 2k(I
+a2 llblk), lbl---7
00 •
Proof. By the boundedness property of generalized functions ([97], p.lll) there exist a constant C > 0 and a non-negative integer k depending on ~such that
IW(b,a)l
~ Cs~p 1(1 + m
2/
(
}m
r{
eibroi/f(am)} I
Composition of Wavelet Transforms
63
z = aro
~ C stato (;) ( ~) lblk-s~r,s (o/(z)) as-2r
~ c'fo (~)a- 2r(a+lbl/ = C'(I +a- 2)k(a+ lbl)k,
(3.7.3)
from which the result follows. We shall need the following lemma for obtaining Abelian theorems for the distributional wavelet transform. In this section we assume that (3.7.4) for some real number fl.
Theorem 3. 7.3. Let ll' E .7 (~) and cfJ E .7' (~) be a distribution of compact support in ~. Then
is a smooth function on
~
x ~+ and satisfies
W(b,a) = O(a.U(l +a+ lbl/), a----> 0, kEN.
(3.7.5)
Proof. Let ll' be in .7(~). then VIE .7(~) and as a function of ro, eibroo/(aro) E 6"(1R), the space of all C"'- functions on Let cfJ E .7'(~), then
~.
iii E .7'(~). Moreover, assume that iii is of compact support K C R
Let A(ro) E @(lR), the space of all coo -functions of compact support, such that A(ro)
=I
in a neighborhood of K. Therefore, W(b,a) =
2~ (iii(ro),eibroo/(aro)) = 2~ (iii(ro),A(ro)eibroo/(aro)).
Then by Theorem 3.7.1, W(b,a) is infinitely differentiable with respect to the variables b and a. By boundedness property, as used in the proof of Theorem 3.7.2, we have IW(b,a)l
= 2~ l(iii(ro),A(ro)eibroo/(aro)
)I
64
The Wavelet Transform
~ Cm~x !~~ ~D~ [.:t(co)eibroijl(aco)] I
t(
~ Cm~x roEKn=O sup :S
~D~-n;qco)D~ (eibroi/f(aco)) I
r)
n
Cm~x ~~~nt (:) st (:) ID~-n;t(co)il(ibt-s (eibroD~i/f(aco)) I
~ C'max sup t r
t ( r) (n) ID~-n.:t(co)llbln-sas+.Uicoi.U n
roEK n=Os=O
~ C"m~x t
n=O
( t r) n
S
(n) lbln-sas+.U
s=O
S
~ C" m~x nt (:) (a+ lblta.U = C"max(l +a+lbiYa.u, r
(3.7.6)
where C" is a positive constant. This gives the required result. The initial value theorem for the distributional wavelet transform (3. 7.1) is given by:
Theorem 3.7.4. Assume that (ji E 9' can be decomposed into
(ji =
C/11
+ C/J-2,
where
C/11
is an
ordinary function and C/12 E g' (IR \ 0) is of order k. Let the real numbers Jl and T/ be such that 2 + k < T/ < 2 + Jl. Finally, assume that
lcol 1-17 i/1( co) E L 1 (IR) and cp 1(co) E L1 ( D, oo )\fD > 0. IJW(b,a) is the distributional wavelet transform of (ji defined by (3. 7.1 ), then lim a2-11W(b,a) = H(T/) lim _!_{ji( co) lcol-l+1J.
a~=
lroi~O
(3.7.7)
2n
Proof. By Theorems 3.7.1 and 3.7.2,
W2(b,a)
=
1 2 7r ( C/J2(co),eibroijl(aco))
is an infinitely differentiable function on IR x IR+ and W2(b,a) = O(ak), as a--->
oo.
Hence
there exists a constant A > 0 such that
Also, since the support of C/J-2 E g' (IR \ {0}) is a compact subset of IR \ {0}, lim eibrol/J2(co)co-I+1J = 0. ro~o
The conclusion of the theorem follows by an application of Theorem 3.6.1 with {ji(co) replaced by
C/11
(co).
65
Composition of Wavelet Transforms
Final value theorem for the distributional wavelet transform is the following:
Theorem 3.7.5. Let 2 < TJ < 2 + Jl. Assume that iii E Y' can be decomposed into iii= t/J1
+ t/Jz,
tPz
E
where t/J1 is an ordinary function satisfying lmiJltfJI (co) E L 1 ( -X,X)VX > 0 and
<£'' (JR?. \ 0). If W(b,a) is the distributional wavelet transform of iii. then lim a2 -11W(b,a) = H (T/ )(2n)- 1 lim eibro iii( co) Irol-l +17. Q--+0
(3.7.8)
lt)-HX)
Proof. By Theorems 3.7.1 and 3.7.3,
Wz(b,a) =
2~ ( f/Jz(m),eibroijl(am))
is a smooth function on lR?. x JR?.+ and
C being a large constant. Since 2- TJ
+ Jl > 0.
a2 - 11 1Wz(b,a)l ~ C a2 - 11 +Jl(1 +
lbll----+ 0
as a----+ 0.
The final result follows by invoking Theorem 3.6.2 with iii( co) replaced by f/Jz(m).
An Application
3.8
We apply the preceding theory to the wavelet transform defined by means of the Mexican hat function l!'(x)
= (1 -
x 2 )e- ~x2 ; so that ij/( co) = (2n) ~ m2 e- !ro2 • Thus
W(b,a)
=
(2jn) 1 1 2 [~ eibro(am) 2 e-~(aro) 2 iii(m)dm.
(3.8.1)
Here (3.8.2) and
H(TJ)
= [~ (2n)~ ro 2 e-~ro 2 lml 1 - 17 dro = 2~( 5 -17ln~r(2- TJ/2), TJ < 4.
Therefore, by a modification of proof of Theorem 3.6.1, for TJ I
e-2ro
2~
(3.8.3)
< 4 and
I
t/)(m) E L (o,oo)\fo > 0, we have lim a2 -11W(b,a) a->=
and by Theorem 3.6.2, TJ lim a2 - 11 W(b,a) a--->0
= 2~( 3 -11ln-~r(2- !l)
lim iii(m)lml-l+17 2 lrol--->0
(3.8.4)
< 4 and m2 iii(m) E L 1( -X,X) VX > 0, we have
= 2~(3- 17 ln- ~ r(2- !]_) lim eibroiii( co) lmi-I+r!. 2 lrol_,=
(3.8.5)
Note that in the present case kernel ij/( co) is exponentially decreasing, hence conditions of validity of initial and final value results are relaxed. Results corresponding to those given in Theorems 3.7.4 and 3.7.5 can be obtained using results (3.8.4) and (3.8.3) respectively.
Chapter4
The Wavelet Transform on Spaces of Type S
4.1
Introduction
In this chapter we consider the wavelet transform (3.1.2) for n = I and write
W(l/') = Wlf'(b,a) = _.!.._ { eibwij)(aw)ij(w)dw, 2n}R
(4.1.l)
where ij denotes the Fourier transform of function q,. The wavelet transform on Schwartz space Y'(IR.) was studied in Section 3.2. The spaces of typeS play an important role in the theory of linear partial differential equations as intermediate spaces between those of
c=
and of the analytic functions. The Fourier transform has been studied on the spaces of type S by Friedman [20] and Gel'fand and Shilov [21]. Nevertheless, there exist band-limited
wavelets with subexponential decay [15], and also infraexponential decay [76]; see Section 4.6. The aim of the present chapter is to study the wavelet transform (4.1.1) on these spaces. Let us recall the definitions of these spaces.
Definition 4.1.1. The space Sa(a
~
0) consists of all infinitely differentiable functions
lf'(x)( -oo < x < oo), satisfying the inequalities Yk,q( 1/') :=sup lx"q,(q) (x) I ~ CqAkJ!
(4.1.2)
xER
where the constants A and Cq depend on the function 1/J. Fork= 0, the expression kka is considered to be equal to I.
Definition 4.1.2. The space sf3 ({3 ~ 0) consists of all infinitely differentiable functions lf'(x)( -oo < x < oo), satisfying the inequalities Yk,q( 1/') := sup lx"l/'(q) (x) I ~ CkBqqqf3 (k, q = 0, I, 2, ... ), xE'llf.
where the constants B and Ck depend on the function 67
If'.
(4.1.3)
68
The Wavelet Transform
Definition 4.1.3. The spaceS~ (a ;;:: 0, f3
;;:: 0) consists of all infinitely differentiable func-
tions f/J(x)( -oo < x < oo), satisfying the inequalities
Yk.q(f/J) := supj./f/J(q)(x)l ~ CAk8qJ!aqqf3(k,q = 0, 1,2, ... ),
(4.1.4)
xEIR
where the constants A, 8 and C depend on the function f/J. The spaces of type S are closely interrelated by means of the Fourier transformation; namely, the formulae (4.1.5) hold. We shall make use of the following inequalities in our investigation: q! ' (q-k).
= k!
(q) ~ t (q) = k!
k
k
k=O
k! 2q
(4.1.6)
and (4.1.7) (See [20, p. 265]). In what follows we shall also need the following similar test function ~
spaces, defined on the upper half-plane H =
x
~+,
called spaces of typeS.
Definition 4.1.4. The spaceSa(~ x ~+), a = ( a 1, a2), a 1, a2 ;;:: 0 is defined to be the space
of all functions f/J E C"' (~ x
Yl.s,k.t(f/J) :=
~+)
such that for all/, s, k, t E No,
ia1bs (
sup (b,a)E!Rx!R+ t+s,;;;k+t
~ )k ( ~
aa
ab
)r f/J(b,a)i
~ ck,rA1AY1a'ssaz,
(4.1.8)
where the constants A I ,A2 and ck.t depend on the testing function l/J. Definition 4.1.5. The space §i3 (~ x ~+ ), ~
of all functions f/J E
Yl,s,k.r(l/J) :=
C"'(~
x
~+)
such that for all/, s, k, t E No,
ia1bs (
sup
= ({31, /32), /31,/32 :? 0 is defined to be the space
(b,a)E!Rx!R+ l+s,;;;k+t
~ )k ( a~b)t f/J(b,a)i ~ Cl,s818~J!f3 1 t 1132 ,
(4.1.9)
aa
where the constants 81,82 and Ct,s depend on the function f/J. Definition 4.1.6. The space sg(~ x
~+),a= ( a1, a2), 73 = ({31, /32), a1, a2, /31,/32;;:: 0, is
defined to be the space of all functions f/J E C"' (~ x
.,1l.s,k,t (A.) 'I'
·= ·
sup (b.a)E!Rx!R+ l+s,;;;k+t
ja1bs
(~) k (~) -, -,b aa
a
1
~+)
II. (b a) 'I' '
such that for all/, s, k, t E No,
I --=:~ CA 1I As2 8kI 8 2 t1a ssazJ!f3t t f32 ' 1
1
1
(4.1.10) where the constants A 1, A2, 81 , 82 and C depend on f/J.
69
The Wavelet Transform on Spaces of Type S
In this chapter we show that, when 1/f belongs to one of the spaces of type S, the wavelet transform is a continuous linear map of the other type of S-space into the same type of S space defined above.
4.2
Wavelet Transform on Spaces of Type S
Theorem 4.2.1. Let 1/f E Sa(IR.). Then the wavelet transform W is a continuous linear map of sa(IR.) into SCI(H) where a= (2a, a) and H denotes the upper half-plane JR. X JR.+. Proof. Let I, s, k, t E No be such that l + s
~
k + t. We have after differentiation and inte-
gration by parts,
2na1b5 =
(:ar (;by
b5 (:by
= bs
ln>ibw~(w) {a' (:a
=
it+k+l+s
m=O m
dw
jl+k+l+s+2m 1m eibw {
J'llf.
y
y
t (~) (s ~
m=O
x
(~) w-mv~-m(wt+k-l~(w))
eiwa.v{vm[ymDl[yk1/f(y)Jldy}dw
l+k+l+s+2m X
t
m=O
'llf.
=
dw
eibw
}'llf.
X
o/(aw)} dw
k_leiway1/f(y)dydw
jl+k+lbs
=
r
keibw(iw)t~(w)ikalwk keibwwt+k-l~(w) (k. eiwayD~/1/f(y)dy) r t (s)v~-m(w+k-l~(w)) (k. eiway(iay)mD~(/1/f(y))dy)
=
X
(wq,)(b,a)
sf n=O
m)
keibww-m(D~w+k-1)~-m-n(~(w)) [k. 1(;) (Dfym)v;-pD~(/1/f(y))dyl t (s) lf (s-m) eiway
dw
= jl+k+l+s+2m
m=O m n=O
X
{
J'llf.
eibww-m
n
(t +k -/)! wt+k-1-n(Ds-m-n~(w)) (t+k-l-n)! w
The Wavelet Transform
70
X
~ (m) [ {eiway m! ym-pDm-pD/ (/lfl(y))dy] dw /=a p JJR (m-p)! Y Y
= t+k+l+s+2m
t (s)sf(s-m)
m=O m n=O X
w
(m) m! p (m-p)!
k!
l-q v;+t-p-qlfl(y)dy] dw
= t+k+l+s+2m
eibw(l
{
}IR
t (s) (s-m)
sf m=O m n=O
n
(t +k-1)! (t+k-1-n)!
+ w2)Mf+k-l-n-m(Ds-m-n~(w)) w
1 x 1 +w2 X
/=o
eiwayym-p m~-p (m+f- p) f=o q (k-q)!
JJR
X
(t+k-1)!
(t + k - I - n)!
eibww+k-1-n-m(JY.'-m-n~(w)) ~
{
JJR
X [ {
X
n
m
Eo
(m) m! p
(m-p)!
k!
m+l-p (m+l- p) q (k-q)!
J;o
eiway(1 +l)ym+k-p-qDm+l-p-qlfl(y)__:!!_] dw. y 1 + y2
[ {
JJR
Now,
III= 2nla1bs ~
(JaY ( Y Jb
(wq,)(b,a)l
t (s) (s-m)
m=O m X
sf n=O
n
(t+k-1)! (t+k-1-n)!
r 1(1 +w2)wt+k-l-m-nlY.'-m-n~(w)I-11+ w
JIR
m
x
Eo
X
[
2
w
(m) m! p
(m- p)!
k!
m+l-p (m+l- p) q (k-q)!
J;o
r1(1 +l)ym+k-p-qvm+l-p-qlfi(Y)I__:!!_] dw. 1 + y2
JJR
y
Using the inequality (4.1.6) we get
III~
t (;)sf (s~m)n!2t+k-llrn 1(1 +w2)w+k-l-m-nv~-m-n~(w)l
m=O
n=O
IR
(4.2.1)
The Wavelet Transform on Spaces of Type S
71
Next, using the property n! ::( nn and (4.1.2) we have
III:::;;
t (s) sf
m=O m X X
r
(s-m)nn2t+k-l (Cs-m-nAt+k-1-m-n }IR n
(t + k -1-m- n /t+k-1-m-n)a + Cs-m-nAt+k-1-m-n+2 (t+k-1-m -n +2)(t+k-l-m-n+2)a)_l_ 1 +w2
f:
X
p=O X
n=O
(m)pp2m mEp (m+l- P)qq2k q q=O p
[.k{Dm+l-p-qA~+k-p-q
(m + k- p- q)(m+k-p-q)a
+ Dm+l-p-qA~+k-p-q+ 2 (m + k- p- q + 2)(m+k-p-q+ 2)a}
1- 2 dy]dw
x -
l+y
: ( t (s) sf (sm=O m X
n
n=O
m) nn2t+k-lc' At+k-l-m-n+2 s,m,n
(t+k-/-m-n+2)(t+k-l-m-n+2)a { _1_
JIR 1 +w2
X
A m+k- p-q+2 (m) pP2m m~- p (m + l - p) qq2k D' m,l,p,q I q 1... p q=O p=O
X
(m+k- p-q+2)(m+k-p-q+2)a ( {
~
1...
: ( n2 t (s) sf (sm=O m
X
X
n
n=O
_:!!_)dw
liR 1 +y2
m) nn2t+k-lc' At+k-l-m-n+2 s,m.n
m ( m ) pP2m m+l-p \"' ( m+ 1 - P) qq2kD' (t +k+2)(t+k+2)a \"' m.l,p,q q 1... 1... p q=O p=O A\m+k-p-q+2)(m+k+2)(m+k+2)a
: :; n2 fo (~) (s _ x'f (s ~ E
m)(s-m)C:,m2t+k-1At+k+2Ai+k+2(t + k + 2 )(t+k+2)a 2m+kmm
m)
n=O
X
p=O
(m) D:,t,p mEp (m + l- p) q q=O p
qq(m+k+2)(m+k+2)a
:::;; n2c~:'
v;:12r+2k+sA (t+k+2)A\s+k+2)(t +k + 2 )(t+k+2)as2s
x (s+l)(s+l)(s+k+2)(s+k+2)ass ~
""'
B. A kAt k2katta .!,/
2 3
'
The Wavelet Transform
72
where A2, A3 and Bs,t are positive constants.
Theorem 4.2.2. Let 1/f E sa (JR.). Then the wavelet transform W is a continuous linear map of Sa(IR.) into Srx(H), where
a= (I+ a, 2a + 3) and H =JR. x JR.+.
Proof. Let l, s, k, t E No, such that l + s
III = 2nla1bs (:a
~
t
m~
r
~
k + t. Then by inequality (4.2.I ), we can write
(:by (Wif')(b,a)l
(ms) s f (s-nm)n!2t+k-/
n~
X D!.,-m-n4)(w)II I w2
+
r I(I +w2)wt+k-/-m-n
k
E
p=O
(m)p!2m mfp (m+l- p)q!2k q q=O p
x [ {I( I +l)ym+k-p-qDm+l-p-q1/f(y)__!!!__] dw. I +y2 y }R
Using the property n!
III ~
t
m=O
~
nn and (4.1.3) we get
(ms) s f n=O
(s
-nm) nn2t+k-l
r(Cr+k-1-m-n + Cr+k-l-m-n+2)
}R
x Bs-m-n(s-m-n)(s-m-n)a_I_
f
I +w2 p=O
(m)pp2m p
x mfp (m + l- p) qq2k [ { (Dm+k-p-q + Dm+k-p-q+2) }R q q=O X
~
B(m+l-p-q) (m + l- p- q)(m+l-p-q)a _I_dy] dw I +y2 I
s ( .5 ) s-m ss-m-n(s- m- n)<s-m-n)a "\""' (s-m) nn2t+k-lc, "\""' t.k.l,m,n n J... m J... n=O m=O X
1m _I_(~ (m)pp2m m~-p(m+l-p)qq2kv' 11>
·~ X
~
I +w2
J... p=O
p
q
J... q=O
m,k,p.q
Bm+l-p-q 1
(m+l-p-q)(m+l-p-q)a { - 1 -dy)dw }R I +y2
f.
(s) s~ (s-m)8~s"amm2m ~
n2ss2z+k-tcm r,k.l,s J... m m=O X
J... n=O
n
B(m+l)(m+l)(m+l)a (m+l- p)(m+l-p)2kv'' m,k,p I
J... p=O
(m) p
73
The Wavelet Transform on Spaces of Type S
Using the property (4.1. 7) we get the estimate
III
~
1C22t+2k+scm Dm BsB(s+l}ss(2+a)ssllesels.sallaesaela5s I k,t ,l,s k.l.s 1C2 2t+2k+sc111 .D111 (BBt e(l+a) 5 )s(Btel+a)l ss(2a+3) 1t(l+a)
= h wereCk.r:=sup a2
= 2a+3.
k,l,s
k,t,l,.')
s.l
I
s+l~k+t
l+a ,A2=SBB,e l+a ,a,=l+a, 1 2 t+2k+s,i" cktlsDkls•At=Bte n2 111
' '
·' '
Theorem 4.2.3. Let 11' E
sg (JR). Then the wavelet transform W is a continuous linear map
of S$(1R) into s3(H), where a= (2a, a),~= ({3 Proof.
Let I, s, k, t E No, such that I+ s
tion by parts,
III= 2nla1b"
t
~
m~ X
X
~
n~
E
p=O
X
(m)p!2m m£p(m+l-p)q!2k q q=O p
dw. rI(I +l)Ym+k-p-qlX?+l-p-qli'(Y)I~] I +y2
}~ ~
y
nn and (4.1.4) we have
t (ms) ~o
m~
k + t. We have after differentiation and integra-
(~)sf (s~m)n!2t+k-llrn I( I +w2)wt+k-l-m-n 1+ W
[
= lR x IR+.
(:by (Wif>)(b,a)l
D'~-m-n~(w)l-1-2
Using the property n!
III~
(:ar
~
+ I,/3 +a+ 1) and H
s f (s-nm)nn2t+k-l
r (c,A\t+k-1-m-n)Brm-n
k
(t + k -[- m- n)(t+k-1-m-n)a(s- m- n)(s-m-n)/3
+ c,A\t+k-l-m-n+2) 8 ;s-m-n)(t +k -l-m _ n + 2 )(t+k-t-m-n+2)a X
X
~
(s-m-n)(s-m-n)/3)1 lw2
[.k{C2A~+k-p-q
+
E
p=O
(m)pp2m mEp(m+l-p)qq2k q q=O p
g;+l-p-q (m + k- p- q)(m+k-p-q)a
X
(m + [- p- q)(m+l-p-q)f3 +C2A~m+k-p-q+2) B~m+l-p-q)
X
(m+k-p-q+2)(m+k-p-q+2)a
X
1 -dy]dw (m+l-p-q)(m+l-p-q)/3}I +y2
1C2
t (~) (s ~
m=O
sf n=O
m) nn 2t+k-tc;A \t+k-1-m-n) B\s-m-n)
74
The Wavelet Transform
X
X
(t +k-1- m- n+ 2)(t+k-l-m-n+2)a(s -m- n)(s-m-n)/3
E p=O
(m) pp2m mEp (m + 1- P) qq2kC~A~m+k-p-q) B~+l-p-q p q=O q
x (m+k _ p _ q+2)(m+k-p-q+2)a(m+ 1- p _ q)(m+l-p-q)f3
~ 1C2 X
fo (;) (.~-m)(s-m)2t+k-tc(~ (s~m)A~+kB~
(t +k+2)(t+k+2)assf3
E
p=O X
Bi+l (m+k +2)(m+k+2)a(m + l)(m+l)/3
~ c" 2t+2k+s ss(l+/3) A'tkB~ (t X
(m)pp2m mfp (m +1- p)qq2kC~Ai+k p q=O q
+ k + 2 )(t+k+2)a (s + l)(s+t) A~s+k) Br+l)
(s + k + 2/s+k+ 2)a (s + l)(s+l)f3 5s.
Using property (4.1. 7) we get the estimate
III~ c"(2A,e2a) 1 (4A,A2e 4 a/(B2el+f3) 1(10B,A2B2e 2ar
x k2ka 1ta 1t(l+f3)ss(l+a+f3)
where
4.3
~
c" D1D~D~D~k2kattalt(f3+1)ss(f3+a)
=
c" D1 [j2D~D~0al ttazllf31 ssf3z,
c', D 1 , D2, D3 and D4 are positive constants.
D
The Wavelet Transform of Generalized Functions
In view of Theorems 4.2.1, 4.2.2 and 4.2.3 the wavelet transform W'T of generalized functions T belonging to (Sa)', (Sa)' and
(S$)' the dual spaces of sa(H), Sa(H)
and S$(H)
respectively, can be defined as follows :
(W'T,tp) := (T,Wtp),
(4.3.1)
where tp belongs to the corresponding testing function space. Using duality arguments we can prove that:
Theorem 4.3.1. The wavelet transform W' is a continuous and linear map of(Srx.(H))' into
(Sa(IR))', (Srx.(H))' into (Sa(IR))' and (S$(H))' into (sg(JR))' respectively.
The Wavelet Transform on Spaces of TypeS
4.4
75
The Wavelet Transform of Tempered Ultradistributions
In this section we shall study the wavelet transform on more general spaces than in previous sections(see[57]). Assume that the sequence {Mp}pENo (also {Np}) of positive real numbers possesses the following properties (1) M~ ~Mp-lMp+l•
(Ia) MpMq
~
pENo
MoMp+q•
p,qE No p,q E N0 ,R > O,H
(2) Mp ~ RHPmino~q~pMqMp-q,
p E No,R > O,H > 0
(2a) Mp+l ~ RHPMp,
(3) Ej=oMj/MH 1 <
>0
oo.
It can be easily verified that M P = pPa, a > 1, satisfies the above requirements. For each sequence { Mp} we define its associated function M(p) on (0, oo) by
M(p)
=
suplog(pPMo/Mp)·
(4.4.1)
p
If (Mol Mp) lfp is bounded below by a positive constant, M(p) is an increasing convex function in log p which vanishes for sufficiently small p and increases more rapidly than
log pP for any pas p tends to infinity. An example is M(p) = pa, 0 < a < I. Definition 4.4.1. The space of all those c=- functions f/J on JR. such that for every compact set K c JR., (4.4.2) for some constants C > 0 and A> 0, is denoted by G"(Mp;IR.).
Definition 4.4.2. The space of all which satisfy (4.4.2) is denoted by
c=
-functions on Q with compact support in K
c JR.
~(Mp;IR.).
Definition 4.4.3. Let {Mk}kENo and {Nq}qENo be the two sequences of positive numbers. An infinitely differentiable complex valued function f/J belongs to the space S{Mk} (JR.) if and only if k, q = 0, l, 2, ...
(4.4.3)
for some positive constants A and Cq depending on l/J; and l/J belongs to the space sl Nq} (JR.) if and only if k,q=O,I,2, ...
(4.4.4)
The Wavelet Transform
76
for some positive constants Band Ck depending on cp; and cp belongs to the space s~Z;j} (JR.) if and only if
k,q
= 0, 1,2,
0
0
0
(4.4.5)
where C,A and Bare certain positive constants depending on cp. Under certain conditions on {Mk} and {Nq}, the following relations hold: §[S{ Mk }]
= s{Md ,,fF[S{ Nq }] = s{Nq} • and §[S{Nq}] {Mk}
= S{Mk} {Nq}'
We shall need similar spaces of functions of two variables with certain restrictions on the order of differentiation. Definition 4.4.4. The space
s{M
I
p +qs
}
(H) is defined to be the space of all functions cp
E
C""(H) such that for alll,s,k,t,p,q E No,
1'/t.s.k,r(C/J) :=
laths(: )k (:b)t cp(b,a)l
sup
a
(h,a)EIR x!R+
~ Ck,rA~A~Mpt+qs•
t+s.;;k+t where the constants A1 ,Az and Ck,t depend on the testing function cp. Definition 4.4.5. The space
s{Nmk+m} (H)
is defined to be the space of all functions cp E
C""(H) such that for alll,s,k,t,m,n E No,
1'/t,s.k.r(C/J) :=
laths (
sup (b.a)E!Rx!R+
~
)k (
ua
~b)t cp(b,a)l ~ Ct,sB1B~Nmk+nt•
0
t+s.;;k+t where the constants 81 ,Bz and Ct,s depend on the testing function cp. Definition 4.4.6. The space S{{~mk+m }} {N pi +qs '
cp
E
gs+hl
} (H)
is defined to be the space of all functions
Coo(H) such that for alll,s,k,t,p,q,m,n E No, 1'/t.s,k,r(C/J) :=
1
la 1bs(: )k(:b) cp(b,a)l
sup (b,a)E!Rx!R+
l+s.;;k+t
ua
u
~ CA~A~B1 B~Mpl+qsNgs+htNmk+nt, where the constants A 1, Az, Bz, Bz and C depend on the testing function cp. Theorem 4.4.7. Let lfl E S{Mn}(IR.). Then the wavelet transform W is a continuous linear
map of s{Mn} (JR.) into s{M2k+,} (H). Proof. Let l,s,k,t E No such that 1+s
~
k+t. Then by inequality (4.2.1) we can write
laths (:ay (:by (wq,)(b,a)l
The Wavelet Transform on Spaces of Type S
~
Using the property n!
nn and (4.4.3) we get
ia 1b.\' (:ar (:by
~
t
(~)sf
m=O X
X
n=O
~
n;2
f
(s ~
m) nn2t+k-/ 1m Cs-m-n (At+k-1-m-n lR
rDm+l-p-q(A~+k-p-q
(m)pp2m mEp (m+l- p)qq2k [ P q=O q }IR
m+k-p-q+2 ) dy ] Mm+k-p-2 +AI Mm+k-p-q-2 I+ y2
t (s)
m=O m X
(w~)(b,a)l
dw M t+k-1-m-n +A t+k-/-m-n+2Mt+k-l-m-n+2 ) I + w2
p=O X
77
m~- P i..
q=O
E
s f (s-m)nn2t+k-Lc;_m-nAt+k+2Mt+k+2 (m)pp2m n=O n p=O P
(m + l - P) qq2kDm+l-p-qA!m+k+2 Mm+k+2 1
q
~ n:2ss2t+k-Lc:~'At+k+2Mt+k+2 m~O (~) :~ e~m)mm2m(m+l)m+t
x2kDm Am+k+2Mm+k+2 -f.. (m) m~- (m + [ - p) s.l
J..
I
p=O
p
J..
P
q=O
q
·
Now using inequalities (Ia) and (2) the last expression can be shown to be
~ Es,iB1B~Mt+2k> where Es.t,BtandB2 are positive constants. The following two theorems are generalizations of Theorem 4.2.2 and Theorem 4.2.3 respectively. Their proofs are similar and exploit the technique used in the proof of Theorem 4.4.7, hence are omitted.
Theorem 4.4.8. Let 11' E
s{Nm} (JR). Then the wavelet transform W is a continuous linear
map of S{Nm} (JR) into S{Nt+ 2,} (H).
78
The Wavelet Transform
Theorem 4.4.9. Let 1f1 E S~~~~ (IR). Then the wavelet transform W is a continuous linear map oifs {Mn} {Nm}
(lTJ)) . m.
S-{N2k+t}
mto {Mt+2s},{J\Ir}
(H)
.
Remark 4.4.10. Using the technique employed in Section 4.3 we can define the wavelet transforms of ultradistributions belonging to (S{Mp} )', (s{Nq} )'and (s~Z~} )'.
4.5
Wavelet Transform of Gevrey Functions of Compact Support
In this section we shall investigate the wavelet transform of Gevrey functions of compact support denoted by Gb(.Q), where .Q is an open subset of JR. First, we recall the definition of this space as given in Rodino [84]. Definition 4.5.1. The function f(x) is in cs(n.) if f(x) E C"(.Q) and for every compact subset K of .Q there exists a positive constant C such that for all p E No and x E K,
lf(Pl(x)l ~ CP+I (p!)". Definition 4.5.2. Assume s > I. We shall denote by Gb(.Q) the vector space of all o~(.Q)
f
E
with compact support in .Q.
We shall assume that the Fourier transform of the wavelet 1f1 satisfies certain general growth conditions and it belongs to the space s;;:;(JR) defined below. Definition 4.5.3. We shall denote by
s;:; .(1R) 1
the space of all functions f(~) E Coo(IR)
satisfying the following condition: for every compact subset K
c lR there exist constants
C, B > 0, and for every £ > 0 there exists a constant Ce such that IDU(~)I ~ C:Cq(q!) 1 (l
for every q E No
and~ E
lR with
I+~~
+ 1~1)-Pqexp[£1~1 1 /s],
Bqs, where s,t,p are real numbers such that s >I,
O~p~l.
Definition 4.5.4. We shall denote by Bs(H) the space of all complex valued infinitely differentiable functions
Ibm (:a
q, (b, a) that satisfy the following condition :
Y(:bY
cf>(b,a)i
~ CeAmBkct (m!)P(k!)q(l!Y(l +a)mexp[ea fs], 1
whereA,B,C,Ce are positive constants, p,q,r ~ 0, I +k+ I> m and£ is any small positive number. Theorem 4.5.5. Jffji
E
from Gb(IR) into Bs(H).
s;;'}s(IR), then the wavelet transform W is a continuous linear map
79
The Wavelet Transform on Spaces of Type S
Proof. Proceeding as in the proof of Theorem 4.2.1, we have I= bm (:ar (:by
(w~)(b,a)
r .b ~ ( a (a )1liRe'
= bm aa )k db = ;m+t X
=
k
eibw
P~ (;) { (:a) k ( :w) PV/(aw)}
q~~-p (m; p) { ( :W
im+l
r eibw [,
}JR.
-~
w~(w)lfl(aw)dw
r 1}(:W) w
m-p-q ~(w)dw
(m){tr=O (k)r (p ~! r).,ap-rwk-rV/(p+k-r)(aw)}
p~m p
xq~~-p (m; p) {(I ~!q)! w -q} ( :w) m-p-q ~(w)dw. III ~ r [, (m) {t (k) ~! ,ap-rlwk-rVI(p+k-r)(aw)l} JJR.p~m p r=O r (p r). 1
x
~
q~~-p (m;p) { (l~!q)! wt-q }1 (:w )m-p-q ~(w)ldw
r [, (m) t (k)
p! ,ap-rwk-rcecp+k-r((p+k- r)!/ JIRp~m p r=O r (p-r). X
(l+lawl)-p(p+k-r)exp[t:lawll/2s]
x w1-qCm-p-q.hexp[-hlwl 1l']dw,
~
h > 0 [84,p.58]
[, (m) r=OE(k)r (p - r) ·,ap-rcecp+k-r((p+k- r)!)t p!
p~m
P
X
wk-r(l + lawl)-p(p+k-r)exp[t:lawl 1/ 2s]Cm.p.h
X
~
[, (m-p)_l_!(l - q) ! q q~m- P
k
[, q~m-p
(m-p)q!iw 1-qexp[-hlwl 1fs]dw q
[, (m) r=OE(k)r!2PCeCp+k-r((p+k-r)!Yap-r r
p~m
P
X
lwlk-p(I
k
+ lawl)-p(p+k-r) exp[t:lawl 1/ 2s]Cm.p.h(m- p)!
x 21w1exp[ -hlwl 1fs]2m-p dw
~ Cecm+kcm,hJl+m t
p~m
(m) r=OE(:)r!((p+k-r)!Y(m-p)!ap-r P
80
The Wavelet Transform
X
k
lwll+k-p(1
+ lawl)-p(p+k-r) exp[elawl 1/ 2s- hlwll/s]dw
~ CeCm,hcm+ki+m f. X
k
p=O
lwll+k-p ( 1 + lawl)-p(p+k-r) exp[elawl 1/ 2s- hlwll/s]dw
~ CeCm,hcm+ki+m f.
p=O
X
t (~)r!((p+k)!) 1m!ap-r
(m) p r=O
t (~)
(m) p r=O
r!((p+k)!) 1m!ap-r
k_lwll+k-Pexp[~(lall/s + lwllfs) -hlwllfs]dw.
for all h > 0 (see [84], p.58). Choosing e = h, we can estimate this expression by ChCmcm+ki+mk!m!((m+k)!l
f.
p=O
where D
= fJR lwll+k-Pexp[-~lwl 1 1s]dw.
t (~)ap-re~ 011'D,
(m) p r=O
Thus
III ~ DChCm(C21+2 )m(C21+l / i (k!y+t (m!y+t (1 + a)me~ 011'. 4.6
Band Limited Wavelets
This section contains examples of wavelets belonging to certain Gevrey classes. The following generalized version of the Paley-Wiener theorem [38, p. 137] will be used in the sequel.
Theorem 4.6.1. Let 0 > 0. Suppose thatfis afunction such that suppf C [-A,A], A> 0, and f E G 5 . Then for every B > 0 there exists a constant CB such that
Definition 4.6.2. A real valued function defined on
~
is said to have
(i) exponential decay if there exist C > 0 and A > 0 such that VxE~;
(ii) subexponential decay if for 0 < e < I, there exists Ce > 0 such that VxE~;
(iii) infraexponential decay if there exist C and A > 0 such that VxE~.
81
The Wavelet Transform on Spaces of Type S
Following the technique ofDziuban'ski and Hema'ndez [15] we construct an orthonormal wavelet in L 2 (JR) with sub-exponential decay and whose Fourier transform has bounded support; such wavelets are called band limited. Spline wavelets have exponential decay. But there is no orthonormal wavelet with exponential decay belonging to C"'(!R) such that all its derivatives are bounded. Now, with X= X[O,l] set Xa
= (1/a)x(x/a).
Then for any sequence a1 ;.;,: a2;.;,: a3 · · · > 0
I:}= 1 aJ < oo, the function
such that a =
cf>k
=
Xa 1 * Xa2 · · · * Xak
belongs to ck-l (JR), has support in [O,a] and converges as k ----+ with support in [O,a], such that fiR cf>(x)dx = I and 2n
IDncp(x)l
= n- 0
By taking an
it follows that
q,
~
oo
to a function
q, E C"' (JR.),
·
aJ ... an E C0 when D > I.
For fixed a> 0, choose a "cut off' function cf>a E C0 for every D > I, where cf>a(x) =
(lja)cf>(xja). Set Ba(x)
= J~oocf>a(t)dt.
Then Ba E C 0 for every D > I. Now, let Sa(x)
= sin(Ba(x)) and Ca(x) = cos(Ba(x)),
and define the "bell" function associated with the interval [tr, 2tr] by
ba(x) = Sa(x -tr)C2a(X- 2tr),
0
Then ba E C 0 for every D > I. Extending ba evenly to [-oo, OJ it follows that the function
li'a defined by
is an orthonormal wavelet in L 2 (JR). Moreover, by Theorem 4.6.1,
lll'a(x)l = Clba(x+ 1/2)1 ~ Ceexp( -lx+ 1/21 11°) ~ Ceexp( -lxl 1-e), x E JR., where 0 <
£
< I and D = (I
-£)-I.
Hence li'a (x) is of subexponential decay.
Also,
IDnl!'a(x)l = Cl(~nei~/ 2 ba(~))"(x)l and ~nei~l 2 ba( ~) E C 0 for every D > I. This demonstrates that there exist band-limited wavelets in L 2 (JR) with subexponential decay. All their derivatives have also subexponential decay. For details of the proof refer to [15].
82
The Wavelet Transform
Modifying the above technique the following result was obtained by Pathak and Singh [76].
Theorem 4.6.3. There exist band-limited coo-wavelets in L 2 (JR) with infraexponential decay. Moreover, all their derivatives have also infraexponential decay.
Chapter 5
The Wavelet Transform on Spaces of Type W
5.1
Introduction
The spaces of W -type were studied by I.M. Gel'fand and G.E. Shilov [22]. They investigated the behaviour of Fourier transformation on the W -spaces. Also W -spaces are applied to the theory of partial differential equations. The wavelet transform on Schwartz space Y(JR) and on spaces of typeS has been studied in earlier chapters. In this chapter we recall characterizations of W -type spaces introduced in [22] and study the behaviour of continuous wavelet transform (4.1.1) over them.
5.2
The spaces WM,a,
wn,{:l and wfi:~
In this section we recall the definitions and properties of Gel'fand- Shilov spaces of Wtype. We also recall the behaviour of Fourier transformation on the spaces WM.a, W 0 ·f:l and
wfi:~ as given in [22]. Let J.L (;) (0 :::::; ; < oo) and ro( 1]) (0 :::::; 1] < oo) be continuous increasing functions such that J.L(O) = O,J.L(;)--+ oo for;--+
oo
and ro(O) = O,ro(TJ)--+
oo
for 1]--+ oo. For x? O,y? 0, we
define M(x)= foxJ.L(;)d;,
M(x)=M(-x),forx
(5.2.1)
Q(y)=Q(-y), for y
(5.2.2)
and Q(y)=
loy CO(TJ)d7],
The functions M(x) and Q(y) are continuous, increasing and convex with M(O) = 0, M(x) --+
oo
for x--+
oo
and Q(O) = 0, Q(y) --+
oo
for y-->
oo.
We have the following
fundamental convex inequalities,
(5.2.3) 83
84
The Wavelet Transform
If the functions
tt (~) and ro (1]) are mutually inverse, that is, tt (ro (1])) = 1], ro(tt (~)) =
~,
then the corresponding functions M(x) and .Q.(y) will be said to be dual in the sense of Young. In this case, we have the following Young inequality
xy for x
~
O,y
~
~
M(x) + .Q.(y),
(5.2.4)
0.
Definition 5.2.1. The space WM.a, a > 0, consists of all complex valued infinitely differentiable functions q, (x), ( -oo < x < oo) which for any o > 0 satisfy
lq,(q)(x)l ~ Cqse-M[(a-S)xJ, q = 0, 1,2, ....
(5.2.5)
where positive constants Cqs depend on function cp(x).
Definition 5.2.2. The space wn,,B, {3 > 0, consists of all entire analytic functions ef>(z), (z = x + iy E q which for any p > 0 satisfy
"" Izkn.( z)I ~c
kp en[(J3+Pl.vl ,
'I'
k-O - , 12 , , ....
(5.2.6)
where positive constants Ckp depend on function ef>(z).
Definition 5.2.3. The space w;:.-~, a > 0, {3 > 0, consists of all entire analytic functions cf>(z), (z =X+ iy E C) which for any O,p > 0 satisfy
ll/>(z)l
~ Cspe-M[(a-S)x]+il[(J3+p)yJ,
(5.2.7)
where positive constants Csp depend on function ef>(z). The following properties are satisfied by the spaces WM,a , wn,,B, w;::~ [22,pp.l2-24]. (1) The operation of differentiation is bounded in WM,a 'wn,,B'
w;::~ and hence is a con-
tinuous operation. (2) The operation of multiplication by
X
in WM,a and multiplication by z in wn,,B' w;:.-~
are bounded and hence are continuous operations.
Example 1. Let M(x) functions
=
x 1fa(x > 0), 0
q,, which satisfy
for certain Cq and a, depending on chapter 4.
q,.
This space coincides with the space Sa discussed in
85
The Wavelet Transform on Spaces of Type W
= y 1f(l-f:l)(O < f3 <
Example 2. Let .Q.(y)
1). The space wn consists of all entire analytic
functions satisfying the inequalities
lz"ct>(z)l
~ Ckexp(blyl)'f(l-f:l)_
This coincides with the space sf:l. Example3. LetM(x) =x 1fa,n.(y) =y 1f(l-f:l) (0
Then the space Wfj consists of entire functions cp (x + iy) satisfying the inequalities
lcf>(x+ iy)l This coincides with the space
~ Cexp[-alxl 1fa
+ blyi 1/(I-f:l)].
sg.
Theorem 5.2.4. If the functions M(x) and .Q.(y) are mutually dual in the sense of Young, then the Fourier operator~: WM,a--+ wn,~ .~: wn,{:l--+ wM,k iscontinuousandWM.a = WQ,~, Wil,f:l = WM 1 • 'lf
Theorem 5.2.5. Let .Q.I (y) and M1 (x) be the functions which are dual in the sense of Young to the functions M(x) and .Q.(y) respectively, then the Fourier operator~: wf:! --+ Wn 1 '
'f
MI·7J
. contmuous . ts an d w:~n M'af:l = wn"~1 • .
Ml,lf
In what follows we shall also need the following similar test function spaces, called spaces of type Wwhich will be used in the study of the continuous wavelet transform. Definition 5.2.6. The space WM,a is defined to be the set of all complex valued infinitely differentiable functions cf>a( CJ)
= cp( CJ,a) E Coo (JR. X JR.+)
which for any 0
> 0 satisfy
l(cfa)k(ja) 1cp(CJ,a)l ~Ck 18 e-M[I~a(a-o)]; k,l=0,1,2, ... ,
(5.2.8)
where positive constants Ckto depend on function cp. Definition 5.2.7. The space W*n,n,{:l,af:l is defined to be the set of all functions cf>a(s) =
cp(s,a), (s,a) E C x JR.+, entire analytic with respect to s
= CJ+ ir which for any p,p' > 0,
satisfy
l( _§_)k"'(s I +a 'I' '
a)l ~Ckpp' en[-r(f:l+p)]+n[-r(af:l+p'J].• k=O , I , 2 , · · ·, '-"
(5.2.9)
where positive constants Ckpp' depend on function cp. Definition 5.2.8. The space wn,n,{:l,f:l is defined to be the set of all functions cp(s,t), (s, t) E C x C, entire analytic with respect to s = CJ + ir, t
= a+ iy which for any PI, pz > 0 satisfy
I(;fs)k(;~dct>(s,t) I ~ CktPIP2en[-r(f:l+Ptl]+il[r(f:l+P2ll; where positive constants Cktp 1p2 depend on function cp.
k, l
= 0, 1,2, ... '
(5.2.10)
86
The Wavelet Transform
5.3 The wavelet transformation on W -spaces In this section we study the wavelet transform (4.1.1) on the spaces WM,a·
wn,f3, and w:f.'~.
In what follows, for convenience, we impose conditions on (j), but one can impose conditions on
f/J, and apply appropriate Theorem 5.2.4 or 5.2.5, and get the desired result.
Theorem 5.3.1. Let M(x) and Q(y) be the functions which are dual in the Young sense. Suppose fi/ E WM,a and (j) E WM,a. then the wavelet transform <Pa(s) I
I
w• 0 ' 0 'a"aa, s =
(j
+ ir,
= cP(s,a)
E
for arbitrary but fixed a > 0, that is, wavelet transform -nnl
__L
('W'I'f/J) (s, a) is a continuous linear map from W 0 •I I a into w· , ,a ,aa
0
Proof. The wavelet transform of a function f/J with respect to the wavelet If/ defined by (4.1.1) is
('W'I'f/J)(cr,a) =cP(cr,a) = -21 7r
Joo eiax{j;(x)fi/(ax)dx.
(5.3.1)
-00
Since (j), filE WM,a,therefore, the wavelet transform can be extended to the complex values of s = cr + ir according to the definition
cP(cr+ir,a) =
_!_joo ei(a+i'r)x{j;(x)fi/(ax)dx 27r _21 !oo eisx{j)(x)fi/(ax)dx. 7r -00
=
-00
Integrating by parts k times, we get
[(is)kcP(s,a)[
=I (~;k l~ eisxD~({j;(x)fi/(ax))dxl = -1 t (k) 100 )k
I(
27r
=
~(-1)k 27r
~
eisx oik-1) {j;(x)DY) fi/(ax)dxl
1=0
l
-00
1=0
l
-00
t (k);·oo t (k) joo
eisx(j)(k-l)(x)alfjl(t)(ax)dxl
_!_(1 +a)k 27r
Now, using Definition 5.2.1, we get
1=0
l
-00
[i1x(j)(k-l)(x)fi!Ul(ax)[dx.
(5.3.2)
87
The Wavelet Transform on Spaces of Type W
C , (k-1)8 18 l f::o (k)c
< ~f. "' 27r
1=
< C' "' ~~ .
1=
e2]n]-M[x(a-8)]-M[ax(a-8')ldx
-=
e2lni-M[x(a-8)]-M[ax(a-8')Jdx
-=
'
q
where 88 , = 2~ E7=o (7)c(k-l)8Ct8'· Now, applying (5.2.3) and Young's inequality (5.2.4), the exponent in the above integral can be transformed as follows:
-M[x(a-8)]+1rxl ~ -M[x(a-8)]+M[x(a-28)]+D. [a_'1"28 ]
~ -M[8x]+n[a_'[28 ], and
~ -M[ax(a- 8')] +M[ax(a- 28')] +D. [ a(a~ 28 ,)]
-M[ax(a- 8')] + lrxl
~
-M[8'ax] +D.
[a(a~ 28 ,)].
Therefore, we get the estimate
III ~ c~88'en[a_'[28]+n[a(a~28')] < C" "'
kpp'e
In the above we set a~ 28
=
I:
n[T(!k+P)]+n['r(a~+P')]
!k + p and
a(a~ 28 ,)
e-M[8x]-M[8'ax]dx
.
= a~+ p' where the quantities p,p' are I
I
arbitrary small together with 8, 8'. Thus cP(s,a) E W*.n,n.a'aa.
Theorem 5.3.2. Let the functions M(x),D.(y) be same as in Theorem 5.3.1. Suppose ljJ E wn.,B and iii E wn,,B. Then wavelet transform ('Wifllf> )( cr,a) is a continuous linear map from wM.I/,6 into WM,I/,6• that is, cP(cr,a) E WM.lj,B·
Proof. Since iii E wn,,B, following the technique of [22, p. 22], the expression for wavelet transform defined by (5.3.1) can be written as cP(cr,a) =
~1= eiu(x+iy)iii(x+iy)lj/(a(x+iy))dx 27r
= ~1= 27r
eiuziii(z)lj/(az)dx,
z=x+iy.
-=
For non-negative integers k, /,after differentiation of (5.3.4 ), we get
C/u/(J) 1cP(cr,a)
(5.3.3)
-=
=
2~ I: eiuz(idiii(z)(jjljl(az)dx
(5.3.4)
88
The Wavelet Transform
= !___ joo eiazl+Lif(z)iji(L)(az)dx. 27r
Now, using inequality lzl 1 ~
-00
izi 1x+ ~2 +1izi 1 and conditions for belonging lj, ijf in wn,/3, we obtain
III= lc/a/(fY~(a,a)l ~ 2~
L:
leiazl ('zik+:2:;izik+L) 1/F(z) II ij!U) (az) Idx
joo e-ay[Ck+L+2 p +Ck+L p]en[y(f3+Pllco 'Len[ay(f3+p')]~ x +1 :>:: C' e-ay+.O[y(J3+p)]+.O[ay(f3+p')] foo ~ ""' klpp' x2 + I :>:: _.!._ ""' 27r
-oo
'
P
•
2
-oo
~ ~lpp'e-ay+.O[y(f3+p+a(f3+p'))] ~
c:lpp'e-ay+.O[y(l+a)(f3+p)]
for all p, p'
>0
for p = p'.
(5.3.5)
Until now y has been an arbitrary number. Using the technique [22, p. 22], let us now choose the sign of yin such a manner that the equality ay = IallYl be satisfied, and absolute value of y so that the Young inequality (5.2.4) becomes equality laiiYI
= .Q[Iyl(l +a)(J3 +P)] +M [(t+a1J(1+Pl] ·
Then exponent in the expression (5.3.5) becomes
+ a)(J3 + p )] =
-cry+ .Q[y(I Replacing 1I (J3
+ p) by ( 1I J3) -
-M [ (l+a1)(1+p) J .
o where o is arbitrary small, we obtain the estimate for
the expression (5.3.5):
I( daa )k(l__)L~(a au ' a)l :>::C" -.. : klo e-M[(i~~)(~-o)] . Hence, wavelet transform ~(a,a) E WM,I//3·
Theorem 5.3.3. Let the functions M(x) and .Q(y) be same as in Theorem 5.3.1. Suppose l
ijf E wn·lJ and If E WM.f3·Then wavelet transform (ll''l'cp)(s,t) extends to an entire function . =a+ q. an d.lt zs. a contznuous . l.znear map firom wnJ . w~n,n.i.~ oif s =a+ rr,t ~-' mto ~-' ~-' Proof. The wavelet transform of a function cp with respect to the wavelet 1JI defined by (5.3.1) is
~(a,a)
= -21
7r
joo eiax/F(x)ij/(ax)dx -00
89
The Wavelet Transform on Spaces of Type W
Since both {j and 1f1 E WM,/3• from [22, p. 20] we have
(cr+i-r,a+iy)
=
__!__1oo ei(a+ir)x{j(x)1oo ei~(a+iy)xlJI(/;)d/;dx; 27r
-oo
-oo
so that
r
__!__ 100 eisx{p(x) ei~txlfl( I;) dl; dx. 27r -oo .J-oo Fork, l E No, after differentiation of (5.3.6), we get
(s,t)
=
(ft )k ( -Js )l(s, t) = 2~ =
i:
ei'x(i4 ;p(x)
i: ei~tx(ixl;
(5.3.6)
)k lJI( I;) dl; dx
__!__ 1oo eisx(ix)l+k{p (x) 1oo ei~rx l;k lfl( I;) d/; dx. 27r -oo -oo
Now using the definitions for (j and lJI, we obtain III= ,;;
,;;
j(frl(-#s) 1(s,t)J
i:
i:
2~ leisxll (ix)l+k{p (x) I lei~txlll;k lJI( I;) Id/; dx 2~ j_: e-•xcokt8e-M[x(f3- 8)] j_: e-rx~Cok8'e-M[~(f3- 8')J dl; dx
~ 17r COk/8 COk8 1oo elrx]-M[x(f3-8)]1oo elrx~]-M[~(/3-8')1 d'" dx -oo -oo o., • 2 1
--=:
(53 7) • •
Applying Young's inequality (5.2.4) in the of exponent (5.3.7),we get lnl- M[x(J3- 8)] ,;; Q[/3~~ 8 ] +M[Ixi(J3- 28)]-M[x(/3- 8)]
,;; n[ 13 _:28 ]-M[ox] and
lrxl; 1- M[l; (J3- o')] ,;; Q[/3~~(5,] + M[lxl; I(J3- 28')]- M[l; (J3- o')]
,;; Q[ /3__1; 8,]- M[/; ((J3- o')- x(/3 - 28') )]. Setting ~ =
b
b
+ p and ~ = + p' where p and p' are arbitrary small together with 0 and 0 1 , inequality (5.3.7) becomes III ,;; __!__Cokt8Cok8'en[{3_:zo] en[/3-~ 8 ']1 00 e-M[ 8xl1oo e-M[~((/3- 8 ')-x(/3- 28 '))] dl; dx 27r -oo -oo
~ C'ktpp'e n[r(b+P )] en[r(b+P')]I 1, "" where I1 denotes the double integral which is estimated as follows:
Il =
i:
e-M[8x]
i:
e-M[~((/3-8')-x(/3-28'))] d/; dx;
(5.3.8)
90
The Wavelet Transform
-1oo -M[ox] 1oo e-M[u] d d -oo e -oo (/3 - 8') - (/3 - 28')x u x -1oo -M[u] 1oo e-M[ox] -oo e du -oo (/3 - 8') - (/3 - 28')x dx =
-A 100 e-M[ox] f3 _ 28 , -oo x- (J dx; wherelJ
f3
=~ 8 ,H(e-M[oxl)(lJ);
o'
=
~~20 ,,A
100 =
-oo e-M[uldu
where H denotes the Hilbert transform.
Since e-M[ox] E LP(IR) for p > 1, from [96, p.275], we have
Since the same LP estimate is valid for first order derivatives, from the Sobolev embedding theorem we have that Hilbert transform H (e-M[ox]) ( (J) belongs to Loo. Hence
IIII
~B(f3,8,8',p).
-
-nn.!..!.
Consequently, it follows from (5.3.8) that (s,t) E W ' 'f3 'f3. Theorem 5.3.4. Let .Ql (y) and M1 (x) be the functions which are dual in the sense of Young
to the functions M(x) and .Q(y), respectively. Suppose 11' E WM 1 ,f3 and -
fixed a E IR+ the wavelet transform (s,a), as a function of s = where A = f3
if E w1J:~- Then for a I
Q
0"
+ i't", belongs toWM1'f, I•I
+ ~-
Proof. By [22, p.24],the expression for the wavelet transform of the function
if E w1J:~ can
be written as
( 0" + i't", a) =
__!___
2n
ei~a(x+iy) 11'( ~) d~ dx;
1oo ei(
-00
(5.3.9) where s = a+ i-r and z = x + iy. Carrying out the transformations used in Theorem 5.3.1 and Theorem 5.3.2, an estimate of the absolute value of (5.3.9) yields
l(s, a) I ~ 2~ j_~ leiszlllP(z) I/_~ lei~azllll'( ~)I d~ dx ,:::: __!___ {""
"' 2n
.I
-00
e-rx-CJyC e-M[(a-o)x]+il[(f3+p)y]100 e-ay~C e-Md(f3-o')~] d~ dx 5' op
°
-00
,:::: _1 C C e-CJy+il[(f3+p)y] 1oo elrx[-M[(a-o)x]1oo elay~[-MI[(f3-o')~] d.t= dx "' 2n op 05' ~ __!__CspC05 ,e-CJy+il[(f3+p)yJe01 la_:'zs le0 [fl~~c5 1 l 1 00 e-M[ox]1oo e-Mdo'~l d~ dx 2n -00
-00
-00
"'
-00
91
The Wavelet Transform on Spaces of Type W
-ay+D.[(f3+p+ fl-;c5t)yj D. I[ a~2c5 J ~ C1 e 0/Y Pe " ~
"
C'
pp'p"e
-ay+D.[(f3+p+~+p")y] ni[-r(~+p')].
a
( set mg f3-2l>' =
- C'
pp'p"'e
-
=
,
e
t.
a
II
73 + p '
I
a-28 =
a + p') I
-Ml [!l+{+P"'] eQI[-r(~+p')]
C~p'p"'e -M [._+~m leni[-r(~+p')J; 1
where A= {3
1 h e-MI[a(±-8")]eD.I[-r(~+p')]. = C' , w ere X+p"' 8, p' ~
=
1
s:"
X- u .
I
Ml,X'f, A.= f3 + %.
Thus
5.4
Q
+~
1
f'
Examples
Let M(x)
= xP lp,Q(y) = fl lq (x > O,y > 0), and write wn.b Wp,a = WM,a, W q,b = wn,b and Wq.b p.a = M.a ·
Note that M and Q are dual to each other if and only if p and q satisfy I I p + I I q Let p
= I.
> I , r > I and let q and s be such that 1Ip + 1Iq = 1Ir+ 1Is = 1. Then the wavelet
transform 1f/"' satisfies the following: (1) For
iii E Wp,a and iii E Wp,a (or 1/f E Wq.Ija and l/J E Wq,lfa), 1f/'I'[Wq,lfa] = W*q,q,(lja),(ljaa), a> 0.
(2) For
iii E Wq,f3 and iii E Wq,f3 (or 1/f E Wp,l//3 and l/J E Wp.Ijf3), 1f/"'[wp,1Jf3l
(3) For
= wp.ljf3·
iii E Wq,l//3 and iii E Wq.f3 (or 1/f E Wp,f3 and l/J E Wq,l/f3), 1f/'I'[Wq,lff3]
(4) For 1/f E Wr./3 and
iii E WJ,'g
= Wq,q.(l//3),(1//3).
(or 1/f E Wr,/3 and l/J E W:]Jf3a),
11'; [Ws,lja] = Ws,lja A.= 'If
r,l//3
r,ljA. '
f.!
1-'
+aiR. 1-'
Chapter6
The Wavelet Transform on a Generalized Sobolev Space
6.1
Introduction
Let II' E L2 (1R.) be an analyzing wavelet which satisfies (1.6.3). We define the translation operator Tb by
Tbl/f(x)=l/f(x-b), hEIR.,
(6.1.1)
= lal- 1/ 2 11' (~),
(6.1.2)
the dilation operator Da by
Da 1/f(x)
a E IR.o
=JR.\ {0}
and a unitary transformation U(b,a): L2 (1R.,dt)----> L2 (1R.,dt) by
U(b,a)l/f(x)
= (TbDal/f)(x) = lal- 1/ 2 11' ( x:b),
(b,a) E JR. x IR.o.
(6.1.3)
The actions of the Fourier transform on the operators Tb and Da are given by (6.1.4) (6.1.5) Then the integral
(Lv,J)(b,a) =
1
frW'I'f(b,a) =
(6.1.6)
yC'I'
yC"'
1 = fr
1
fr(f,U(b,a)l/f)
1 1m JT::T
yC'I' y
ial
f(t)l!'((t- b)/a)dt
(6.1.7)
IR
defines an element of L 2 (JR. x IR.o, d~~a) . Moreover, L"': L2 (1R.,dt)----> L2
(JR. x JR.0 , d~qa) 93
is an isometry [45, Theorem 2.2, p. 30].
94
The Wavelet Transform
The Fourier transform of L"' with respect to its translation argument is given by
(L"'f)(·,a/'(~) =
;,___
yC"'
alal-l/Zo/(a~)f(~).
(6.1.8)
The operator L"' is also called wavelet transform with respect to analyzing wavelet II'· This is a normalized form of the operator W"', considered earlier. In this chapter, we extend the wavelet transform L"'(f), which we defined on L2 (JR,dt), to generalized Sobolev space B~k and interpret its image set as the space U((lRo, !f!i),B~k) abbreviated by Wp~k· This work generalizes some of the results contained in [81].
6.2 The generalized Sobolev space B~k In this section we recall definitions and properties of certain function and distribution spaces introduced by Bjorck [5]. Let Jlt be the set of continuous and real valued functions w on JRn satisfying the following conditions:
(I) (6.2.1) (2)
r (I+~~ w(~) l)n-t-1 d~ < oo,
}JW.n
(6.2.2)
(3) (6.2.3) for some real number a and positive real number b. We denote by Jltc the set of all w E Jlt satisfying condition w( ~) = Q( I~ I) with a concave function Q on [0, oo). We suppose
w E Jltc from now on. We denote by Sro the set of all functions lfJ E L 1(JRn) with the property that lfJ and and for each multi index
($ E C"'
a and each non-negative number A we have Pa.A.(l/J) =sup eA.ro(x)IDalfJ(x)l < oo;
(6.2.4)
xEIR"
1ra.A.(lfJ) = sup eA.ro(~)IDa($(~)1 < oo.
(6.2.5)
~EJRn
The topology of Sro is defined by the semi-norms Pa,A. and 7ra,A.· The dual of Sro is denoted by S~, the elements of which are called ultra-distributions. We may refer to [5] for its
The Wavelet Transform on a Generalized Sobolev Space
various properties. We note that for ro(;) = log(1
95
+ 1;1), Sro
reduces to!/, the Schwartz
space. We also recall the definition of test function space D 00 • The space Dro is the set of alief> in L 1(!Rn) such that cp has compact support and lll/>IIA.
llct>IIA. =
< oo for all A> 0 and
f l~(;)leA.ro(~ld;.
}JRn
(6.2.6)
Now, let ro E .4c. Then Kro is defined to be the set of positive functions k in IRn with the following property. There exists A > 0 such that (6.2.7)
< oo. Then generalized Sobolev space B~.k(IRn) is defined to be the space of all ultra-distributions f E S~ such that
Let roE .4c. k E Kro and 1 ~ p
(6.2.8) and 11/lloo.k = esssupk(;)IJ(;)I.
(6.2.9)
Note that the space B~,k(IRn) is a generalization of the Hormander space Bp.k(IRn) [38] and reduces to the space Bp.k (IRn) for ro = log( 1 + I; I). For k(;) = esro( ~) and I ~ p ~ oo;
the generalized Sobolev space studied by Pathak [63]. An inner product in B~k is given by (6.2.10) The space B2.k(1Rn) reduces to Sobolev space Hs(IRn) fork(;)= (1
+ 1; I2 )S1 2 • The Sobolev
space Hs(IRn) consists off E Y'(!Rn) such that (6.2.11) where s E JR.
6.2.1
Examples
1. Dirac
o belongs to W(IRn) for s < -n/2.
2. Let g(x) = Eaaao(a)(x- a), aa E C. Suppose that, for each e > 0, there exists Ce. such that laal ~ Ceelal(a!)-lfd, 0
< d
96
The Wavelet Transform
Then from [84, p.118], we have
18(~)1 = l[aa~aeia~l ~ Deexp(el~ld), a
for every
e > 0 and suitable constant De. Therefore, g(x) E B~k(IRn),
wherero(~)=l~ld,
But g(x)
O
tf. H~(IRn) for every s E IR, because g not a tempered distribution. > s, where s E IR, is fixed. Consider the convolution of
3. Let mE No, such that m+ 1/2
m + 1 characteristic functions:
g := X[o.t] Then, from [2, p.487],
* ··· * X[o.I]·
g(~) = e-i(m+l)~/ 2 cinJ% 2) r+l, and g E H'.
4. Let g be the same as in Example 3. For every j = 0, 1, 2, ... ,define the wavelet q,UJ by
(i)Ul (~ ) =
g( ~ ) Jrog,J(~)
with 00
ro8 .j(~) = [. (1 +2 2 1(~ +2kn) 2 Yii(~ +2kn)l 2 . k=-oo
Since there are constants
c1,c1 > 0 such that c1 < Wg,J < c1, it follows
that q,Ul(~) E
Hs(IR), cf.[2, p. 488].
6.3 The wavelet transform on generalized Sobolev space B:,k In this section we define the space llf(b,a)llwf = 1~ p
w;: of all measurable functions f
d ) (liRor (llf(b,a)ll:,k) laiP;+l
ljp
on IR x !Ro such that
< oo,
(6.3.1)
< oo, a E IR\ {0} = !Ro.
Theorem 6.3.1. Assume that analyzing wavelet satisfies the following admissibility condition:
c
= lfl.p
r lo/(~W dj: < I
lJR
I~
~
00
(6.3.2)
.
Let (Liflf)(b,a) be the wavelet transform of the function f E
B~.k
with respect to the ana-
lyzing wavelet If! satisfying (6.3.2). Then (6.3.3)
97
The Wavelet Transform on a Generalized Sobolev Space
where Ap = (C'I')-PI 2 clfl,p is a positive constant depending upon p and 'I'· Proof. Assume at first that f E S00 (1R.). Then from (6.3.1), we have
ko II(L'I'f)(b,a)ll~,k lal(:~)+l ko (k lk(~)IPI(L'I'f)(b,a)i\(~)IPd~ lal(:~)+l) 2 = ko (k lk(~)IP (~'I') pf laiP! 2 Io/(a~)IPIJ(~)IPd~) lai(:/~)+I
II(L'I'f)(b,a)ll~: = =
1 = (-
c"'
= (
)p12 liR.or lo/(u)IP dull/liP lui
p.k
~"') p/ c"',pll!ll~,k = Apll!ll~,k· 2
Since by [5, Theorem 2.2.3, p. 384] Sw is dense in
B~k·
the above result can be extended
to all f E B~,k·
6.3.1
Adjoint of L"'
Let Wk and Bk denote the spaces Wl and sr,k respectively which are the special cases of the spaces W{ and B~k for p = 2. If k( ~)
=I, then Wk is denoted by W1. Assume that 'I' is
real valued. From (6.3.3) it follows that the operator L"' with integrable and admissible 'I' and p
= 2,
is a linear isometry from Bk to Wk. For linear isometry between Hilbert spaces we know that U * U = id and UU* is the orthogonal projection onto range(U) (which is closed), where U* denotes adjoint of U. Therefore, the transform L"' is inverted onto its range, by its adjoint L; and that an element g E Wk lies in range(L"') if and only if L"'L; =g. Suppose that
L; is the adjoint of transform L"'. Now, we derive an explicit expression for L~: Wk----> Bk.
We assume that f E S00 , and set (6.3.4)
and (6.3.5) Setting up a scalar product on Wk:
(Q, r)k =
1m
IRo
da (Q(·,a), r(·,a)}k--z a
(6.3.6)
98
The Wavelet Transform
we get
(Ly,f,g)k= f f 1\(a,~)d~d~· 1JR0 JJR a
(6.3.7)
Applying two times Schwarz's inequality, we get
r JJRrIA(a,~)ld~d~a ~ 1JRr0 IILI{/f(·,a)llkil(g(·,a))ll/~a liRo ~ IIL"'/IIwk llgllwk'
(6.3.8)
which allows the change in the order of integration in (6.3.7). We have from (6.3.5) and (6.3.7),
(Liflf,g)k = { (k(~)) 2 j(~)d~ { ~C (D-al/f)'\~)(g(·,a))A(~)d~· lJR 1JR0 y '"'If/ a
(6.3.9)
We denote the inner integral by Ag( ~) and estimate lAg I to conclude that Ag E L2 (1~, dt). Now,
-
Ag(~)
= 1m
I A A da IC(D-al/f) (~)(g(·,a)) (~)2.
~Y'"'Ifl
(6.3.10)
a
Using Schwarz's inequality in (6.3.10), we have
IAg(~)l 2 ~ 1JRf l(g(·,a))t\(~Wd~· a
(6.3.11)
0
Therefore,
f IAg(~)l 2 d~ ~ f f l(g(·,a))t\(~)l 2 d~ d~
JJR
lJR 1JR0
a
=
llg(·,a)ll~1 •
Consequently, there exists an AgE L2 (l~,dt) with (6.3.12) Using (6.3.12) in (6.3.9), we have
(Liflf,g)k = =
k(k(~)) j(~)(Ag)A(~)d~ 2
(f,Ag)k.
(6.3.13)
Thus Llfl and A are adjoints of each other on Bk. Next we determine an explicit form of Ag(x) using the fact that the integral
fJR.o fJR(D-ali')A(~)(g(·,a))t\(~)d~ '1ft exists. We have
99
The Wavelet Transform on a Generalized Sobolev Space
Therefore
Ag(x)
6.3.2
=
B~,k -approximation
;1r 1m y Cll'
1m
IR0 IR y
1 lfl (b-x) dbda JT::T g(b,a)2 .
Ia I
a
(6.3.14)
a
of wavelet transforms
We will now determine the B:,k -distance of two wavelet transforms with different basic wavelets and different argument functions in order to study the dependence of the transform on its wavelets and its argument. 00 k, Theorem 6.3.2. For admissible and integrable lf/1, lf/2 and f, g E B p,
IILII'J(b,a) -LII'2 g(b,a)llp,k
"lal 112
(II }c;-
;.;t
IIIII,,.+
I
;.;t
11/-gll,>) ·
Proof. We have
IILytJ(b,a) -Lyt2 g(b,a)llp,k ::s;IILII'1f(b,a) -LII'zf(b,a)llp,k
+ IILyt2 /(b,a) -Lyt2 g(b,a)llp.k·
(6.3.15)
Now,
IILII'J(b,a) -LII'zf(b,a)llp,k
=
(k
I(LytJ(b,a)
-Lytzf(b,a))"'(~)IPik(~)IPd~) l/p
~ (J.I )c..1al 112 o/i(a~)j( ~)- ;;;-lal 112 ¥ii(a~ lfC ~)I' lk( ~)I'd~)
~ (J.wt'lk( ~)I'll@ I'd~ I( Jb- Jb) (a~f) 11, Now, using inequality
I/P
(6.3.161
100
The Wavelet Transform
we have
so that (6.3.17)
Using (6.3.17) in (6.3.16), we have
IIL.,f(b,a) -L.,f(b,a)ll,> <;;
11;..- ~t IIJII,,•.
lal 112
Similarly, we can write
II L.,f(b, a) -
L.,g(b, a) II p,k <;;
II
t
}c;,
Ia 1112 11! - g llp,k ·
Invoking (6.3.18), inequality (6.3.15) yields
IIL,,J(b, a) - L.,g(b, a) II,> <;; lal 112
1 ;.. - ~ '11 }c;,
+ lal 11
t
t
(6.3.18)
(6.3.19)
llfll,,k
IIJ-cll,,,.
This completes the proof of Theorem 6.3.2. The following result is a consequence of Theorem 6.3.2.
Corollary 6.3.3. Let lfl and f be as in the preceding theorem, then
6.4
Asymptotic behaviour for small dilation parameters
Let us recall the relation (6.1.7): L 11,J(b,a)
=
1 11"
1
10
v elf' v Jal
1m f(t)lf!((t-b)/a)dt.
(6.4.1)
lR
In what follows we assume that lfl is real valued and a
> 0.
Let us write lf!a(x) =
~ 1{1 G),
(6.4.2)
The Wavelet Transform on a Generalized Sobolev Space
101
and use the notation:
1\y,f(b,a)
r (b-t) = (lfla *f)(b) =;• J!Rf(t)lfl -a- dt.
(6.4.3)
Notice that in previous chapters 1\IJf was also denoted by Wx, where x(x) = lfl( -x). From (6.4.1) and (6.4.3), we have
(lfla*f)(b) = (1\IJff)(b,a) =
yrc; -;-LIJff(b,-a).
(6.4.4)
Let P(D) = E'J=oCjDj be a differential operator, where Cj E C and Pis polynomial. Define
(P(c;)f =
m
E IDjP(c;)l
(6.4.5)
2.
j=O Then, by Hormander [38, p. 10], we have (i) f E B~k ==? P(D)f E B:k/P
(ii) IIP(D)fllp,k/P:::;;
11/llp,k/P"
Lemma 6.4.1. Let f E B~,k and lfl E L 1 (~)with fiR lfl(t)dt
= 1. Then
(i) 1\IJff(·,a)---. f(-) in B~,k as a---. 0; (ii)
(6.4.6) Proof. In view of (6.4.3) and (6.1.8), we have (i)
lllfla * f- fll~,k = ~ l(lfla * f- f)'\c;)IPik(c;)IPdc; =
~ l(lfla *f) 11 (c;) -J(c;)IPik(c;)IPdc;
=
(c;) ~ I( C: )
=
~ lif/(ac;)J(c;) -J(c;)IPik(c;)IPdc;
=~I
1/2
((LIJff)(b,a)) 11 (c;) -J(c;) 1/2 (
diJf
) 1/2
lp
lk(c;)IPdc;
lp
lal 1/ 2 if/(ac;)J(c;) -J(c;) lk(c;)IPdc;
= ~ lk(c;)IPIJ(c;)IPil- i/f(ac;)IPdc;
= ~ lk(c;)j(c;)IPII- i{l(ac;)IPdc;
The Wavelet Transfonn
102
=
kII(a,~)IPd~,
where II(a,~)l = lk(~)j(~)(1- if/(a~))l. Then we have lima ..... o II( a, ~)I= 0 a.e. Let us now set M = sup~EIR 11
-
i/f(a~) I, which is independent of a.
Then
Now, applying the dominated convergence theorem, we have
(lfla*f)=l\lflf(·,a)--"f(·) in B~,k as a--'>0. (ii) Let {fn}nEN E S00 (1R.) converge to fin B~.k· Now, operating the differential operator P(D) on both sides of equation (6.4.3), we have the following equality.
Since S00 (1R.) in dense in
B~k(IR.)
and the operators 1\IJI and P(D) are continuous, hence
limits of the above three are equal in B~,k· Theorem 6.4.2. Let f E B~,k and lfl E B~.k nL 1 (JR.) with fiR lfl(t)dt = 1 and P(D)lfl E L 1 (JR.).
Then lim
a-->0
where cd
=
1 a 112 LP(a-'D)lflf(b,a)- v~P(D)/11 _ 0, cd =
1
p,k/ p
CP(D)IJI•
Proof. By virtue of Lemma 6.4.1 (ii), we have
LP(a-'D)IJif(b,a)
Now, we estimate
=
~ 1\P(a-lD)IJif(b,a)
=
~ 1\IJIP(D)f(b,a)
=
~P(D) 1\lfl f(b,a).
The Wavelet Transform on a Generalized Sobolev Space
103
=II a1~2 ~P(D)(Aytf)(b,a)- ~P(D)ftk/P I
=Jed IIP(D)[Aytf(b,a)- f]llp.k/P I
~Jed II Aytf(b,a)- f(b)JIIp.k.
by (6.4.6), and according to Lemma 6.4.I (i),
Aytf(·,a)---+ f(·) as a---+ 0. Therefore,
Setting
ro(c;) =log( I+ lc;l),k(c;) =(I+ lc;I 2 YI2 ,P(c;) =
c;k,k = 0, I,2, ... , and p = 2, we
arrive at the following result involving Sobolev space H'' obtained by Reider [81, p.884]
Corollary 6.4.3. Let f E H 5 (1R),s E IR, and lfl E H.B(JR)nL, (IR),/3 E No, with fJR.lfl(t)dt = I and Dklfl E L1 (IR) at least fork= {I, ... ,/3}. Then
lim~~~; LyY'f(·,a)- vCk ~df(-)11 s-k =0, a + 2
a--->0
where Ck = Cvkyt and Dk fork = 0, I, 2, ... , denote generalized derivatives.
6.5
Local convergence
In what follows we study the local convergence of the wavelet transform defined in local B~.k -space.
Definition 6.5.1 For n an open subset of IR, we define
called local generalized Sobolev space.
Lemma 6.5.2. f E (B~.k)toc(n) {::} f ·I/) E B~k 'v'I/J E Dw(n), where Dw(n) is the Bjorck space of C"' -functions of compact support inn defined by (6.2.6.)
Proof. Let f E
(B~k)Loc(n) and
1/J E Dw(n), then f agrees with an element of B~k on the
support of 1/J, hence 1/Jf E B~k· Conversely, suppose 1/Jf E B~,k for alii/) E Dw(n) and no is an open subset with compact closure inn. Choose 1/J E Dw(n) with 1/J =I on no, then f agrees with 1/Jf E B~k on no.
The Wavelet Transform
104
Definition 6.5.3. Let Un}nEN be a sequence in (B:,k)Loc(Q) and f E (B:,k)Loc(D.). Then {fn}nEN converges to fin (B~.k)Loc(Q) if and only if 111/>fn- fllp,k converges to zero for any If> E Dro(D.). Now, let us assume that
suppf = [-T, T] =I,
(6.5.1)
and consider (6.5.2) For 0
<
£
< T, let le denote the closed interval [-e,e]. Then there exists aTe
E De(/0 )
which is identical to I on le.
Lemma 6.5.4. (i) Te(·)f(·) E B~,k(JR.);
(ii) Te(·)f(·)--+ fin (B~,k)Loc(/0 ) as£--+ T; (iii)Te( · )/( ·)
= f(-) on (B~,k)loc(J2) ·
Proof. (i) See Lemma 6.5.2. (ii) Let If> E D 00 (r>). For sufficiently large et/1 with 0 all
£
with et/1
~ £
< et/1 < T, we have supp(lf>)
~lefor
< T.
This implies that 1/>Td =If>! in B~.k(IR.) for et/1 ~
£
< T.
(iii) It is clear that both Td and fare in (B~,k)Loc(J2). Let a test function If> E De(J2) act on the distribution Tef.
(Te(t)f(t), 1/>(t)) = (f(t), Te(t)lf>(t)) = (f(t), 1/>(t)). This proves (iii).
Theorem 6.5.5. Let f satisfy (6.5.1) and (6.5.2). Let lfl be defined as in Theorem 6.4.2 and
Te as above. Then
) 12 LP(a-1D)lfi(Tef)(·,a)--+
~P(D)f
in
(B:,k)Loc(l~).
Proof. First we conclude that P(D)f = P(D)Td in (B~k)toc(J2). Let 1/> E Dro(J2). Then by the Leibnitz rule, we have
(P(D)Tef,l/>) =
L Ca
a;;.o
(t (~) i=O
1
(Da-if,DiTelf>))
105
The Wavelet Transform on a Generalized Sobolev Space
a;;,o
=
([CaDaf,l/J)
= (P(D)f,cp). Therefore,
P(D)Tef = P(D)f. Now, let l/J E Sro(IR.) and T41: B~k ~ B~k be the multiplication operator defined by Trpf
=
(/J f. Then T41 is continuous. Let (/J E Dw(.fi)
c Sro(IR.). Then, using Theorem 6.4.2, we have 11(/J(·) al1/2LP(a-1D)Ifi(Tef)(·,a)- (/J(·) ~P(D)fllp.k/P
~IIT4111""11 a 1112 Lp(a-1DJifi(Td)(·,a)-
;,C P(D)(Tef)ll
v~d
_, pA/P
which completes the proof of Theorem. The next theorem provides the order of convergence and the method of proof is similar to that used in the proof of [81, Lemma 4.9]
Theorem 6.5.6. Let f be two times continuously differentiable in (b - e, b + e) for small
e > Oand bE R Let lfl E B~k(JR.)nL 1 (1R.) with fiR lfl(t)dt =land supplfl= [Tt, Tz]. Then for a > 0 sufficiently small, we have !r-f'(b) + O(a),
a- 312Lvlflf(b,a) =
vCt
where f' denotes ordinary derivative off, and Ct = CDifl· Proof. For small a, set [b+aTt,b+aTz]
=
I(b,a) and M(a) =
sup~E 1 If''(~)l. We have,
by mean value theorem,
la- 312Lvlflf(b,a)- !r,J'(b)l vC1
=
~
~ =
!r--11\lf/Df(b,a)- J'(b)l
vCt
!,! r
v C, a
llfl(t-b)llf'(t)-J'(b)ldt a
J1(b.a)
!r,!M(a) f
v C, a
K(a,b,J)a,
where K(a,b,J) = }JM(a) f~2 llfi(Y)IIYidy.
J1(b.a)
llfl(t-b)llt-bldt a
106
6.6
The Wavelet Transform
Example
As an illustration of the approximation property of the wavelet transform in Sobolev space H 5 we quote the following example from [81].
Let
l l
f () X=
1~
lxl
~ 1.5
2+x-I~x<0
(6.6.1)
2-xO~x~1
0
elsewhere.
Then f E H 5 (lR). Its distributional derivative is given by
Df = 8(x+ 1.5)- 8(x-l.5) + g(x),
(6.6.2)
1 -l~x
(6.6.3)
where
0
elsewhere.
Choose
I -l~x
(6.6.4)
elsewhere.
Wavelet Transform 00 80 40 20
w
0
,
-20 -40· -60 ·OU
100
~
~
--
0 0
~· a
Fig. 6.1: Approximation of Df by wavelet transform.
The Wavelet Transform on a Generalized Sobolev Space
This is a translated version of the Haar wavelet.
107
Then by Corollary 6.4.3,
,.;c;;,a- 312 Lvtf(b,a) approximates (6.6.2) for small a. This approximation of Df by the wavelet transform for certain values of b and a (small) is given by Figure 6.1.
Chapter 7
A Class of Convolutions: Convolution for the Wavelet Transform
7.1
Introduction
The convolution product holds a central place among the various modes of function compositions. It is a powerful mathematical tool used in symbolic calculus, approximation theory, Fourier series, and solutions of boundary-value problems. Associated with each important integral transform we have a special type of translation and convolution operator. These translation and convolution operators are used in the development of the theory of the integral transform and to solve the associated boundary value problems. The aim of the present chapter is to consider a general integral transform and define generalized translation operator (GTO) and the corresponding convolution. It is shown that the application of the general integral transform to the convolution of two functions yields the product oftheir transforms. Convolutions associated with important integral transforms are shown to be special cases of the present generalized convolution. Thus a unification of the existing results on convolutions is proposed. Wavelet transform has emerged as a powerful tool in analysis, but no convolution associated with the wavelet transform has been investigated as yet, as far as known to the author. Using the aforesaid technique, translation and convolution for the wavelet transform are defined and their properties are given. A brief study of the translation and convolution associated with the discrete wavelet transform has also been done.
7.2
The generalized translation and convolution
In this section we present a general unified approach to study convolution for a wide class of integral transforms. Various convolutions, mentioned below, need detailed investigations 109
110
The Wavelet Transform
in their own right. For this purpose, we consider the general integral transform:
~ = (lll'f)(x) f(x)
=
1b a f(t)lfl(x,t)dt, x E C
(7.2.1)
where a may take value 0 or -oo and b may be +oo, so that most of the common integral transforms are special cases of (7.2.1). Assume that it is possible to recover f(y) from its transform (Ill'f)(x) by means of the inversion formula (7.2.2) We know that associated with each integral transform there is an appropriate translation, which leads to the definition of the corresponding convolution. The integral transform converts the convolution of two functions into the product of their transforms. Hence, we define convolution of two functions f and g by means of the relation: (7.2.3) Using (7 .2.1) it can be expressed as follows:
(f~g)(z) = /~ 1
[1b f(x)lfl(t,x)dx 1b g(y)lfl(t,y)dy] (z)
= /~ 1 [1b f(x)(1b g(y)lfl(t,x)lfl(t,y)dy)dx]
(z).
The change in order of integration can be justified under different conditions on
lfl. One simple situation is where f, g E
L 1(a, b)
f, g and
and lfl(t, ·) E L""(a, b).
Let us assume that there exists a basic generalized function D(x,y,z) such that
lfl(t,x)lfl(t,y)
=
1b D(x,y,z)lfl(t,z)dz.
(7.2.4)
In some cases D(x,y,z) is a measure and in other cases it turns out to be a distribution. The right-hand side integral exists if lfl(t,·) E L""(a,b) and
1b ID(x,y,z)ldz
~C.
(7.2.5)
Then
(f~g)(z) = /~ 1
=1~ 1
[1b f(x) (lb g(y)dy 1b D(x,y,u)lfl(t,u)dx)] (z) [1b lfl(t,u) (1b (1b D(x,y,u)f(x)dx)g(y)dy)du] (z).
(7.2.6)
The change in order of integration is valid if J, g E L 1(a, b), lfl(t, ·) E L"" (a, b) and if (7 .2.5) holds.
A Class of Convolutions: Convolution for the Wavelet Transform
Now define the generalized translation
'rz
Ill
by
j'b -rzf(y) := f*(z,y) := la D(x,y,z)f(x)dx.
(7.2.7)
Then, by (7.2.5), (7.2.8)
and
(f~g)(z) = /~ 1 = /~ 1
[lb lfl(t,u) (lb (-rzf)(y)g(y)dy) du] (z) [/"'
(1b (-rzf)(y)g(y)dy) (t)J (z).
This yields the generalized convolution:
(f~g)(z) := 1b(-rzf)(y)g(y)dy.
(7.2.9)
This definition makes sense under different conditions on f and g. In particular, if lfl(t, ·) E L=(a, b), (7.2.5) holds, f E L 1(a, b) and g E L=(a, b), then JHg E L=(a, b). Also, if lfl(t, ·) E L=(a, b), (7.2.5) holds and J,g E L 1(a, b), then JUg E L 1(a, b).
We close this section by deriving some important properties of D(x,y, z) and the generalized convolution. From (7.2.4) it follows that D(x,y,z) is given by
D(x,y,z) = /~ 1 [lfl(t,x)lfl(t,y)](z),
(7.2.10)
where a:::;; x,y,z:::;; b. Clearly, D(x,y,z) is symmetric in x andy, but in general, not symmetric in all the three variables x,y and z. Setting t =to a constant, (7.2.4) yields
lb D(x,y,z)dz = 1,
(7.2.11)
where
D(x,y,z) provided lfl(to,x) lfl(to,y)
=
D(x,y,z)lfl(to,z)/(lfl(to,x)lfl(to,y))
(7.2.12)
i= 0. In some of the cases specific forms of D(x,y, z) are available
in the literature and in some cases D(x,y, z) is known to be symmetric in all the three variables x,y and z. Now, an important property of the convolution
~is
obtained from (7.2.9), which also fol-
lows from (7.2.3). Indeed, from (7.2.9) for f,g E L 1(a,b), we have
lyt(f~g)(t) =
lb lfl(t,z) (lb(-rzf)(y)g(y)dy) dz
112
The Wavelet Transform
1b lfl(t,z) (1b (1b D(x,y,z)f(x)dx) g(y)dy) dz = 1b f(x) (1b g(y)dy 1b lfl(t,z)D(x,y,z)dz) dx = 1b f(x) (1b g(y)dylfl(t,x)lfl(t,y)) dx 1b f(x)lfl(t,x)dx 1b g(y)lfl(t,y)dy
=
=
= (Ilflf)(t)(Ilflg)(t).
(7.2.13)
The proof of the above identity needs detailed investigation. In the case of wavelet transform we shall investigate it in sections 7.4-7.6.
7.3
Special cases
In this section we discuss various known and unknown special cases of (7.2.1), (7.2.7), (7.2.9) and (7.2.10). For each case we give appropriate D(x,y,z) and then convolution 'W can be defined using (7.2.7) and (7.2.9). (i) Fourier Convolution
The Fourier transform off E L 1 ( -oo, oo) is defined, for lfl(x, t)
ff:(f)(x)
=
i:
f(t)e-ixtdt,
= e-ixr, x, t E ffi., by (7.3.1)
and inverse Fourier transform is given by
f(t) = g:-I(ff:J)(t).
(7.3.2)
Then by Zemanian [97, p. 188],
D(x,y, z)
=
g;-t [e-ixr e-iyr] (z)
I ~oo =-
27r =
. e- 1.(x+y ) 1e!Ztdt
-00
c5(z-y-x).
(7.3.3)
Hence,
(rzf)(y) = /_: c5(z-y-x)f(x)dx =!(z-y).
(7.3.4)
Therefore,
(!Ug)(z) = j_:t(z-y)g(y)dy,
(7.3.5)
113
A Class of Convolutions: Convolution for the Wavelet Transform
whence §(f~g)(t) =
§(f)(t)§(g)(t).
(7.3.6)
(ii) Laplace Convolution The Laplace transform off on (O,oo) is defined by
L[f](s)
fo"" e-stf(t)dt,s E C,
=
(7.3.7)
provided the integral is convergent. In this case
lfl(s,t) =e-st' D(x,y,z)
=
L -I [e-s(x+y)](z) (7.3.8)
=O(z-x-y) by Zemanian [97, p. 352]. Therefore,
('rzf)(x)
=
fo"" o(z-x-y)f(y)dy= /_~ o(z-x-y)H(y)f(y)dy (7.3.9)
= H(z-x)f(z-x),
where H(y) is the Heaviside unit function. Hence
(f~g)(z) = fo"" ('rzf)(x)g(x)dx = fo"" H(z -x)f(z -x)g(x)dx = foz f(z -x)g(x)dx.
(7.3.10)
Clearly, L(f~g)(s)
= L(f)(s)L(g)(s).
(7.3.11)
(iii) Mellin Convolution The Mellin transform off defined on (O,oo) is given by
(Mf)(s)
fo"" ts-lf(t)dt,
=
sEC,
provided the integral exists. Here
lfl(s,t)
=
ts-1'
and we take
D(x,y,z)
=
M- 1[~-ly-"'](z)
= M- 1 [(xjyy-ly- 1](z)
(7.3.12)
114
The Wavelet Transform
=
o(z -xjy)y- 1•
(7.3.13)
Then
(rxf)(y)
fo'" f(z)o(z-xjy)y-
=
1dz =
f(xjy)y- 1;
(7.3.14)
so that
(f~g)(x) =loco f(xjy)g(y)y- 1dy.
(7.3.15)
Hence M[f~g](s) =
(Mf)(s)(Mg)(s).
(7.3.16)
(iv) Hankel Convolution Hirschman [35], Cholewinski [ 10], Haimo [28] and others have studied Hankel convolution for the following form of the Hankel transform.
F(x)
=
(hyf)(x)
=loco j(xt)f(t)dCJ(t),
(7.3.17)
where 1
I
I
j(x) = 2Y-'1r(r+ 2)x!-YJr-1 (x), and Jr-~ (x) is the ordinary Bessel function of order y-
dCJ(t) =
t2Y I
2r-'1r(r+
!)
!. and
dt.
The inverse Hankel transform is also given by
h~/ (F)(t) =loco j(xt)F(x)dCJ(x).
(7.3.18)
In this case
D(x,y,z)
=loco j(xt)j(yt)j(zt)dCJt =
23y- i (n)-lf2[r( r+
~ )]2(r(y))-l (xyz)-2y+l [L\(x,y,z)fY-2, y > 0,
(7.3.19)
where L\(x,y, z) is the area of triangle with sides x,y, z, if such a triangle exists and 0 otherwise. We note that D(x,y,z)
~
0 and that D(x,y,z) is symmetric in x,y,z and we have the
formula
loco j(zt)D(x,y,z)dCJ(z) =
j(xt)j(yt).
(7.3.20)
Moreover, setting t = 0, we get
loco D(x,y,z)dCJ(z) =
l.
(7.3.21)
A Class of Convolutions: Convolution for the Wavelet Transform
115
In this case we have
l"'
D(x,y,z)!(z)dcr(z)
(7.3.22)
(f~g)(x) =loco (rxf)(y)g(y)dcr(y);
(7.3.23)
(rxf)(y)
=
and hy(f~g)(y) =
(hy/)(y)(hyg)(y).
(7.3.24)
Convolution for Watson transform [77], Kw and lw transforms [98], Kontorovich-Lebedev transform [24], Mehler-Fock transform [25], Fourier-Jacobi transform [89] can similarly be obtained as special case of (7.2.9). In the following examples we show that convolutions related to certain discrete transformations can also be derived as special cases of (7.2.9). (v) Legendre Convolution
The Legendre transformation [86] off E L 1(-I, I) or C( -I, I) is defined by
P[f](n)
=
F(n)
= /_',
Pn(t)f(t)dt
(7.3.25)
with the inversion formula 00
f(t) =p-I [F(n)](t) =
L F(n)Pn(t).
(7.3.26)
n=O
Define
D(x,y,z) =p-I [Pn(x)Pn(Y)](z) 00
=
L Pn(x)Pn(y)Pn(z);
(7.3.27)
n=O
so that 1 /_ D(x,y,z)Pn(z)dz 1
Setting n = 0, and using the fact that P0 (x)
1
1
-I
= Pn(x)Pn(y).
(7.3.28)
= 1, we get
D(x,y,z)dz = 1.
(7.3.29)
Moreover,
(rxf)(y) = /_', D(x,y,z)f(z)dz.
(7.3.30)
(f~g)(x) =
(7.3.31)
Then 1
/_ 1(rxf)(y)g(y)dy
The Wavelet Transform
116
and
P[J~g](n) =
i
1 1 Pn(x)
(i
1 1('rxf)(y)g(y)dy)
dx
= i>(y)dy i 1/n(x)(rxf)(y)dx
i =i =i =i =
=
1 1 g(y)dy 1 1 g(y)dy
1 1 g(y)dy
i i i
1 1 Pn(x)
(i
1 1 D(x,y,z)J(z)dz)
dx
1 1 J(z)dz.{ 1 Pn(x)D(x,y,z)dx
1 1 J(z)Pn(y)Pn(z)dz
1 J(z)Pn(z)dz 1
i
1 g(y)Pn(y)dy 1
(Pf)(n)(Pg)(n).
(7.3.32)
For various properties of Legendre convolution see [86]. (vi) Dual Poisson-Laguerre Convolution The dual Poisson-Laguerre transform [II] off E L(O,oo) is defined by
PL(f)(n) =F(n) =loco f(x)Lia)(x)d/\(x),
(7.3.33)
with the inversion formula
f(x)
=
(PL)-l [F(n)]
=
EF(n)da)(x)<J(n),
(7.3.34)
n=O
where
Lia) (x) = r(p )r( a+ I )da) (x), p(n) = n!fr(n +a+ I) e-xxa d/\(x)= r(a+I/x a(n) = [r(a + I)p(n)r 1• We set
D(x,y,z) = (PL)- 1 [da)(x)Lia)(y)](z) =
ELia)(x)Lia)(y)Lia)(z)<J(n);
(7.3.35)
n=O
(rxf)(y) =loco D(x,y,z)J(z)d A (z)
(7.3.36)
117
A Class of Convolutions: Convolution for the Wavelet Transform
and
(!Ug)(x) =
~o=(rxf)(y)g(y)dA(y).
(7.3.37)
As in the previous case we get
PL(!Ug)
=
PL(f)PL(g).
(7.3.38)
(vii) Chebyshev Convolution Chebyshev transform off E L~ (-I, I) or C[ -I, I J is defined by ~[f](k) =
Ij' f(u)Tk(u)w(u)du,
~ = f(k)
1C
(7.3.39)
-1
where
Tk(x)=cos(kcos- 1 x), xE[-I,I], is the kth degree Chebyshev polynomial and w(x) =(I -x2) 112. The inverse Chebyshev transform is given by ~-
,~
I~
z=~
1C
1C k=l
[f](x) = - f(O) +-
L f(k)Tk(x).
(7.3.40)
Set (7.3.41) Then
j_', D(x,y,z)!(z)dz,
(7.3.42)
(!Ug)(x) = /_', ('rxf)(y)g(y)dy
(7.3.43)
~(JUg)= (~!)(~g).
(7.3.44)
(rxf)(y)
=
and
Conjecture 3.1. Let f E L~( -I, I) or C([-I, I]) and lxl ~ 1. Then for y E [-I, I],
(ry/)(y) =
~ {! ( xy+ V(I -x2)(I-y2)) + f
( xy-
V(I -x2)(I-y2))}
as given in [9, p.l66]. (viii) Sturm-Liouville Convolution Let I denote any open interval a
< x < b where a may be
-oo
and b may take the value oo.
Let { lfln (x)} ;=O be the orthonormal system of a Sturm-Liouville boundary value problem. Assume that the Sturm-Liouville transform off is defined by
S{f}(n) = F(n) =
1b f(x)lfln(x)dx.
(7.3.46)
118
The Wavelet Transform
Then (under certain conditions) inversion formula is 00
f(x) = s- 1 (F)(x) = L,F(n)1!fn(x).
(7.3.47)
0
For conditions of validity of the formula we may refer to [57], [98]. In this case we define
D(x,y,z) = s-l [l/ln(x)l/fn(Y)](z),
(7.3.48)
1 D(x,y,z)!(z)dz 1
(7.3.49)
(f~g)(x) = 1b('t"xf)(y)g(y)dy.
(7.3.50)
(-rxf)(y) = { and
7.4
Convolution for the wavelet transform
The above approach can be used in the development of a theory of convolution associated with the wavelet transform. For fixed and arbitrary p
~
0, and for any real number a =f. 0, the
continuous wavelet transform (CWT) of a signal f(t) E LZ(JR) with respect to the wavelet
q, E LZ(JR) is defined by (Wq,f)(b,a) =
l:f(t)~b;a(t)dt,
(7.4.1)
where
=
iai-p
(7.4.2)
As noted earlier, the present general form (7 .4.1) of the wavelet transform has been studied by Kaiser [41, pp. 63-70]. It may be pointed out that for p
J:'oo Iq, (u) jdu
is unaffected by the scaling operation, and for p
fected but II
=
II q, liz. The choice p =
=
=
I, the integral
! ,f.:_"oo Iq, (u) jdu is af-
0 is convenient in the study of orthonormal bases
of wavelets [41, Chapter 8]. Some of the properties of this transform has already been investigated in Section 2.3. A reconstruction formula for (7.4.1) with respect to the wavelet
q, is given by [4l,p.68]: (7.4.3)
where 0 < Cq, = joo
-oo
iii'( coW dm < oo. lml
(7.4.4)
A Class of Convolutions: Convolution for the Wavelet Transform
119
For validity of the reconstruction formula in U(JR), 1 < p < oo, we may refer to Perrier and Basdevant [79]. We define the basic function for the present case. Assume that D(x,y,z) is such that
Wtf>[D(x,y,z)](b,a)
=
l~ D(x,y,z)I/Jb;a(x)dx
= ll'b;a (Z )eb;a (y);
(7.4.5)
so that formally by (7.4.3),
D(x,y,z) = c;'l~l~ ll'b;a(z)Bb;a(y)I/Jb;a(x)lai 2P- 3 dbda. Clearly, D(x,y,z) is symmetric in all the three variables when
(7.4.6)
e = 11' = 1/J and 1/J is real
valued. The translation 't"x is defined by
('t"xf)(y) = f*(x,y) =
{~ D(x,y,z)J(z)dz
= Ci 1 l~l~l~ ll'b;a(z)Bb;a(Y)ll'b;a(x)J(z)lai 2P- 3 dbdadz.
(7.4.7)
(7.4.8)
Using (7.4.3) this can be expressed as (7.4.9) The associated convolution is defined by
(f~g)(x) = {~ f*(x,y)g(y)dy = {~~~ D(x,y,z)f(z)g(y)dzdy
(7.4.10)
= Ci 1 l~l~l~l~ ll'b;a(z)Bb;a(Y)I/Jb;a(x)J(z)g(y)iai 2P- 3 dbdadzdy. Using definition (7.4.1) we can write it as
(f~g)(x) = Ci 1 l~l~ (Wv,J)(b,a)(Weg)(b,a)I/Jb;a(x)lai 2P- 3 dbda = wl/>-l
[(W11,f)(b,a)(Weg)(b,a)](x);
(7.4.11) (7.4.12)
so that Wtf>[f~g](b,a)
= (W11,f)(b,a)(Weg)(b,a).
(7.4.13)
From representation (7.4.10), we conclude that if e = 11' = 1/J, then the following commutative and associative properties hold: (7.4.14)
120
The Wavelet Transform
(f~g)~h = f~(g~h).
Remark 7.4.1. D(x,y,z), 7:xf(y) and
(7.4.15)
f~g can be defined by other expressions similar to
(7.4.6), (7.4.8) and (7 .4.10) respectively. Then result of type (7 .4.13) will still hold. The following differentiability results can also be established.
For j,g E Y'(JR.), the
Schwartz space of rapidly decreasing functions, we have
where dxf(x)
D~(f~g)(x) = (Dk f~g)(x) + (f~Dkg)(x),k E No,
(7.4.16)
d~(f~g)(x)
(7.4.17)
= (dk f~g)(x) + (f~dkg)(x),k E No,
= x(d / dx)f(x).
Example 7.4.2. Let lfl(t)
· I 2 1 = B(t) =
(Morlet wavelet).
Then
iJ'b;a(t)
. ('-b) = laJ-Pe'OJo a -'1I ('-b)2 a .
Hence, by (7.4.6), we have
D(x,y, z) =
c; I
i: i:
eiOJo(b+x-z-y)fae-[(z-b)2+(y-b)2+(x-b)2]/(2a2) Jal-p-3 dbda
= 2Ci 1 /_: !~ tp-t cos( root(b+x- z- y) )e- 12 [(z-b) 2 +(y-b) 2+(x-a) 2ll 2dbdt =
c;'r(I + p /2)2t+p/2 .
.
i: [(z-
bf + (y- b)2 + (x- afrt-p/2
coJ(b+x-z-y) 2
x 1F1 (I+ pj2, l/2,- 2[(z _ b) 2 + (y _ b) 2 + (x- a)Z] )db, p > 0, by [18, p. 15(14)], where 1Ft (a;b;x) is confluent hypergeometric function.
7.5
Convolution for the discrete wavelet transform
Let lfl E L2 (JR.); set
lflm.n ( x ) = a0-m/2 lfl ( a0-mx- nbo) , m, n = 0 , ± 1, ±2, ... , ao, bo > O. Define the discrete wavelet transform off E L2 (JR.); by
J(m,n)
= (W11,f)(m,n) = (!, lf/m,n)·
(7.5.1)
If the wavelets { lf/m,n} form an orthonormal basis of L2 (JR.) then we have the reconstruction formula 00
t(x)
= w;' [J(m,n)](x) =
E m,n=-oo
Rm,n)lflm,n(x),
(7.5.2)
121
A Class of Convolutions: Convolution for the Wavelet Transform
and if the wavelets constitute a frame of L2 (1R), then 00
f(x)
=
Wi 1 [J(m,n)](x)
=
L
J(m,n)ij/m.n(x),
(7.5.3)
m,n=-oo
where ij/m,n is the dual frame [96, pp. 539-540]. From [96, Theorem 32.3, p. 540] we know that if the Fourier transform iji of lfl satisfies (7.5.4) with
a > 0 and r > a+ I' then { lf/m,n} constitute a frame.
Now, let us define formally our basic generalized function D(x,y,z) (resp. D(x,y,z)) as follows:
D(x,y,z)
=
Wi 1 [lflm,n(x)lflm,n(Y)](z)
=
L lf/m,n(x)lflm,n(Y)lflm.n(Z), m,n=-oo
00
(7.5.5)
in case { lflm,n} form an orthonormal basis, and by
wi' [lflm.n(x)lflm,n(Y)](z) = L lflm,n(X)lflm,n(Y)ij/m,n(Z),
D(x,y,z) =
00
(7.5.6)
m,n=-oo
when {lflm.n} form a frame, provided that the series (7.5.5) and (7.5.6) converge. This will happen when Cm.n = lflm.n (x) lf/m,n (y) E 12 . Then
1:
D(x,y,z)lf/m,n(z)dz = lf/m,n(x)lflm.n(Y)
(7.5.7)
D(x,y,z)ij/m,n(z)dz = lflm.n(x)lflm.n(Y)·
(7.5.8)
and
1:
Next, we define the translation ('t' and i" resp.) by
(rxf)(y)
=
f*(x,y)
=
(i"xf)(y) = J*(x,y) =
j~ D(x,y,z)f(z)dz,
1:
D(x,y,z)f(z)dz.
(7.5.9)
(7.5.10)
Finally, we define the convolution
(f~g)(x) =
1:
f*(x,y)g(y)dy
(7.5.11)
122
The Wavelet Transform
and
(f"g)~(x) = j_~J*(x,y)g(y)dy
(7.5.12)
for the aforesaid two cases respectively. Proceeding as in the derivation of (7 .4.13) one can easily show that (7.5.13) and
Wiii(!Ug)(m,n) 7.6
= (Wijif)(m,n)(Wijig)(m,n).
(7.5.14)
Existence theorems
First we obtain boundedness results for the basic function D(x,y,z).
Theorem 7.6.1. Let (1
+ lwiP)cp(w) E U(IR), (J E U(!R), ~ + ~ =
1 and (1
+ lwiP)ljl(w) E
L 1(IR),p ~ 0. Then
ID(x,y,z)l ~ zP+~c;'IY- zl-b lx- zl-~-p liB llqll (1 + lwiP)cp(w) liP II (1 + lwiP)ljl(w) lit, where cl/> is given by (7.4.4).
Proof.We have
[D(x,y,z)[
~ c;'[f.f.ia[-P'I' (~) WPB (y:b )ta[-P~ ( x:b )ta[ p-3dadbl 2
[kk e~b) (y~b) (X~ ~ c;' kk IV'(u)IIB (y:z lief> (
= c;'
ljl
cp
(J
+u)
b
}a~-p-3dadbl
x:z +u) llai-P- 2 dadu.
Using Holder's inequality, we get
ID(x,y,z)l ~ c;'
k
lll'(u)ldu
(k le (y:z
+u) lq lal- 2da) ljq
X(k lcp (X: z + U) lp lal-pp- 2da )'jp
k
=
c;'ly-zl-' 1qlx-zl- 11p-pi1BIIq
x
(k lcf>(w)llw-uiPPdw) lfp
=
zPc; 11Y- zl-' 1qlx- zl-l/p-p II Bllq
lll'(u)ldu
klll'(u)
ldu
123
A Class of Convolutions: Convolution for the Wavelet Transform
(
x
~ ltf>(w)IPiwiPPdw+ ~ ltf>(w)IPiuiPPdw ) I
I
1/p
I
= zP+-pC¢ 1 IY- zl_q_P lx- zl--p-P liB llqll (1 + lwiP)q>(w)IIP
x 11(1 + lwiP)q>(w)III· Theorem 7.6.2. (i) Let
VIE L 1 (JR),I/> ELP(!R), BE U(IR),p,q > 1,0 < p < 1 and~+~=
1 +p. Then
~ ID(x,y,z)ldz::;; C¢ 1C(p,p)lx- YI-p I V'IIIIII/>IIPIIBIIq,
(7.6.2)
where C(p, p) is constant.
VIE L 1(JR), (1 + lviP-l )tf>(v) EL 1(IR), (1 + lvlp-I )B(v) EL 1(JR), and p ~ I. Then ~ ID(x,y,z) ldz::;; c;Izp-llx- YI-p [111/>(v)vP-IIIIII Bill+ IIB(v)vP-IIIIIII/>11 I] I 'I'll I·
(ii) Let
(7.6.3) Proof.(i) We have
~ ID(x,y,z)ldz::;; C¢ 1 ~~~IV' ( z:b) lie (Y: b) Ill/> ( x:b) llai-P- 3dadbdz = C¢ 1 ~~ ie
(Y: b) Ill/> ( x: b) llai-P- a~ l'l'(t)ldtdadb 3
(7.6.4) Now, using Hardy-Littlewood-Sobolev inequality [47, p. 98] we get
IID(x,y,z)III where ~
+~
(ii) For p
~
:S:C¢ 1C(p,p)lx-yl-pii'I'IIIIII/>IIPIIBIIq, for O
= 1 + p.
1, from inequality (7.6.4) we have
~ ID(x,y,z)ldz::;; C¢ 1 11 VIlli IX- YI-p~~ IB(u)lltf>(v)llv- ulp-ldvdu :S: X
[~ 11/>(v)llviP-ldv ~ IB(u)ldu+ ~ IB(u)llulp-Idu ~ ltf>(v)ldv] C¢ 1 IIV'III2p-llx- YI-p
= c; lzp-llx- YI-p [111/>(v)vP-IIIIIIB III+ II B(u)uP-IIIIIII/>III]II 'I'll I· Theorem 7.6.3. (i) Let 1 +p. Then
VIE LP(JR), If> EL 1(JR), BE U(IR),p,q >
I ,0 < p < I and~+~=
124
The Wavelet Transform
EL 1(JR.), (1 + lulp-l )B(u) EL 1(JR.), (1 + lulp-l )lJI(u) EL1(JR.) and p ~ 1. Then ID(x,y,z)ldx ~ Ci 11Y- zi-p [illJI(u)uP-IIIIII eIII+ II B(u)uP-IIIIIIll'll d114'111· (7.6.6)
(ii) Let cp
k
The proof is similar to that of Theorem 7 .6.2. Theorem 7.6.4. (i) Let lJI EU(JR), cp
EU(JR), BE L 1(JR),p,q > 1,0 < p < 1 and~+~=
1 +p. Then
k
ID(x,y,z)ldy
~ Ci 1C(p,p)lx-zl-pllll'llqii4'11PIIBII1·
(7.6.7)
(ii) Let BE L1(JR.), (1
+ luiP-I )cp(u) EL1(JR.), (1 + lu1P- 1)lJI(u) EL 1(1R) and p ~ 1. Then ID(x,y,z)ldy ~ Ci 11x- zi-p [illJI(u)uP-IIIIIIC/' III+ llcf'(u)uP-IIIIIIll'll dII eIII· (7.6.8)
k
Next we obtain certain boundedness results for f~g.
EL 1(JR.), lJI ELP(JR), B EU(JR),p,q > 1,0 < p < l,f E L'(JR) and g E L1 (JR),r,r' > 1, ~ + fr +p = 2. Then llf~glll ~ c; 1C(p, p, r) IIC/'IIIIIll'llpll Bllqll!llrllgllr,
Theorem 7.6.5. Let cp
1, ~
+~
=
p+
(7.6.9)
where Ccp is given by (7.4.4) and C(p,p, r) is a constant. Proof. We have
kl(f~g)(x)ldx ~ k (k k k ~k k (k ID(x,y,z)ll!(z)ldz) lf*(x,y)llg(y)dy) dx
=
lg(y)ldy
lf*(x,y)ldx
lg(y)ldy
dx.
Therefore, by Theorem 7 .6.3 (i) we get
kl(f~g)(x)ldx ~
Ci 1C(p,p)II4'11111BIIqllli'IIP
=C- 1C(p,p)IIC/'IIIIIBII 1P
q
k
lg(y)ldy
llll'll f f P J'R.
}'R.
k
l!(z)lly-zi-Pdz
l!(z)llg(y)ldydz. IY- ziP
Now, invoking Hardy-Littlewood-Sobolev inequality [47, p.98], we obtain
In the sequel we shall use the following theorem whose proof can be found in [51, p. 75]. Theorem 7.6.6. Let f E L'(r > 1),0
(Iaf)(x) = Q(a)Ka(f)(x) =
k
lt-xla-lf(t)dt,
(7.6.10)
A Class of Convolutions: Convolution for the Wavelet Transform
125
where Q(a) = 2r(a)sin n2a ,0
for some K > 0. Theorem 7.6.7. Let (1 + luiP)(j)(u) E LP(IR), B(u) E U(IR), ~+~=I, (I+ luiP)lfl(u) E
L1(IR),f E Loo(IR) and g E L'(JR), r >
l. Then
11/Uglls ~ c; lzP+ ~ Ma,f3KII (J llq llgllrll/llooll (I + luiP)q, (u) liP II (I + luiP)lfl(u) III, where
(7.6.12)
i =~+P-I> 0,0 < p < 1,0 < 1- ~ < p <~and Ma,f3 = [B(a,/3) +B(a, 1- a- /3) +B(/3, I - a- /3)]
(7.6.13)
with a = I - p - !p > 0 ,,.,a = 1 - !q > 0. Proof.Using Theorem 7.6.1, we have
I(!Ug)(x)l
=
lk.k.
D(x,y,z)f(z)g(y)dydzl
~ c;lzP+~ IIBIIqii(I + luiP)q,(u)IIPII(I + luiP)lfl(u)llt
x
kk 1/(z)llg(y)IIY- zl-~ lx-zl-~-p
dydz.
Since f E Loo(IR),
I(!Ug)(x)l
~ c;lzP+~ IIBIIqll/llooii(I + luiP)q,(u)IIPII(I + luiP)lfl(u)III, x
k,lg(y)l (k,ly-zl-~lx-zi-~-Pdz)dy.
(7.6.14)
Now, from [7, pp. 397-398] we know that
kIY- zi-I/qlx-
zl-l/p-p dz = =
kIY- zi(I-I/q)-tlx- zi(I-I/p-p)-tdz kly-zl/3-llx-zla-tdz
=
Ma,f3lx- Yla+/3-l
=
Ma,{3lx-yi-P,
where Ma.f3 is given by (7.6.13). Hence, from (7.6.14), we have
I(JUg) (x) I ~ c; lzP+~ Ma,f311 (J llq 11/llooll (1 + luiP )if' (u) liP I (I + luiP) lfl(u) lit' X
klg(y)llx-
YI-Pdy
=Apft-p(lg(y)l)(x),
(7.6.15)
126
The Wavelet Transform
where Ap =
c;'2P+t Ma.f311 Bllqll!llooll(l + luiP)q,(u) liP II (I+ luiP)ljl(u) Ill· Then using in-
equality (7 .6.11 ), we get (7 .6.12). Theorem 7.6.8. Let
V' E L1 (!~) nU(JR), BE L 1(JR) nU(JR),I/' E LP(JR),p,q > 1,0 < p <
t
1, ~ + ~ = p +I. Assume further that f E L'(JR),g E L'' (JR), + 1 = ~ + ~ + p, ~ + ~ = 1
and ( 1 -
!!!
r '
I-
!i) < p < 1. Then r'
II (!~g) lit ::.;; c; I C(p, p) Ill/' llp(ll e lltlllflllq) l/m(ll e llq II VIlli) l/m' Kll/llrllgllr'·
(7 .6.16)
Proof. For~+~= 1,
l(!~g)(x)l = lk.k D(x,y,z)!(z)g(y)dydzl :=.;;
X
=
x
(k.k
1/(z)lmiD(x,y,z)ldydz) l/m
(J~k. lg(y)lm'ID(x,y,z)ldydz) l/m'
(k 1/(z)lm (k. ID(x,y,z)ldy) dz) l/m (k. lg(y)lm' k(ID(x,y,z)ldz)dy) l/m'
Using Theorem 7.6.2 (i) and Theorem 7.6.4 (i), we get
l(!~g )(x)l ::.;; c;'C(p,p )Ill/' lip( liB II II V'llq) 11m(IIB llqiV'IIt )11m' X
(Ia (lfl)m(x)) 1/m (Ia (lgl)m' (x)) l/m'.
E L1sfm(JR), ]'} = 7- a, a= I- PJa(lglm')(x) E L1s' /m' (JR), '{f, = 'fr- a and±+~= I. Now, using Holder's inequality and then applying (7.6.11) we get (7.6.16)
By (7.6.11), la(lflm)(x)
Let (I+ lroiP)B E U(JR),I/' E LP(JR),~ + ~ = 1 and (I+ lroiP)V' E ~ 0. Assumefurtherthat f E L'(JR),g E Lr' (JR), r, r' > 1,q > r > 1,0 < ~- p < ~
Theorem 7.6.9.
L1(ffi.),p
t
and + I = ~ + ~ + p. Then
where K > 0. Proof. In view of Theorem 7.6.1 we have
lf*(y,z)l::.;;
.k
lf(x)IID(x,y,z)ldx
A Class of Convolutions: Convolution for the Wavelet Transform
127
I
I
~ C¢ 12P+p 111/JIIpii(I + lwiP)e(w)llqll(l + lwiP)vr(w)lltiY- zl_ii_P
x/l(lfl)(z). q
By (7.6.11),
/1 q
(lfl)(z)
E
U(JR),
t = ~- ~ > 0. Therefore,
l(f~g)(z)l = l.kf*(y,z)g(y)dy' ~ C¢ 1 2P+~ 111/JIIPII(l + lwiP)e(w)llqll(l + lwiP)vr(w)lltll(lfl)(z) q x
_klg(y)lly-zi-~-Pdy
~ C¢ 1 2P+~ 111/JIIPII(I + lwiP)e(w)llqii(I + lwiP)vr(w)lltllq (l!l)(z) X
/Lp(lgl)(z), p
where /!_p(lfl)(z) P
E
u' (IR), ~·'
= ~-.!. + p r
p
> 0.
Now, using Holder's inequality and applying (7 .6.11) again we get
llf~gllt ~ C¢ 1 2p+~ Kill/' llpllgllr' llfllrll (1 + lwiP)8(w) llqll (I+ lwiP)'I'( w) lit· Theorem 7.6.10. Let 1/J E L 1 (JR) nco(!R), 8 E U(!R), II' E U(!R), p, q
> 1, 0 < p < I, t +
~ = p + l. Assume further that f E U(JR),g E Lr' (IR), r, r' > I and~+~+ p = 2. Then Wq,(f~g)(b,a) =
Proof.
(Wvd)(b,a)(Weg)(b,a).
(7.6.18)
By Theorem 7.6.5., f~g E L 1 (JR). Therefore, for basic wavelet 1/J E L 1 (!R) n
co(JR), Wq,(f~g)(b,a) exists. Hence using (7.4.10) and (7.4.5) we can write
.k
Wq,(f~g)(b,a) = (f~g)(x)I/J ( x: b )iai-Pdx =
_k 1/J (x:b)iai-Pdx _k_kD(x,y,z)f(z)g(y)dzdy
=
.k
.k !(z)g(y )dzdy .k D(x,y, z)l/1 ( x: b) lal-p dx
= .k .k f(z)g(y)dzdyiifh;a(z) {Jb;a(Y) =
(Wv,f)(b,a)(Weg)(b,a).
7.7 An Approximation Theorem An approximation of g
E
U(!R) using wavelet convolution is given. For this purpose we
need the following theorem due to Okikiolu [52, p. 39].
128
The Wavelet Transform
Theorem 7.7.1. Let g E L'(JR),r >I. Then IIKa(g)- gllr----+ 0 as a----+ where Ka(g)
=~(~~and la,Q(a)
Theorem 7.7.2. Let L 1 (JR) ,f E
0+,
are given by (7.6.10).
(I+ luiP)IfJ(u)
L oo (JR) , and g E L' (lR), r
E U(JR),B E U(JR),~
> I.
+~
= 1,(1
+ luiP)lJI(u)
E
Then
lll(f~gp)l-lglllr----+0 as p----+1-,
where gp
=
ApQ(l-p)"
Proof. By inequality (7.6.15) and Theorem 7.7.1, we have
Remark 7.7.3. It is interesting to establish analogous existence and approximation theo-
rems for discrete wavelet transform.
Chapter 8
The Wavelet Convolution Product
8.1
Introduction
In this chapter we consider the n-dimensional wavelet transform defined by (3.1.1) and its variant (3.1.2). Using representation (3.1.2) a convolution associated with the wavelet transform is defined in terms of ordinary (Fourier) convolution ( 1.3.4.). In terms of the Fourier convolution, wavelet transform (3.1.1) can be expressed as follows: (8.1.1)
(W11,f)(b,a) = (/* Oa)(b), where
(8.1.2) Existence of convolution and some of its basic properties are summarized below; these will be used in the sequel.
II!* gil'
~
ll!ll,llgll'
(8.1.3)
[42, p.62]
and
(!*g)/\= jg [42,p.l58].
(8.1.4)
If f,g,h E L 1 (l~n), then we also have
(8.1.5) (ii) Iff E U(!Rn),g E U(!Rn),
i + ~ = l, l ~ p,q ~
oo,
then f
* g exists a.e.,
is bounded
and continuous, and
llf*glloo ~ IIJIIpllgllq 129
[42,p.64].
(8.1.6)
130
The Wavelet Transform
and (8.1.7) (iv) Let f E U(!Rn),g E U(!Rn), I ~ p,q <
oo,
~
+ ~ - 1 ?: 0. Then
f * g exists a.e. and
i +~-I and we have Young's inequality
f*g E U(!Rn), where~=
(8.1.8) Suppose in addition that 1 ~ p,q,r
~
2. Then from Theorem 1.3.8, ~
r'
I
I
(! *8)1\ = fg E L (!Rn),;: +? = l.
(8.1.9)
(v) A more general form of Young's inequality is also given [47, p. 90 ]: Let/ E LP(!Rn),g E U(!Rn) and hE U(!Rn) with p,q,r >I,
lknf(x)(g*h)(x)dxl
i + ~ + ~ = 2, then
~ Cp,q.r.nllfllpllgllqllhllr·
(8.1.10)
Motivated from the Fourier transform property (8.I.9) of the convolution product"*" defined by (1.3.4) associated with the Fourier transform we define formally the convolution product associated with the wavelet transform (8.I.I) by means of the relation: (8.1.1I) and obviously we have f~g
= gUf (commutativity).
This in turn shows that the product of two wavelet transforms could be a wavelet transform. In this chapter we shall obtain conditions for the existence of the convolution
'U', study its
properties and give some applications. Furthermore, existence results for the wavelet transform of convolution can also be obtained by using the analysis of Perrier and Basdevant [79], and Rieder [81 ].
8.2
The Wavelet Convolution Product
Using property (8.1.9) of the Fourier transform, from (8.1.1) we get (8.2.I)
The Wavelet Convolution Product
131
In view of property (iv) given in Section 8.1, this relation holds in
U(!Rn), II' E U(!Rn), ~ = ~+~-I ~ 0, ~ +
u' (!Rn)
when
f
E
fr = I, I ~ p, q, r ~ 2.
Hence by (8.1.11 ), and using a variant of property (8.1.9), we get §[Wlfl(f~g)](co)
= §[(Wlflf)(b,a)(Wlflg)(b,a)](co)
[§-
1(
lji(aco)f(co)) §- 1 ( lji(aco)g(co)) J (co)
=
§
=
(2n)-n [ lji(a.)f(.) * lji(a.)g(.)] (co).
(8.2.2)
Using (8.2.1) this gives the important relation between the two types of convolutions:
lji(am)(f~g)i\ = (2n)-n [lji(a.)j(.) * lji(a.)g(.)] (co),
(8.2.3)
whence formally we get
(f~g)(z) provided lji( am)
=
(2n)-n §- 1 [ ( lji(a.)j(.) * lji(a.)g(.)) (co)jlji(aco)] (z).
(8.2.4)
=f. 0. This can be re-written as
(f~g)(z) = =
X
(2n)-Zn { { ~(a 11 ) lji(a(C0-11))i(11)g(C0-11)eizwd17dW }JRn }JRn II'( am) (2n)-2n
r r
lJR• liR•
~(a11) lji(a(C0-11))
vr( am)
(k.n k.. ei(z-y)w-i(x-y)1) f(x)g(y)dxdy) d17dco.
(8.2.5)
Now, set
x(co, 11) = lji( 11) lji( co -17) I lji( co), lji( co) =1- o
(8.2.6)
and define the basic generalized function:
Da(x,y,z) = (2n)-Zn { { ei(z-y)w-i(x-.v)1lz(aco,a17)dcod17. }JRn }IR.n
(8.2.7)
Ba(u, v) = (2n)-Zn { { eiuw+iv11z(aCO,a17)dcod17, }IR.n }JRn
(8.2.8)
If we write
then clearly we have
Da(x,y,z) =Ba(z-y,-x+y).
(8.2.9)
Now, from (8.2.5) it follows that
(f~g)(z) = }JRn { { Da(x,y,z)f(x)g(y)dxdy }JRn
(8.2.10)
132
The Wavelet Transform
= f
f Ba(u,v)f(z-u-v)g(z-u)dudv.
(8.2.11)
}JRn }JRn
Next, we define the wavelet translation
by
't"z
(-rzf)(y) :=
f
}JRn
Da(x,y,z)f(x)dx;
(8.2.12)
so that the wavelet convolution can be expressed as
(f~g)(z) = }IRn r (-rzf)(y)(y)g(y)dy.
(8.2.13)
Example 8.2.1.Let l!f(t) be the Morlet wavelet having Fourier transform:
Then by (8.2.6),
x(co, 1J) = (2n) 112 exp( -1] 2 + ~1]- (1/2)%). Moreover, from (8.2.7) we have
8.3
Existence Theorems
In this section we give conditions of validity of the wavelet convolution. Assume that
f
E L 2 (JRn),
and l!f E L 2 (JRn) satisfies the admissibility condition (8.3.1)
Let W 2 denote the Hilbert space:
W2 (JRn
X
JR+) = {f(b,a):
~+ ~n lf(b,a)l 2 a-ldadb <
00 }
possessing the norm
ll.f(b,a)llw2
:=
(~n (~+ lf(b,a)l a- da) db) 2
1
112
Now, from (8.1.1) we have
r r
}IR i }JRn
1Wvd(b,a)l 2 a- 1dadb =
r (}IRn r IU* 8a)(b)l db) a- da
JJR+
2
1
(8.3.2)
133
The Wavelet Convolution Product
= (2n)-n = (2n)-n
k+ (k.. lf(;)O,(;)I 2d;) kn lf(;)l 2d; (k.+ li/f(a;)l 2a- da) a- 1da 1
= Clfl(2n)-n f lf(;)l 2d; }JRn
=CV'II!II~Therefore, we have Plancherel formula for the wavelet transform: (8.3.3) The following Parseval relation also holds [45,p.30]. For J,g E L2 (1Rn) and for real wavelet
1f1 E L 2 (JRn) satisfying (8.3.1) we have
f
}JRn
f(x)g(x)dx
= (CV')- 1 f
f
}JR+ }JRn
Wlflf(b,a)Wlflg(b,a)a- 1dadb.
(8.3.4)
Theorem 8.3.1 Let f,g E L2(1Rn), and let 1f1 E L2(1Rn) be a real wavelet which satisfies (8.3.1 ). Then (8.3.5)
Proof. Applying (8.3.3) to (8.1.11) we get
JC';IIf~gll2
= IIWV'(f~g)llw2 = II(Wlflf)(Wlflg)llw2 =
(k.n (k.+ I(Wlflf)(b,a)(Wlflg)(b,aWdaja) db)
112
.
(8.3.6)
Since g E L 2(1Rn) and 1f1 E L 2(1Rn), from (8.1.1) and (8.1.6) we have (8.3.7) Applying (8.3.7) to (8.3.6) and then using (8.3.3) we get
JC';IIf~gll2 ~ a-n/2ll8ll2lllfiii2IIWV'JIIw2 ~ a-n/2ll8ll2lllfiii2JC';IIfll2·
(8.3.8)
Corollary 8.3.2. Let f,g,h E L2(1Rn) and 1f1 E L2(1Rn) be a real wavelet which satisfies (8.3.1). Then (8.3.9)
134
The Wavelet Transform
Theorem 8.3.3. (i) Assume that f E U(IRn),g E F(!Rn), 1 < p,r
F'(!Rn) 'p!
u' (IRn) n
+ l_p' = l 'r! + ? = l. Then 1
(8.3.10) (ii) Assume that J,g E L 2 (1Rn) and 1f1 E L 2 (1Rn) is a real wavelet which satisfies (8.3.1).
Then (8.3.11) Proof.(i) Using (8.l.l) and (8.1.6) we have IWlfl(f~g)(b,a)l =
I(Wlflf)(b,a)(Wlflg)(b,a)l
=I(!* Oa)(b)(g* Oa)(b)l ~ a-n(l/ p+l/r) II/IlP lllflllp'llgllrlllflllr'· (ii) In view of (8.3.4) we have
~~. (~, IWlfl(f~g)(b,a)ldaja) dbl = ~~. (k+ (Wlflf)(b,a)(Wlflg)(b,a)dafa) dbl =
~~. (~+ (Wlflf)(b,a)(Wlflg)(b,a)a- 1da) dbl
=elf!
~
I~. f(x)g(x)dxl
CV'II!IIzllgllz.
Theorem 8.3.4. Assume that f, g E L 1(!Rn) and ij) E Co (IRn)
n L 1(!Rn). Then (8.3.12)
Proof. From (8.2.3) we have
s~p Iij/( am) Is~p I(f~g )A (am) I
~ (2n)-n ~. Iij)(a17 )f( 17) Is~p Iij/(a( co -17) )g( co -17) Id17 ~ (2n) -n sup 18( ')I sup Iij/(a') I f Io/( a11 )j( 11) ld77. S
S
}~n
Therefore, sup I(f~g )t\ (co) I ~ (2na) -n llilloo II o/lloo llflloo II o/ll1· w
The Wavelet Convolution Product
8.4
135
Wavelet Convolution in a Generalized Sobolev Space
In this section, we investigate the boundedness of the wavelet convolution in the generalized Sobolev space studied by Hormander [38], which appears to be the most suitable space for this work. Definition 8.4.1 Let k be a weight function defined in !Rn (not necessarily temperate). Then the space B p,k, 1 ~ p <
oo, consists of all distributions u E .9'' such that uis function and
llullp,k = ( ~" lk(.;)u(.;)IPd.; ) In what follows we set k(.;) = Iij/( a.;) I, for fixed a > 0. Theorem 8.4.2. (i) Let f E
B1,k
and g E
Bp,k,P ~
I~
< oo.
(8.4.1)
1. Then
llf~gllp,k ~ (2n) -n llfllt,ki!gllp.k·
(ii) Let f E
Bp,k
and g E
Bq,k,
1 ~ p,q <
(8.4.2)
oo, -!J + ~- 1 ~ 0. Then
llf~gllr,k ~ (2n) -n llfllp,ki!gllq.k, ~r =
_!_
p
+! q
1.
(8.4.3)
Proof. In view (8.2.3) we have
llf~gllp.k = (~n lk(.;)(f~g)"(.;)IPd.;) l/p = (2n)-n
(~" I(ij/(a. )j(.) * ij/(a. )g(.) )(.;) IP d.;) 1/p.
(8.4.4)
Then using (8.1.7), we get
llf~gllp,k ~ (2 7r) -n II ij/(a. )j(.) Ill II ij/(a. )g( .) liP' which yields (8.4.2). Also, by an application of (8.1.8) to (8.4.4) we get (8.4.3).
8.5
Approximations in LP and
Bp.k
Spaces
The wavelet convolution is at first used to approximate certain LP -functions. Theorem 8.5.1. Let 0 ~ q, E C()(!Rn), fiR" cf'(x)dx = 1. Set cf'E(x) = e-nq,(x/t:) and g~(x) =
((2n)n /iif(ax))cf'c(x),t: > 0, ij/(x) =J 0. (i) Assume that 0 =J ij/( m) is continuous function and j is a continuous function of compact
support. Then (8.5.1)
The Wavelet Transform
136
uniformly for all z E IR.n as e
---->
0+.
(ii) /fO-/= ij/(ro) E Co(IR.n) and j(ro)ij/(aro) E U(!R.n), then
(f~g~) E C~(JR.n) and ij/(aro)(f~g~)i\----> ij/(aro)j in LP(JR.n), I ~ p 0+. (8.5.2) Proof From (8.2.3) we have
ij/(aro) (f~g~- f)!\( ro) = [ ( o/(a. )j(.) * cf'e(.)) (ro)- ij/(aro )f( ro) J .
(8.5.3)
Therefore,
llo/(a.)(f~g~- !)/\liP~ II (o/(a.)f(.) * cf'e(-)) (ro)- ij/(aro)j(ro)IIP.
(8.5.4)
Now, applying Theorem 1.3.2 of Horrnander [37, p. 17] to (8.5.3) and (8.5.4) we get (8.5.1) and (8.5.2) respectively. An approximation property of the wavelet convolution is now given in the space Bp,k·
Theorem 8.5.2. I~p
Let
g~
be the same as in Theorem 8.5.1.
If f
E
Bp,k(JR.n),
< co and ij/( ro) -!= 0, then f~g~---->
f
Proof. The proof follows from (8.5.2) using Definition 8.4.1 (cf. Horrnander [38], p. 12).
Chapter 9
Asymptotic Expansions of the Wavelet Transform when lbl is Large
9.1
Introduction
The wavelet transform off with respect to the wavelet 'I' defined by (2.1.1 ), for n = I, is given by
(W1 ,J)(b,a)=Jal-~ f~J(t)'l'c~b)dt, a~O.
(9.1.1)
Using change of variables b = x,a- 1 = y, and setting
W(x,y) = IYI-~ (W11,J)(x,y- 1 ), 'I'( -y)
=
g(y),
(9.1.1) can be expressed in the form
W(x,y) =
J~ f(x- u)g(yu)du,
(9.1.2)
which is a hybrid of usual (Fourier-type) and Mellin-type convolutions. In what follows we shall assume that x > 0, y > 0. The analysis can be suitably modified in case x or y or both are negative. We shall assume further that f(t) and g(t) are even (the analysis can be modified for other cases; see (9.1.26)) locally integrable functions on
(0, oo) and possess asymptotic series representations in descending powers oft when t-> oo and derive asymptotic expansion of W(x,y) when x---> oo andy belongs to an open subset of (O,oo), under different assumptions on the asymptotic forms off and g. In (9.1.26) we shall see that for fixed y > 0, W(x,y) can be expressed as a sum of a Laplace convolution transform and series involving generalized Stieltjes transforms. Therefore, to derive the asymptotic expansion of (9.1.2) we shall exploit the techniques used for the two types of integral transforms (Li and Wong [46], Wong [93], Wong and McClure [94]). Asymptotic behaviour of Laplace convolution (9.1.6) has also been investigated by Handelsman and Lew [29] and Riekstins [83] under the assumption that f(t) and g(t) have asymptotic series representations in descending powers oft as t
--->
137
oo.
138
The Wavelet Transform
For simplicity we assume that = f(t)rv [ast-s-a, t->oo, O
(9.1.3)
s=O
and =
L hst-s-{3,
g(t) "'
t
--->
oo, 0 < J3 ~ 1.
(9.1.4)
s=O
We shall use the following definition of non-commutative convolution fog due to Riekstins [83] and the Laplace convolution f*g [46].
r/2
= Jo
f(x-t)g(t)dt
(9.1.5)
(f*g)(x) =lox f(x- t)g(t)dt.
(9.1.6)
(fog)(x)
and
The advantage of the convolution o over the convolution *·as noted by Li and Wong [46] is that in (9.1.5) f should be locally integrable on the open interval (0, oo) whereas in (9.1.6) it should be so on the semi-closed interval [0, oo ). Therefore fog may be defined for distribution fin Ltoc(O,oo) and distribution gin Lt;c(l~). the class of functions which are locally integrable on lR and which vanish on ( -oo,O). From (9.1.5) and (9.1.6) it follows that rx/2
(f*g)(x) =
lo
rx/2
f(x-t)g(t)dt+
Jo
g(x-t)f(t)dt
= (fog)(x) + (gof)(x).
(9.1.7)
We recall some of the properties off o g from [46, pp. 1541-1544]. Iff and g are (m- I)-times differentiable functions on (O,oo) and g E Lt;c(!R), then for x>O,
(! o!Y"g)(x)
=
1 m-1 Dm(f o g)(x) +D"'-l-f[f(x/2)d g(x/2)]. 2 f=O
L
(9.1.8)
The proof can be given by induction on m, cf. [46, pp.l541-1542]. In view of the property
o=
DH, from (9.1.8) we have
t-s-a oDj 0
for s, j E No and 0 <
a
~
= Djt-s-a = (-l)j (a+ s)jf-s- j-a,
(9.1.9)
1.
In what follows we shall use the incomplete Beta integral defined by Bx(a,b) =lox ta-l (1-t)b-ldt
(9.1.10)
Asymptotic Expansions of the Wavelet Transform when
where a> 0, b
lbl is Large
139
> 0 if x ~ 1, and b can be negative if x <
I.
Using the relation a
(-I)i . a (f3)J D 1t-,.,
.
t-,.,-J =
from definition (9.1.8) it follows that for 0
(9.l.II)
< f3 < 1 and s E No, (9.1.12)
where
(9.1.13) For f3 = I, the relation .
t- I -J
=
(-1 )i
.
I
- - D1+ 1ogt J!
(9.1.14)
is used to derive the identity (9.1.15) where h ·(a)= (a+s)J2a-l+s_ (a-s+I)J+I;: (a) s,J
ks,j(a)
•1
•1
J.
J.
=(j+ I) (a- ~,+s)j ~s(a)-
(a - I
1·
+i~ f=O
~s)J+l1Js(a)-
J.
[(j+I) (~-1+s)e~s(a)£
(J+ 1-t')e
-(j)£ (j-£)f+I
(a+s)e 2a-I+s]
0
'
':>s
,
(9.1.16)
(a :.s)J2a-l+s1og2 J.
(a+I+s+£)J-f-1 2a+s+f (£+ I)j-f
'
(9.1.17)
with ~o(1)
~s(a)= for 0
I
= log2,
a-I+s
(2a-I+s_l)
< a < I, s = 0, I, ... , or a = I, s = I, 2, ... , and ['12 1J 5 (a)=
Jo
(1-u)-a-slogudu
(9.1.18)
(9.1.19)
140
The Wavelet Transform
for 0 < a
~
I and s = 0, I, ....
The convolution is left distributive in the following sense [46, Lemma 4, p. 1544] (9.1.20) This relation holds for g1 ,g2 E LL:;c(IR) such that g1 is n-times differentiable, g2 ism-times differentiable and f is N = max(m,n)-times differentiable. For each n > I , we set
n-1 f(t)
=
L ajl-j-a + fn(t)
(9.1.21)
}=0
and
n-1 g(t)= [bJl-j-~+gn(t)
(9.1.22)
}=0
and we write .fo(t) = f(t) and go(t) = g(t). Following Wong [93, p.296] we define inductively fn,o(t) = fn(t) and
fn.j+l(t)=-
l
oo
(-})J+lloo . .1 (u-t) 1.fn(u)du, j=O,l, ... ,n-1. (9.1.23)
fn.j(u)du=
I
}•
I
Similarly, gn.O (t) and gn,J+ 1(t) are defined. For 0 <
a
~
1, the distribution associated with fn (t) is defined by
Un,cp) = (-l)n(fn,n,cp(n)) =(-It
r
lo
.fn.n(t)cp(n)(t)dt,
(9.1.24)
for cp E Y(O,oo), the Schwartz space of rapidly decreasing functions on (O,oo). Let Sy denote the operator
(Syg)(x) = g(yx).
(9.1.25)
Then using (9.1.6),(9.1.21 ),(9.1.22) and (9.1.25) we can write (9.1.2) as follows:
W(x,y)
=
1x f(x- u)g(yu)du + 1
00
f( -v)g[(x+ v)y]dv
+ 1 f(x+v)g(-yv)dv 00
=
(! * Syg)(x) + :~ b_.y-s-~ 1oo (x+ v)-s-~ f(v)dv
+
E
asy+a-11oo (xy+v)-s-ag(v)dv
s=O
0
+ 1oo f(v)gn[(x+v)y]dv+ 1 g(yv)fn[x+v]dv 00
Asymptotic Expansions of the Wavelet Transform when
lbl is Large
141
(9.1.26) since by assumption /(v) = f(v). The second and third integrals are generalized Stieljes transforms, and the first term is the Laplace convolution off and Syg. The last two integrals exist since f and g are locally integrable on ~Modifying the results of Li and Wong [46] we derive the asymptotic expansion of T1, and those of Wong [93, pp.295-30 1] are used to find the asymptotic expansions of Tz and T3. The asymptotic expansion of the generalized Stieltjes transform (9.1.27) can be found in [91]. In order to make the analysis complete in itself we first derive asymptotic expansion of (9.1.27) for p > 0 using the distributional technique of Wong [93, pp.298-301], where the special case p = 1 has been considered. To accomplish this we need the following two lemmas of Wong [93, pp.296,297].
Lemma 9.1.1. For 0 < a < 1 and n ;?; 1, the identity n-1
n-1
(!, cp) = [. aJ(t_;J-a, cp)- [. CJ+1 (oUl, cp) + (fn, cp) }=0
(9.1.28)
}=0
holds for any cp belonging to the Schwartz space .9'(0, oo) and ( -1 )i+l Cj+l =
.1
J.
.
M[f;J + 1],
(9.1.29)
where M[f;z] denotes the Mellin transform defined by
or its analytic continuation.
Lemma 9.1.2./f a= 1 in (9.1.3), then for each integer n;?; 1 and for any cp E Y'(O,oo), n-1
(f,cp)
= [. aJ(t.; 1- 1,cp))=0
n-1
[. cj+ 1(oUl ,cp) + Un.l/>),
(9.1.30)
)=0
where (9.1.31)
142
The Wavelet Transform
9.2
Asymptotic Expansions of the Generalized Stieltjes Transform
Let us take the following element of Y (Schwarz space) in Lemma 9.l.I and in Lemma 9.1.2:
(9.2.I) where t E (O,oo) and z is a complex variable in the cut plane largzl < n. We also define
t! =
0
{
t~ 0
t;. t
> 0,
where A E C. Then (t~a-J,t/>TI(t)) __I_ {"" t-a -(a) 1 1o
(!!:...)} dt
e-111 dt (t-z)P
= -1
{""t-at ( j) ( -17 )i-r e-TII ( -p )(-p-I) ... ( -p- r+ I )(t + z)-p-r dt r=O r (a)J Jo
--->_I_ (-I )i (p) · {"" t-a (t + z) -p-i dt
(a)J
1
lo
as 11 ---> 0.
Evaluating the last integral by means of the formula [ 19, p. 233, (8)], we get lim (t~a-J ,tf>TI(t)) = -I)( ( -I)iz-p-J-a+l r(l- a)r(~ ~ j +a- I)' a 1 r p
71 _,o
(9.2.2)
where p + j > I - a > 0. Furthermore, as in the above, using (9.1.14), we have
. -1-J I . looo (}+I) e-111 hm (t+ ,t/>1J(t)) = - 7 hm (Iogt)D 1 -(-)-dt 1)-->00 1! TJ--->0 0 t +z p (-I)i {"" Iogt = p(P)J+I Jo (t +z)P+}+I dt ( -1 )i r(p + j)
. z-P- 1 [Iogz-r-lfl(p+j)]· ' see the Appendix. Here ydenotes the Euler constant and lfl(z) = r'(z)/r(z).
= -j!
r(p)
(9.2.3)
We also note that
.
. looo e-111 f(t) looo ( f(t)) ( ) dt =
hm (!, t/>11 ) = hm
11-.o
t+z P o by Lebesgue's convergence theorem, where a+ p > 1. TJ->O
In view of (9 .1.24 ),
oo
o
Iim(fn,t/>1)) = Iim(-IY lo fn,n(t)JY; ( e
TJ-->0
TJ-->0
0
-TJI
) dt
t +z p
dt, t+z P
(9.2.4)
Asymptotic Expansions of the Wavelet Transform when
lbl is Large
143
=(-It{"" fnn(t)(-p)(-p-1)···(-p-n+l)(t+z)-p-ndt
lo
=
'
(P)n ~o= fn,n(t)(t+z)-p-ndt.
=lim 1)-->0
L,
j-1 (
-I)
1.
q=O
(9.2.5)
Tjj- 1-q(p)qz-p-q
q
= (p)j-1Z-p-j+ 1, j = 1,2, .... Substituting the values of (·, q,11 ) as 11
(9.2.6)
0 from (9.2.2), (9.2.6) and (9.2.5) into (9.1.28) we
---+
obtain
S ( )-n~1 f z -
. -p-j-a+1r(l-j-a)r(p+}+a-l) r( ) p
!-- a}z
J=O
n-1
(
)
+ L,(-l)j ~/M[f;j+l]z-p-j+Dn(z) j=O
where p > l -
(9.2.7)
].
a > 0, and Dn(z) = (P)n ~o= fn.n(t)(t +z)-p-ndt,
(9.2.8)
with fn,n(t) defined by (9.1.23). Also, if we substitute the values of(·,·) from (9.2.3), (9.2.6) and (9.2.5) into (9.1.30) and use the value of cj+ 1 from (9.1.31) we obtain the asymptotic expansion for a= I, _
Sj(Z) =
n-1
(-l)j
j=O
].
.
E aj-.,-(P)jz-P-J[logz- r- 'I'(P + j)] n-1
- L,cj+1(P)jZ-p-j+Dn(z), j=O
(9.2.9)
where On in the same as given by (9.2.8) and cj+ 1 by (9.1.31).
9.3
Asymptotic expansions for Tz and T3
The integral in Tz is the generalized Stieltjes transform of order p = s + f3 of f(v). Substituting the values of SJ(x) and Dn(x) from (9.2.7) and (9.2.8) in the series for T2 we derive T -~1n~1b -s-{3 -j-s-{3{ .r(j+s+a+f3-1)r(l-j-a) 1-a 2 - 1... 1... sY X a1 r( + f3) X
s=OJ=O
S
144
The Wavelet Transform
+ ( - 1)i (·~ + J3)j M[f;j + 1]} + Rn,l (x,y),
(9.3.1)
J.
where a + J3 > 1, 0 < J3 ~ 1, 0 < a < 1,
Rn.I(x,y) = Y!.bsY-s-~(x+J3)n {"" (
:n,n)~:~+ndu
Jo x u
s=O
(9.3.2)
and from (9.1.23),
fn,n(u)
=
( -1)n {"" (n-1)! Jo (-r-u)n-lfn(-r)d-r.
s
To find the asymptotic expansion of 1 (x) when (9.2.9) instead of (9.2.7) and get
Tz
(9.3.3)
a = I and 0 < J3 ~ I we apply formula
n-ln-1 [ (-1)1 =.?;Eo bsY-s-~ arJl-(s+ j3)j{logx+ 1J1(1 + j)- l!'(s+ J3 + j)} +cj+l (s + J3)J] x- j-s-~ + Rn,l (x,y),
(9.3.4)
where l!'(z) = r'(z)/r(z) and cj+ 1 is given by (9.1.31). Similarly substituting the asymptotic expansion of S8 (xy) we get T _n~ln~l 3 -
/.., /..,
s=OJ=O
a_,.y
-j-1 -j-s-a{br(j+s+a+f3-I)r(1-j-J3)( )1-~ X J r(s + a) xy
+( -1 )i (s +.~)} M[g; j +I]}+ Rn,z(x,y), 1· where a + J3 > 1, 0 < a ~ 1, 0 < J3 < I, n-1
f+a-1
Rn •z(x,y) = f;;Q ~ a.,.(s+ a)nY ( -1)n
gn,n(u) = (n- 1)! For 0 <
a
~
0
gn,n (U ) (xy+us ) +an + du,
(-r-ut- 1gn(-r)d-r.
(9.3.6)
(9.3.7)
1 and J3 = 1, we proceed as in the case of Tz and find that
=fa]; a.,l+a-l n-ln-1
T3
{""
Jo
looo
(9.3.5)
[
(-I)i br-]!-(s+ a)j{log(xy) +I!'( I+ j)- l!'(s +a+ j)}
)](xy )-j-s-a + Rn,zx,y, ( ) d * (-1) 1 ( +j+J-.,-s+aj J.
(9.3.8)
where * . [ ( -1 )i dJ+I =hm gJ+IJ+I(t)+-. -ajlogt ] . r-o ' J!
(9.3.9)
Asymptotic Expansions of the Wavelet Transform when
lbl is Large
145
9.4 Asymptotic Expansions of f*Syg and W(x,y) In this section we modify the technique of Li and Wong [46] and derive the asymptotic expansion of
(! * Syg)(x) = (! oSyg)(x) + (Sygo f)(x)
(9.4.1)
when x----> oo andy belongs to a bounded open subset of (0, oo ). The formula (9.4.1) holds for any
f and g in Ltc(lR).
Using (9.1.26) and collecting the asymptotic expansions of Tt, Tz and T3 we derive the asymptotic expansion ofW(x,y). In view of Lemma 9.1.1 we have g(yt) =
where dj+l
n-l
n-l
j=O
j=O
L bj(yt)-13- j- L dj+IY- j-l oU) (t) + 8n(yt),
( f)J+I
= -=-y--M[g;j +I].
(9.4.2)
Applying (9.1.12) and (9.1.9) we get n-l
(t-s-a oSyg)(x) =
L
bjy-f3-jes,j(a,{3)x 1-a-{3-s-j j=O n-l dj+IY-j-l ( -1)j(a+s)jx-a-s-j j=O
-L
+ (t-s-a ogn(yt))(x). Since the distribution 8n
= Dngn,n
(9.4.3)
using (9.1.8) we can write
(t-s-a ogn(yt))(x) =(t-s-a oy-nD78n,n(ty))(x) =D~(t-s-a
oy-ngn,n(ty))(x)
+ y~n :~ D~-1-£ {
G
rs-a
D~8n.n
c;) }.
(9.4.4)
Since 8n,n is locally integrable on [0, oo) using (9.1.5) we can express (t-s-a o Y-n8n,n(ty))(x)
r112
= xl-a-sy-n Jo
(1 - u)-a-s 8n.n(xyu)du.
(9.4.5)
The local absolute integrability of 8n,n allows us to differentiate within the integral sign n-times and get the value of the first term on the right-hand side of (9.4.4). Thus we have (t-s-a ogn(ty))(x) =y-n [x-ne?\x,y)
+~fa D~-l-f {G) -s-a D~gn,n ( ~)}] ,
(9.4.6)
The Wavelet Transform
146
where
(9.4.7) and the rate of decay of e} 1l (x,y) can be shown to be independent ofn [46, p. I545]. Hence, (9.4.3) gives
n-1 (1-s-a og(ty))(x)
L bjy-f3-Jes,j(a,f3)x 1-a-{3-s-J
=
)=0
n-1
L (-I)i+ldJ+I (a +s)}Y-1-Jx-a-s-J
+
}=0
+ y-n
[x-ne;'l (x,y)
+~ :~ D~-l-£ {G) -s-a v;gn,n ( x;)} ]·
(9.4.8)
Therefore, from (9.1.2I) and (9.4.8) we have n-1 (foSyg)(x) = as(t-s-a oSyg)(x) +(in oSyg)(x)
L
s=O
n-ln-1
= L L asbjes,J(a,f3)y-f3-Jxl-s-j-a-{3 s=Oj=O
n-ln-1
+ L Las( a +s)JdJ+I ( -y)-1-lx-s-J-a s=Os=O
+ r~:~ asy-nD~-I-j {
Grs-a
D{gn,n
(x;)}
n-1
+ L as(xy)-ne}'l(x,y) + (fn(t) og(ty))(x), s=O
where 0 < a : : ; I and 0 < f3 < I. Similarly, using (9.I.22) and (9.I.28) for 0 < f3 ::::;; I and 0 < a < I we get
n-ln-1
(Sygof)(x)
=L
L ajbses,J(f3,a)x 1-s-}-a-f3y-s-{3
s=OJ=O
n-ln-1
+ L L (-I)j+l bs(/3 + s)jCj+IX-s-{3-iy-s-{3 s=O}=O
(9.4.9)
Asymptotic Expansions of the Wavelet Transfonn when
+~
lbl is Large
147
%k hsy-!3-sv~- -j Gr/3-s D~fn,n G)} 1
{
n-1
+x-n
L hsY-s-/3 eP) (x) + (gn(ty)
0
f(t))(x),
(9.4.10)
s=O where
(9.4.11) Adding (9.4.9) and (9.4.10) we obtain n-ln-1
T1 = (f*Syg)(x) =
L L asbj[es,j(a,f3) +ej,s(f3,a)]x1-a-J3-s-jy-J3-j
s=Oj=O
n-1n-l
+L +
L (-1)j+l (a+s)jasdj+lx-a-s-jy-j-1
s=Oj=O n-1n-1
L L (-1)j+l ({3 + s)jbsCj+IX-/3-s- jy-!3-s
s=Oj=O 8
+
L Rn,m(x,y),
(9.4.12)
m=3 where
_ 1 n-ln-l -/3-s (n-1-j) Rn,4(x,y)- s~k hsY Dx
2
{
X (2)
a -p-S
j X Dxfn,n
n-1
Rn,s(x,y)
=
L as(xy)-neJ!l(x,y),
s=O n-1
Rn,6(x,y)
L bs(x)-ny-!3-se,}2l(x),
=
s=O Rn,?(x,y)
=
Un(t) og(ty))(x),
Rn,s(x,y)
= (gn(ty) o f(t))(x).
}
(2) ,
148
The Wavelet Transform
Now, rearranging each of the double sums in (9.4.12), (9.3.1) and (9.3.6) to give a single sum, and truncating each of the single sums after n terms, we get following expansions for
T1, T2 and T3. n-1 T1
=
s
L La_,._ jbj[es- j,j( a,f3) + ej.s- j(/3, a)]y-J3- jxl-a-J3-s + Rn,9(x,y)
s=Oj=O n-1 s + [(-1)j+ 1(a+s- j)jG.~-jdj+IY-I-j[a-s+Rn,IO(x,y) s=Oj=O n-1 .1 + -1 )j+l (/3 + S - j)jhs- jCj+IY-/3-s- jx-13-s + Rn,ll (x,y), s=Oj=O
L
L L(
where
n-1
2n-2
Rn.9(x,y)
L L
=
as-jbj[es-j,j(a,f3)+ej.s-j(f3,a)]y-J3-jxl-a-J3-s,
s=n j=.1-n+ I 2n-2 n-1 R,.w(x,y)
=
L L
(-1)H 1 (a+s-j)jas-jdj+IY-I-jx-a-s,
L L
(-1)j(f3+s-j)jh.,·-jCj+IY-J3-s-jx-J3-s,
s=n j=s-n+l 2n-2 n-1 Rn,JI(x,y) =
s=n j=s-n+l ·b _r(a+f3+s-1)r(1-a-j) -J3-s+j l-a-J3-s T2 --"~If{ L. L. a 1 s- 1 f.! • y x
r(p
s=O j=O
+(-1)jbs-j (/3
+S -
) )
+~,- j)j M[J;j+ 1]y-J3-s+j[J3-s} ].
+Rn.l (x,y) +Rn,I2(x,y) where
R
n,l2
( )- 2~ 2 "~1 X,y - £... . £... s=n J=s-n+l
{
·b _r(a+f3+s-1)r(1-a-j) -/3-s+j 1-a-J3-s a} S-j r(f.!+ _ ') Y X 1-'
S
)
+( -1 )jhs-j (/3 + ~~- j)j M[f;j + 1]y-J3-s+jx-J3-s}. J.
T =;=,'fa
!-s=OJ=O
3
£...
. {b. r(a + f3 +s-I)r(1- f3- j) I-J3xl-a-J3-s J r(a+s- ') y
S-j
)
+(-1)j(a+~,- j)j M[g;j+ 1J}y-l-jx-a-s 1·
+Rn,2(x,y) +Rn.13(x,y), where R
n.J3
-2~2
X,y ) - £... £... aS-J s=n j=s-n+ 1
(
;=,1
·{b·r(a+f3+s-1)r(I-f3-j)( )'-J3 J r( + - ') XY a
S
J
Asymptotic Expansions of the Wavelet Transform when
lbl is Large
149
Substituting the above values of T1, T2 and T3 in (9.1.26) we get the asymptotic expansion ofW(x,y).
Theorem 9.4.1. Let 0 < a,{j < 1 and a+ {3 > 1. Assume that f and g are even, locally integrable functions on (O,oo) possessing properties (9.1.3) and (9.1.4) respectively. Then for fixed y belonging to an open subset of (O,oo), we have W(x,y) =
n-1
n-1
s=O n-1
s=O
L As(y)xl-a-{3-s + L Bs(y)x-a-s
+ L Cs(y)x-/3-s +Rn(x,y),
(9.4.13)
s=O where ·b· A ( ) - f { b . . r(a+{j+s-l)r(l-a-j) -/3-s+j sY s- 1a 1 r(s- j+/3) y +as- 1 1
f;:o
r(a+f3+s-1)r(1-{j-j) ] x [ r(a+s-j) +(es-j,j(a,{j)+ej.s-j({j,a)) y
-/3-j}
Bs(Y) = t(-1)jas-j(a+s- j)j [~M[g;j+ 1]-dj+l] y-l-j j=O J. Cs(Y)
=
t(
-1 )jb.,·-j(/3 +s- j)j
j=O
[~M[f;j + 1]- Cj+I] y-f3-s+j ].
13
Rn(x,y)
=
L Rn,m(x,y) +Pn(x,y) + Qn(x,y), m=l
dj
=
(-1)j . (j -1)!M[g;}], Cj
Pn(x,y) =
(-1)j
.
= (j -1)!M[f;J],
fo'"" f(v)gn[(x+ v)y]dv,
and Qn(x,y)
= J;' g(yv)fn(x+ v)dv.
Remark 9.4.2. The cases 0 < {3 < 1,a
=
1;0
=
1;a
= {3 =
1 can be treated
separately using Lemma 9.1.2, equations (9.3.4), (9.3.9) and suitably modifying asymptotic expansions of f o Syg and Syg of. Following Li and Wong [46] error bounds for these asymptotic expansions also can be found.
150
9.5
The Wavelet Transform
Asymptotic Expansion of Mexican Hat Wavelet Transform
As an illustration of the aforesaid analysis we consider asymptotic expansion of (9.1.1) with lfl(t) Then g(y)
= ( 1 - t 2 )e- 12 12 ,
= lfl( -y)
the Mexican hat wavelet.
does not satisfy (9.1.4), but the aforesaid technique using Lemma
9 .1.1, gives asymptotic expansion of Mexican hat wavelet transform for b 1f1 E
--->
0+. Clearly,
Y(O,oo), hence we can apply (9.1.28) and get asymptotic expansion of
r
1 Jo f(t) (Wv,J)(b,a) = y'a
( I - (t-b) -a-
2
)
e- ('-b)2 a dt as b---> 0 +.
(9.5.1)
Here for the sake of convenience we assume that b > O,a > 0. Also, assume that f(t) satisfies (9.1.21) and that
f(t) = O(ea12 ) for some a> 0.
(9.5.2)
In (9.1.28) choose
(9.5.3) Now, for 0 <
a < I , using (9 .1.11) and integrating by parts, we have (t;_a-j,
=
-a3f2fo=t-a-jD2e-H'~b)2dt
= - a3f2
(a)j
r t-avj+2 [e-H'~h/] dt.
lo
Using Rodrigues' formula and series representation for Hermite polynomial [80] we can write the last expression as
lo=
2 -a312 t -a (-J)H e _l(t-b)2 ! a H ·+ 2 (t-b) - - dt 1 ( a)j o ( J2a)H2 J2a
-a3f2 [j/2]+1 (-1/(2+ j)! = (a)j k!(2+ j - 2k)!
k
1 )j+2
x ( - J2a
r
Jo
(J2
I ('-b 2 )j-2k+2 t-ae-z ...,) ---;;-(t-b) dt
-a3/2 [j/2]+1 (-J)k+j( 2 +J")! . 1(h)2 =-2-ka2k-2J-4e-z a
(a)j
L
k=O
k!(2+ j-2k)!
Asymptotic Expansions of the Wavelet Transform when
lbl is Large
151
2+}-2k (2+]'-2k) (-1)j-mb2+}-2k-m XL m Jor tm-ae-'lt 1a +tb/a dt. m=O
I 2
X
2
2
Now, evaluating the above integral by means of the formula [18, p. 16 (21)]:
i
oo
.o
I
2
2
_r-le-'lax -f3xdx = a-sf 2l(s)e(f3 f4a)D_s(f3/Va),Re a> O,Re s
> 0,
(9.5.4)
where Dv(x) denotes parabolic cylinder function, we find that (r-a-1 q,) + '
X
IJ/2]+ I
(a) 1
k=O
(2
+ 1') 1. rka2k-2j-4 k! ( 2 + j - 2k) !
L (
2+}-2k -1 )i-m (2+J'-2k) b2+ j-2k-m am-a+ I m=O
X
E
3/2
= _ ~e-!(hfa) 2
m
f'(m- a+ 1)D_(m-a+l)( -bja).
Since, by [16, p. 267 (31)], Dv(x) = 2vf 2e-!x2lfl( -V /2, 1/2;x2 /2),
(9.5.5)
and by [ 16, pp. 262-263], lfl(a c· x)
''
=
1(1-c) + O(ixi 1-Rec) 0 < Re c < 1,x---> 0+, 1(1+a-c) '
(9.5.6)
we get t-a-j
(+
_
-i(b/a) 2 [i/2]+1 j-2k+2
,i/')-e X
k~ fo
f'(m- a+ 1) y'7irk-t(m-a+l)a2k-2j+m-a-3f2b2+j-2k-m
qm 2a + 1)
+ O(b3+}- 2k-m). Moreover,
( _ 1)i+m+l (j +2)! k!m!(2+}-2k-m)!
(9.5.7)
152
The Wavelet Transform
= a-2j-5f2e-lf2(bfa) 2
U/2]+1 (-I)k+j+l( .+ 2)'
L
1 1 • (ajV2,fkb-2k+j+2. • 1 k.(J +2- 2k).
k=O
(9.5.8)
In view of (9.1.24), error term: Rn
:= (fn,cfJ) = (-I)n {"" fnn(t)cfJ(n)(t)dt
lo
=
'
(-I)n+'fo''" !n,n(t)a3f2 (:rr+2 e-H'~b)2dt a-n-1/2 {co
=-
Jo
2n/2+1
'('-b)2
fn,n(t)e-2
a
Hn+2
(t-b) V'ia dt,
(9.5.9)
where fn,n is given by (9.1.23). Finally, from (9.1.28), (9.5.7) and (9.5.8) we obtain for fixed a E (O,oo), n-1 [j/2]+1j-2k+2
(Wv,f)(b,a) = e-!(bfa)2
L L L
j=O k=O
m=O
(-I )j+m+l (j + 2) !r(m- a+ I )y']r2-k-:l;(m-a+l) 1 k!(2+ j - 2k-m)!m!r(m 2a+ I)
x [a·~~~~~~~--~~~~~~----xa2k-2j+m-a-3f2b2+ j-2k-m + O(b3+ j-2k-m)] 1
-e-2(bja)
2
n-I[j/2]+1c· ,(-I)k+j+l(J'+ 2)'
L L
J+
j=O k=O X
.
k!(j+2-2k)!
2ka2k-2j-5/2b2+j-2k +Rn,
(9.5.10)
where Rn is given by (9.5.9).
Appendix
We demonstrate the evaluation of the integral
r
Jo
logt d p >0, (t +z)j+P 1'
used in (9.2.3). We know that
loo
co
1a ( ) +pdt = za+l-j-p B(a+ I,}+ p- a -I), t+z J
Differentiating formally with respect to
loo
co
talogt ( ) .+Pdt t +z J
a> -1, j
+ p- a- 1 > 0.
a we get .
= za+I-J-P [Iogz B(a+ 1,} + p- a +()daB( a+ I,}+p- a- I)]
-1)
Asymptotic Expansions of the Wavelet Transform when
lbl is Large
= za+l-j-p [logz B(a + 1,} + p- a- 1)
+ ( .1
rJ+P
) { r' (a+ I )r(j + p - a - 1)
-r(a+I)r'(J+p-a-1)}]. Now, let a
----+
0, then
{""'
logt. dt (t + z)i+P
lo
= zl-j-p [lo z r(j +p-I) g
r(j + p)
+r(J~p) {r'(I)r(J+p -I) -r'(J+p -I)} J 1-j-p
= .2
[logz-r-1Jf(J+p-I)], J+P- 1
where y is Euler's constant.
153
Chapter 10
Asymptotic Expansions of the Wavelet Transform for Large and Small Values of a
10.1
Introduction
Asymptotic expansion with explicit error term for the general integral
I(x) =
fo'"" g(t)h(xt)dt,
(l O.l.l)
where h(t) is an oscillatory function, was obtained by Wong [91], [93] under different conditions on g and h. Then the asymptotic expansion for (2.1.3) can be obtained by setting
g(t)
= eibt f(t)
for fixed bE JR. Let us recall basic results from [93] which will be used in
the present investigation. Here we assume that g(t) has an expansion of the form 00
g(t)rv [csts+A.-l as t---->0, s=O n-1 = Csts+A.-1 + 8n(t), s=O 1. Regarding the function h, we assume that as t
L
where 0 < A ~
( 10.1.2) ---->
h(t) = O(tP), p +A> 0, and that as t
---->
0+, (10.1.3)
+oo, 00
h(t) "'exp(inP) where -r
-I- 0 is real, p ~
I and 0 <
f3
~
L bst-s-/3,
(I 0.1.4) s=O I. Let M[h; z] denote the generalized Mellin trans-
form of h defined by M[h;z] = lim rtz-lh(t)exp(-etP)dt. E--->0+}0
(10.1.5)
This, together with (10.1.1) and [93, p.216], gives
n-1 I(x)= [csM[h;s+A]x-s-A.+Dn(x), s=O 155
( 10.1.6)
156
The Wavelet Transform
where Dn(x)
=
lim {"" gn(t)h(xt)exp( -EtP)dt. e-.0+ Jo
(10.1.7)
If we now define recursively h0 (t) = h(t) and h(-j)(t) = - ["" h(-j+ll(u)du, j = 1,2, ... , then conditions of validity of aforesaid results are given by the following [93, Theorem 6, p.217]:
Theorem 10.1.1. Assume that (i) g(m) (t) is continuous on (0, oo ), where m is a non-negative integer; (ii) g(t) has an expansion of the form (10.1.2), and the expansion ism times differentiable; (iii) h(t) satisfies (10.1.3) and (10.1.4) and (iv) and as t---> oo,t-f3gUl(t) = O(t-l-e) for j = 0, I, ... ,mandforsome E > 0. Under these conditions, the result (10.1.6) holds with (1 0.1.8) where n is the smallest positive integer such that It
+ n > m.
The aim of the present chapter is to derive asymptotic expansion of the wavelet transform given by (2.1.3) for large values of a, using formula (10.1.6). We also obtain asymptotic expansions for the special transforms corresponding to Morlet wavelet, Mexican hat wavelet and Haar wavelet.
10.2 Asymptotic expansion for large a In this section using aforesaid technique, we obtain asymptotic expansion of (W'I'f)(b,a) for large values of a, keeping b fixed. We have (W'I'f)(b,a)
=
'{! [~ eibwfii(aro)i(ro)dro
=
Va { {"" eibwfii(aro)j(ro)dro 2n lo
+
l""
e-ibw fii( -aro )j( -ro )dro}
= Va 2 7r (/1 +h), say.
(10.2.1)
Let us set
h(ro)
= l[i(ro).
(10.2.2)
Asymptotic Expansions of the Wavelet Transform for Large and Small Values of a
157
Assume that 00
fi/( m) ""' exp( i-rmP) L brw-r-{3,
{3 > 0, m ---+ +oo,
't" =/:-
0, p ;:;::, I,
(10.2.3)
-r=O
and 00
f( m) ""'
L csms+A.- 1 as m---+ 0.
(I0.2.4)
s=O where 0 < A. ~ I. Also assume that as
m ---+ 0,
h(m) = fi/(m) = O(mP), p +A.> o. Then, as
( I0.2.5)
m ---+ 0, __
g(m) := eibw f(m)""'
"bw)r LCsWs+A.-1 L -~-,oo
oo
s=O
r=O
(
r.
_ ~ ~ (ibY s+A.-1+r - i...t i...t C5 - - W r! s=Or=O } m-'+A-1 (ib)r E Es ----;:y-Cs-r 00
=
{
s=O r=O 00
=
Ldsms+A.-1, s=O
(10.2.6)
where
d.,.
=
s (ibY -,-Cs-r· L r=O r.
(10.2.7)
For each n ;:;::, I , we write
n-1 g(m) = I'.dsms+A.- 1+gn(m). s=O
(10.2.8)
The generalized Mellin transform of h is defined by
r= r=
wz 1- 1h( m )e-ew dm M[h;zi] = lim e-->O+ lo wz 1- 1fi/(m)e-ewdm. = lim e-->O+ lo
(10.2.9)
Then by (IO.l.6),
n-1 h(a) = LdsM[h;s+A.]a-s-A. +8~(a), s=O
(10.2.10)
where (10.2.11) Also, from (I 0.2.5) we have
h(-m)=O(mP), W---+0, p+A.>O
(l0.2.I2)
158
The Wavelet Transform
and (10.2.13) Hence /z(a)
=
n-1
L,ds(-ly+A+IM[h(-ro);s+A]a-s-A. +D;(a), s=O
(10.2.14)
where (10.2.15) Finally, from (I 0.2.1 ), (I 0.2.1 0) and (I 0.2.14) we get the asymptotic expansion:
(Wvd)(b,a)=
't: {~ds(M[~;s+A.] + (-I y+A+I M [ fi/( -ro);s+
A]) a-s-A.+ Dn(a)},
(10.2.16)
where
r
r
Dn(a) = lim gn(ro)h(aro)e-E(J)dro+ lim gn(-ro)h(-aro)e-E(J)dro. E->O+lo E->O+lo
(10.2.17)
Since g(ro) = eibwj(ro), the continuity of j(ml(ro) implies continuity of g(ml(ro). Using Theorem
10.1.1
we get the following existence theorem for formula
(10.2.16).
Theorem 10.2.1. Assume that (i) j(ml(ro) is continuous on ( -oo,oo), where m is a nonnegative integer;(ii) j( ro) has asymptotic expansion of the form (I 0.2.4) and the expansion ism times differentiable,(iii) fi/(ro) satisfies (10.2.2) and (10.2.3) and (iv) as
ro
---7
oo, w-/3 j(j) (ro) = 0( ro-l-E) for j = 0, 1, 2, ... , m and for some e > 0. Under these
conditions, the result (10.2.16) holds with
(10.2.18) where n is the smallest positive integer such that A + n > m.
In the following sections we shall obtain asymptotic expansions for certain special cases of the general wavelet transform.
10.3
Asymptotic expansion of Morlet wavelet transform
In this section we choose
159
Asymptotic Expansions of the Wavelet Transform for Large and Small Values of a
Fig. 10.1: Morlet wavelet (with
roo= 7).
25
2.0
1.5
1.0
0.5
-10
-5
Fig. 10.2: Fourier transform of Morlet wavelet (with
roo= 7).
Then from [14, p. 373] we have -(ro-"'o)2
iji( ro) = V2ice -----y- , which is exponentially decreasing. Therefore, Theorem 10.1.1 is not directly applicable, but a slight modification of the technique works well.
160
The Wavelet Transform
Assume that f has an asymptotic expansion of the form (10.2.4). In this case we have
h(ro) = ij/(ro) (10.3.1) and
h(ro)
= 0(1)
as ro----> 0.
(10.3.2)
Then from (1 0.2.1 0) and (1 0.3.1 ), we get
lt(a)
E
d.,.M [ J2ne -<w-zwo)2 ;s+A.] a-s-A.
=
+
!ooo gn(ro)J2ne -(aw-wo)2 e-ewdro,
Jim
(10.3.3)
£--->0+ 0
where
M [J2ne -<"'-z"'ol2 ;s+A.] =
J2'1r fooo ros+A.-Ie -<w-zwol2 dro .,2
roo
2
= J2ne---i Jo ros+A.-te-~+w~dro. Evaluating the last integral by means of formula [14, (31), p. 320]: fooo _K-l e-(x2 12 )-f3xdx =
eU32 14 lr(s)D-s(f3), Re(s) > 0,
where D-v(x) denotes parabolic cylinder function, we get
M [ J2ne -<"'-;"'o)2 ;s +A.] = J2ne=# r(s +A. )D-(s+A.) (-roo), s +A. > 0.
(10.3.4)
From (10.3.3) and (10.3.4), we get
It (a) =V21re=#
E
dsr(s +A. )D_(s+A.)( -roo)a-s-A.
s=O
+ fooo gn(ro)J2ne -(awzwo)2 dro.
(10.3.5)
Similarly, we get _.,2 n-l
h(a) =J2ne-?
+
i
oo
.o
L dsr(s+ A.)( -1y+A.-I D-(s+A.)(roo)a-s-A.
s=O
gn( -ro)J2ne-
(aW~WtJ)2
dro.
Finally, using (10.2.1),(10.3.5) and (10.3.6) we get -wij 4 (W11,f)(b,a) =e-
n-1 [.
s=O
dsr(s+ A.) [D-(s+A.)( -roo)
(10.3.6)
161
Asymptotic Expansions of the Wavelet Transform for Large and Small Values of a
(10.3.7) where
v'a
Dn(a) =
l"' gn(ro)e-(aw-~) dro+v'a fo', gn(-ro)e-(aw+~) dro. 2
2
Using Theorem 10.2.1 we get the following existence theorem for formula ( 10.3.7).
Theorem 10.3.1. Assume that j( ro) satisfies conditions of Theorem I 0.2.1. Then the result (10.3.7) holds with Dn(a)
= (-I)ma-m+l/21~ g~m)(ro)
where n is the smallest positive integer such that It
10.4
(e -(aw-od)
(-m)
dro,
+ n > m.
Asymptotic Expansion of Mexican Hat Wavelet Transform
In this section we choose (see Figs. 1.5, 1.6)
lfl(t)
= (1-tz)e-tz;z_
Then from [14, p.372] (10.4.1) so that ( 10.4.2) Assume that f has an asymptotic expansion of the form ( 10.2.4 ), and satisfies 2
~
f(ro)
=
O(eaw ), ro--->
+oo,
( 10.4.3)
for some cr > 0. Therefore,
g(ro) := eihwj(ro) = O(eaw\ ro---> +oo.
(I 0.4.4)
Then by (10.2.9) and (10.4.1), we get n-l
It (a)=
L, d_,.M[Jliro 2e-w 12;s + A.]a-s-A. + 8~ (a), 2
s=O
where
( 10.4.5)
The Wavelet Transform
162
and (10.4.6) Similarly, we get
h(a) = vfn2(A+I)/ 2
%
ds( -I)s+A.-I2s/ 2r
( s+ ~ + 2)
a-s-A.+ s;(a),
(10.4.7)
where (10.4.8) Finally, using (10.2.1 ), (10.4.5) and (10.4.7), we have
(Wy,f)(b,a)=
2(A.+I)/2
Vii
n-l
Eds2sf2r
(
A +2)
s+ 2
{1+(-1r+A.-I}a-s-A+I/2
+ Dn(a),
(10.4.9)
where
Dn(a) = 2 3/ 2vln
fo', gn(ro)(aro) 2e-(aro) 2/ 2dro.
(10.4.10)
Existence theorem for formula (10.4.9) is as follows:
Theorem 10.4.1. Assume that
j( ro) satisfies conditions of Theorem 10.2.1. Then the result
(10.4.9) holds with
Dn(a) =
(~I) m 23/2vln /_~ g~m)(ro)((aro)2e-(aro)2/2/-mldro,
where n is the smallest positive integer such that A + n > m.
10.5
Asymptotic expansion of Haar wavelet transform
In this section we choose (see Fig.l.l, 1.2) 1 o:::;t
1/f(t)= { -Ilj2:::;t
0
otherwise.
Then from [14,p.368],
fi/(ro) =4ie-iro/2sin2roj4 = (J)
Although the condition {3
~(I-2eirof2+eiro).
> 0 is not satisfied in this case but the result
remains valid, cf. [91, p.753].
(10.5.1)
(J)
(I 0.1.6)-(1 0.1. 7)
163
Asymptotic Expansions of the Wavelet Transform for Large and Small Values of a
Clearly,
h(ro) = O(ro), as ro---+ Oo Assume that
j( ro)
(100502)
has an asymptotic expansion of the form ( 1002.4 )o
Using (100201) and (100501) we get /I (a)
=
r
eibw
lo i a
= -F(b)
i( ro)-1- (1- 2eiawj2 + eiaw)dro aro
-2i
. ~ eiaw !ooo e•bwf(ro)-dro, eiaw/2 ~ !ooo e'bwf(ro)--dro+i 0
o
o
aro
aro
where F(b) = (" eibwf(ro) droo
lo
ro
Then, (1 Oo2o6) and the generalized Mellin transform formula [93, Lemma 2, pol98]:
M[eit;z]
=
ei7rz/2r(z)
give
I,(a)
=~F(b)- ~ !ooo [.~dsros+A.- 2 +gn(ro)] eiaw/ dro 2
+ _ai
r
Jo
=!_F(b)
[t. {'E
dsros+A.-2 + gn(ro)] eiawdro
s=l
+ 2i a
a
dsr(s+A. -I)(aj2)-s-A.+Iei7r(s+A.)/Z
s=l
-(2ifat !ooo g~n)(ro)eiawf2dro}
+ !_ { - nE dsr(s+A. -1)(a)-s-A.+lei7r(s+A.)/2 a
s=l
+(i/a)n !ooo g~n)(ro)eiawdro} o
= !..F(b) + i a
n-l
L dsr(s +A. -
1 )a-s-A. (2s+A. - 1)ei7r(s+A.)/ 2
s=l
+ (i/at+l !ooo g~n)(ro)(eiaw _ 2n+leiawf2)droo
( 100503)
Notice that for existence of the Mellin transform in the above case we have to assume that
do=Oo Similarly,
The Wavelet Transform
164
n-1
+i
L d.vr(s +A- 1)a-s-A. (-1 y+A.-I (2s+A.- 1)ei7r(s+A.)/2
s=l
+ (ija)n+l fo"" g~n)( -m)(e-iaw _ 2n+le-iawf2)dm.
(10.5.4)
Finally, from (10.2.1),(10.5.3) and (10.5.4) we get
(W'I'f)(b,a) = ~f(-I)(b)
+ Dn(a),
(10.5.5)
where
and
Dn(a) = (i/at+l '{;
fo"" g~n)(m)(eiaw -2n+leiawf2dm.
(10.5.6)
Existence theorem for (10.5.5) is as follows:
Theorem 10.5.1. Assume that]( m) satisfies conditions of Theorem 10.2.1. Then the result (10.5.5) holds with
Dn(a)
= (ija)m+l '[; .l~ g~m)(m)(eiaw _ 2m+leiawf2)dm,
where n is the smallest positive integer such that A + n > m. 10.6
Asymptotic expansion for small a
Let us set a = I/ c, where c --+
+=. Then
(W'I'f)(b,a)
= Va {"" ceibcwj(cw)fi/(m)dm =
2n
1-oo
Va
[c .k{"" eibcwj(cm)fi/(m)dm+c lo{"" /hcwj(-cm)fi/(-m)dm]
2n
= Va 27r(/J+h), say.
(10.6.1)
For fixed b E JR, set (10.6.2)
165
Asymptotic Expansions of the Wavelet Transform for Large and Small Values of a
Then h plays the role of kernel of the integral transform. In this section we assume that 00
j( ro) "' [. brw-r-{3, ro----> +=, {3 > 0.
( 10.6.3)
r=O Then
L brro-r-{3, 00
h( ro) ,...., eibro
b =/:- 0.
(10.6.4)
r=O Let 00
i/f(ro)rv[csros+A.-i, W---->0, O
=
L csros+A.-i + i/fn(ro),
n ~ l.
( 10.6.5)
s=O Also, assume that
(10.6.6)
so that
h(ro)=O(roP), ro----.0, p+A.>O.
(10.6.7)
Then proceeding as in Section I 0.2, we get n-1
It =c[csM[h;s+A.]c-s-A.+D~(c),
(10.6.8)
s=O where
r
(10.6.9)
= c L Cs( -l)s+A.+!M[h( -ro);s+A.]c-s-A. + s;(c),
(10.6.10)
D~(c) =
lim c if!n(ro)h(cro)e-Ewdro. E->O+ lo
Similarly, n-1
[z
s=O where (10.6.11)
Finally, from equations (I 0.6.8) and ( 10.6.1 0), we get
(W11,J)(b,a) =
':{! {1 cs(M[h;s+ A.]+ (-1)s+A+lM[h( -ro);s + A.])as+A.-i + Dn(a)},
(10.6.12)
where
Dn(a)
=
lim
r if!n(aro)h(ro)e-Earodro+ E->O+}o lim r if!n( -aro)h( -ro)e-Earodro. (10.6.13)
E->O+}o
!66
The Wavelet Transform
Theorem 10.6.1. Assume that (i) ijl(m)(m) is continuous on(-=,=), where m is a non-
negative integer; (ii) ijl( m) has asymptotic expansion of the form (1 0.6.5) and the expansion is m times differentiable; (iii) f(m) satisfies (10.6.3) and (10.6.6) and (iv) as m
----+ oo,
w-!3 ijl~j) ( m) = 0( w-l-e) for j = 0, I, ... ,m and for e > 0. Under these condi-
tions the result (I 0.6.12) holds with Dn(a)
=
(-l)mam j_~ ijl(m)(am)(eibwj(m))(-m)dm,
(10.6.14)
where n is the smallest positive integer such that A + n > m. Example 10.6.2. Let us consider asymptotic expansion for small values of a, of the Mexican hat wavelet transform discussed in Section 10.4. In this case
= V2ii(amfe-(aw)2/2
ijl(am)
= V2ii
f: (-1 yazr+z w2r+2 r=O n-1
2rr! a2r+2w2r+2 =V2iiJ;(-1Y 2rr! +o/n(am);
(10.6.15)
and j( m) satisfies (10.6.3) and (10.6.6). Then, following the technique of proof of ( 10.6.12) and using formula [DkfY'
(W111.f)(b,a)
= (im/f(m),k E No [57, p.
39], it can be shown that
J;
1 n-1 ( 1)r { ;r! M[eibwj(m);2r+3]
= ..j2n
+ M[e-ibwj( -ro);2r+3] }a2r+S/Z + Dn(a) =-
V2iifa 2;r!D2r+2 .f(b)a2r+5/2 + Dn(a),
(10.6.16)
where Df denotes generalized derivative off and
Dn(a)
=,;a lim r eibw i( (1)) 1/fn (am )e-E(J) dm 21C E-->0+ JE-->0+ (10.6.17)
Example 10.6.3. Asymptotic expansion for small a of the Morlet wavelet transform can be obtained as follows. Here
lj/(m) = V2iie-(w-~)2/2
= V2iie-wJ;z
E(-1Ym2s E(~my s=O
2ss!
r=O
r!
Asymptotic Expansions of the Wavelet Transform for Large and Small Values of a
= ..fiie-w1il2
167
E
dso.i
s=O
~ dsro s + =-----( vr;:c L.ne -w1J/2 1... l!'n ro ) ,
=
(10.6.18)
s=O
where [s/2]
ds =
(_I )i (J.)s-2}
L 21 J.. s }=0 1(
_ 02
Assume, as in the previous example, that
.) 1 by [80,p.57(7)].
(10.6.19)
1 .
j( ro)
satisfies (I 0.6.3) and ( 10.6.6). Then, pro-
ceeding as in the above example, we find that (10.6.20)
n-1 ..fiie-ro&/2
=
L dsas+lf2e-isnf2(Ds f)(b) + Dn(a),
(10.6.21)
s=O
where (10.6.22) Example 10.6.4. Consider now the Haar wavelet transform. In this case
(10.6.23) Then we find that
(Wy,f)(b,a)
=
E
y'a {n-1 ·r+l ( I ) 2; l r! I - 2r-l M[eibw j(ro);r]
+( -I)r-1 M[e-ibw J( -ro); r]} ar-1 + Dn(a) = - nE r=2
~(I- 2r~l) vr-lf(b)ar-1/2 + Dn(a),
(10.6.24) (10.6.25)
where (10.6.26)
The Wavelet Transform
168
10.7 Asymptotic expansion for small a continued In this section we obtain asymptotic expansion of the wavelet transform given in the form (9 .1.1) when a
-+
instead of j and
0+. Naturally, in this case we have to impose conditions on
f
and 11'
fi/.
Now, Jet us write (9.l.l) in the form:
(Wy,f)(b,a) = c 1 / 2 [~ f(t + b)l!'(ct)dt, where c
(10.7.1)
= I I a -+ +oo and b is assumed to be a fixed real number.
Then setting g(t)
= f(t +b)
and h(t)
(Wy,f)(b,a)
= c 112
= l!'(t), we have
[fooo g(t)h(ct)dt+ [ 0 g(t)h(ct)dt]
= c 112 [lt
oo
+h] (say).
(10.7.2)
Assume that g(t) satisfies (10.1.2) and h(t) satisfies (10.1.3) and (10.1.4). Then from (10.1.6) it follows that
n-l It = I:CsM[lji;s + l.]c-s-A. s=O
+ 0~ (a),
(10.7.3)
where
O~(a) =
r= 8n(t)o/(tla)e-EtP dt;
Jim
e-.o+lo
(10.7.4)
and n-1
h =
L Cs( -l)s+A.-l M[ 11'( -t);s +A.]+ o;(a),
(10.7.5)
s=O where
o;(a)
=
Jim
r= 8n( -t)l!'( -tla)e-EtP dt.
e-.o+lo
(10.7.6)
From (10.7.2), (10.7.3) and (10.7.5) we get
n-l (W'I'.f)(b,a)
=
L c M[lji;s+l.]a +A.-If2 5
5
.1'=0
n-l
+ L c5 ( -l)s+A.-l M[l!'( -t);s + A.]as+A.-l/2 + On(a),
(10.7.7)
s=O where
On(a) = a- 1/ 2 { lim
r= 8n(t) IJI(t I a)e-et'' dt + e-->0+ lim r= 8n( -t) 11'( -t I a)e-etP dt}. lo
e-.O+ lo
(10.7.8)
169
Asymptotic Expansions of the Wavelet Transform for Large and Small Values of a
Example 10.7.1. Let us find again asymptotic expansion of Morlet wavelet transform for small a, using the above technique. Here
lf!(t)
=
eiWot-t 2/2.
Suppose that g(t) = .f(t +b) satisfies (1 0.1.2). Then from (10.7.3), using (9.5.4), we get n-1
h = [. CsM[lJi;s+ A.]c-s-A. + D~ (a) s=O n-1 =
L Cs Jo{"" eiWot-tz/2ts+A.-1dtas+A. + D~ (a)
s=O
0
n-1
= [. c5 r(s+A.)e-cofii4 D_ 5 _A.( -i~)as+A. + D~ (a).
(10.7.9)
s=O
Similarly, n-1 = [ . cs( -Iy+A.- 1r(s
h
+ A.)e-cofii4 D_s-A. (i~)as+A. + <5J(a).
(10.7.10)
s=O
Therefore, n-1
W(b, a)
=
L Csr(s +A. )e-cofi14 as+A.-If2(D_s-A. ( -i~) + (- I)s+A.-
1D-s-A.
(i~))
s=O
(10.7.11) where
Dn(a)
= a-1/2
{Ia= 8n(t)e-iWo(t/a)-(t-a)2/2dt + ~o= 8n( -t)eiWo(t/a)-(t+a)2/2dt}. (10.7.12)
10.7.1
The case A.= I
This is an interesting case in which asymptotic expansion can be expressed in terms of moments of the wavelet. Let 1f1 be real valued. Assume that f(t) possesses asymptotic expansion of the form (I 0.1.2) with A. = I; so that n-1
.f(t)
=
L Csl
5
+ .fn(t).
s=O
Then
(Wvt.f)(b,a) =a- 112 j_~ f(t +b)lf!(t/a)dt =a- 112
[lo= .f(t+b)lf!(tja)dt+ ~o= .f(-t+b)lf!(-t/a)dt]
(10.7.13)
170
The Wavelet Transfonn
(10.7.14) where
Jlk
= j_~rklJI(t)dt
is the k-th moment of the wavelet lJI and
On= (-I tan-l/2 j_~ J~n) (t) lJI(-n)
(10.7.15)
c
~b) dt.
(10.7.16)
If we assume that the mean value and first (n- 2) moments of lJI vanish,
j_~ tklJI(t)dt =
0,
0:::;; k:::;; n- 2
( 10.7.17)
and the (n- I )th moment exists and
j_~ tn-llJI(t )dt "I 0,
(10.7.18)
then
and asymptotic expansion coincides with Taylor series expansion of f; hence Cn-I = j(n-l)(b) /2 Tn"=T)! and On= O(an-I ). Thus we derive the result
(wll' f)(b ' a)=
j(n-l)(b) I I - Qn-l/2+0(an-lf2) (n- I)! r-n I '
which has been obtained by Rieder [82] in his study of high frequency behaviour of continuous wavelet transforms.
Bibliography
1. K.F. Andersen, On the range and inversion of fractional integrals in weighted spaces, Proc. Roy. Soc. (Edinburgh) 92A ( 1982), 51-64. 2. F. Bastin and P. Laubin, Regular compactly supported wavelets in Sobolev spaces, Duke Math. Jour. 87 (3) (1997), 481-508. 3. B.L.J. Braaksma and B. Meu1enbeld, Integral transforms with generalized Legendre functions as kernels, Composito Mathematica 18 (1967), 235-287. 4. B. L. J. Braaksma and A. Schuitman, Some classes of Watson transforms and related integral equations for generalized functions, SIAM J. Math. Anal. 7 (6) (1976), 771-798. 5. G. Bjorck, Linear partial differential operators and generalized distributions, Ark. Math. 6 (1966), 351-407. 6. S. Bochner and K. Chandrasekharan, Fourier Transforms, Princeton University Press, 1949. 7. P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, Acad. Press, New York, 1971. 8. P. L. Butzer and R. L. Stens, The operational properties of the Chebyshev transform I, Funct. Approximate Comment. Math. 5 (1977), 129-160. 9. P. L. Butzer and R. L. Stens, Chebyshev transform methods in the theory of best algebraic approximation, Abh. Math. Sem. Univ. Hamburg. 45 (1976), 165-190. I 0. F. M. Cholewinski, A Hankel convolution complex inversion theory, Mem. Amer. Math. Soc. 58 (1965). 11. F. M. Cholewinski and D. T. Haimo, The dual Poisson-Laguerre transform, Trans. Amer. Math. Soc. 144 (1969), 271-300. 12. C.K. Chui, An Introduction to Wavelets, Academic Press, 1992. 13. I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, Pennsylvania, USA, 1992. 14. L. Debnath, Wavelet Transforms and their Applications, Birkhiiuser, 2002. 15. J. Dziubanski and E. Hernaridez, Band-limited wavelets with subexponential decay, Canad. Math. Bull. 41 (4) (1998), 398-403. 16. A. Erdelyi (Editor), Higher Transcendental Functions, Vol. I, McGraw-Hill Book Co., Inc., New York, 1953. 17. A. Erdelyi (Editor), Higher Transcendental Functions, Vol. II, McGraw-Hill Book Co., Inc., New York, 1953. I 8. A. Erdelyi (Editor), Tables of Integral Transforms, Vol. I, McGraw-Hill Book Co., Inc., New York, 1954. 19. A. Erdelyi (Editor), Tables of Integral Transforms, Vol. II, McGraw-Hill Book Co., Inc., New York, 1954. 20. A. Friedman, Generalized Functions and Partial Dfferential Equations, Prentice-Hall Inc., Englewood Cliffs, NJ, 1963.
171
172
The Wavelet Transform
21. I. M. Gel'fand and G. E. Shilov, Generalized Functions, Vol. II, Acad. Press, New York, 1968. 22. I. M. Gel'fand and G. E. Shilov, Generalized Functions, Vol. III, Acad. Press, New York, 1967. 23. H. J. Glaeske, On the convolution structure of Hermite transforms - a survey, in Proceedings of International Conference on Geometry, Analysis and Applications (Ed. R. S. Pathak), World Scientific, Singapore, 2001,217-225. 24. H. J. Glaeske, A. convolution connected with the Kontorovich-Lebedev transform of generalized functions, Math. Z. 193 (1986), 67-78. 25. H. J. Glaeske and A. Hess, On the convolution theorem of the Mehler-Fock transform of generalized functions, Math. Nachr. 136 (1988), 119-129. 26. M. L. Glasser, Some useful properties of Hilbert transform, SIAM J. Math. Anal.l5(5) (1984), 1228-1230. 27. R. R. Goldberg, Certain operators and Fourier transform on L 2 , Proc. Amer. Math. Soc. I 0 ( 1959), 385-390. 28. D. T. Haimo, Integral equations associated with Hankel convolutions, Trans. Amer. Math. Soc. 116 (1965), 330-375. 29. R. A. Handelsman and J. S. Lew, Asymptotic expansion of Laplace convolutions for large argument and tail densities of certain sums of random variables, SIAM J. Math. Anal. 5 ( 1974), 425--451. 30. Eleonor Harboure, Roberto A. Macias and Carlos Segovia, Roundedness of fractional integral operators on U' with different weights, Trans. Amer. Math. Soc. 285 (2) (1984), 629-647. 31. G. H. Hardy and E. C. Titchmarsh, A class of Fourier kernels, Proc. London Math. Soc. 36 (1933), 116-155. 32. E. Hernandez and G. Weiss, A First Course on Wavelets, CRC Press, Boca Raton, Florida. 1996. 33. D.L.J. Herson and P. Heywood, On the range of some fractional integrals, J. London Math. Soc. (2), 8(1974), 607-614. 34. P. Heywood, On a modification of the Hilbert transform, J. London Math. Soc. 42 (1967), 641645. 35. I. I. Hirschman Jr., Variation diminishing Hankal transforms, J. Analyse Math. 8 (1960/61 ), 307336. 36. M. Holschneider, Wavelets: An Analysis Tool, Clarendon Press, Oxford, 1995. 37. L. Hormander, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, Berlin Heidelberg, 1983. 38. L. Hormander, The Analysis of Linear Partial Differential Operators II, Springer-Verlag, Berlin Heidelberg, 1983. 39. D. S. Jones, Infinite integrals and convolution, Proc. Roy. Soc. London A 371 (1980), 479-508. 40. D. S. Jones, The Theory of Generalized Functions, Cambridge University Press, Cambridge, 1982. 41. K. Kaiser, A Friendly Guide to Wavelets, Birkhauser, 1994. 42. W. Kecks, The Convolution Product and Some Applications, D. Reidel Publishing Co., Dordrecht, Holland, 1982. 43. H. Kober, A modification of Hilbert transforms, the Weyl integral and fractional integrals, J. London Math. Soc. 42(1967), 42-50. 44. H. Kober, New properties of the Weyl extended integral, Proc. London Math. Soc. (3) 21 ( 1970), 557-575. 45. T. H. Koornwinder (ed.), Wavelets: An Elementary Treatment of Theory and Applications, World Scientific, 1993. 46. X. Li and R. Wong, Error bounds for asymptotic expansions of Laplace convolutions, SIAM J. Math. Anal. 25 (1994), 1537-1553. 47. E. H. Lieb and M. Loss, Analysis, Narosa Publishing House, New Delhi, 1998. 48. I. Marrero and J. J. Betancor, Hankel convolution of generalized functions and their applications,
Bibliography
173
Rendiconti de Matematica Y dell'a Sue applicazion, Series VII 15(1995), 351-380. 49. A.C. McBride, Fractional Calculus and Integral Transforms of Generalized Functions, Pitman, 1979. 50. G .0. Okikiolu, A generalization of the Hilbert transform, J. London Math. Soc. 40 (1965), 2730. 51. G. 0. Okikiolu, Fourier transform and the operator Ha, Proc. Camb. Phil. Soc. 62 (1966), 73-78. 52. G.O. Okikiolu, Fractional integrals of the Ha-type, Quart. J. Math. 18 (2) ( 1967), 33-42. 53. J.N. Pandey, The Hilbert Transform of Schwartz Distributions and Applications, Wileyinterscience, New York, 1996. 54. R.S. Pathak, Some fractional integrals of generalized functions, Prog. of Math. 24 {1/2) ( 1990), 129-142. 55. R.S. Pathak, Fractional integrals on D~ -spaces, J. Fractional Calculus 9 (1996), 93-102. 56. R. S. Pathak, Generalized Sobolev spaces and pseudo-differential operators on spaces of ultradistributions, in: M. Morimoto and T. Kawai (Eds.), Structure of Solutions of Differential Equations, World Scientific, Singapore (1996), 343-368. 57. R. S. Pathak, Integral Transforms of Generalized Functions and Their Applications, Amsterdam: Gordon and Breach, 1997. 58. R. S. Pathak, Continuity and inversion ofthe wavelet transform, Integral Transforms and Special Functions 6(1-4) (1998), 85-93. 59. R. S. Pathak, Asymptotic expansion of the wavelet transform, in Industrial and Applied Mathematics (Eds. A. H. Siddiqui and K. Ahmad) Narosa Publishing House, New Delhi ( 1998), 43-50. 60. R. S. Pathak, Abelian theorems for the wavelet transform, in Wavelets and Allied Topics (Eds. P. K. Jain et al.), Narosa Publishing House, New Delhi, 2001. 61. R. S. Pathak, The wavelet transform in Fourier space, in Analysis and Applications (Eds. H. P. Dikshit and P. K. Jain) Narosa Publishing House, New Delhi (2002), 221-233. 62. R. S. Pathak, The wavelet transform of distributions, Tohoku Math. J. 56 (2004 ), 411-421. 63. R. S. Pathak, Wavelets in a generalized Sobolev space, Computers and Maths. with Applications, 49 (2005), 823-839. 64. R. S. Pathak and M. M. Dixit, Continuous and discrete Bessel wavelet transforms, J. Comput. and Appl. Math. 160 (2003), 241-250. 65. R. S. Pathak and M. M. Dixit, Bessel wavelet transform on certain function and distribution spaces, J. Analysis and Appl. 1 (2003), 65-83. 66. R. S. Pathak and C. P. Pandey, The wavelet transform in a generalized Sobolev space, J. Indian Math. Soc. 73 (3-4) (2006), 235-247. 67. R. S. Pathak and C. P. Pandey, Chebyshev wavelet transforms, J. Wavelet Theory and Appl. I (I) (2007), 13-30. 68. R. S. Pathak and G. Pandey, Calderon's reproducing formula for Hankel convolution. Inter. J. Math. and Math. Sci. volume 2006, Article ID 24217, pages 1-7. 69. R.S. Pathak and G. Pandey, Wavelet transform on spaces of type W, Rocky Mountain J. Math. 39 (2) (2009), 619-631. 70. R. S. Pathak and Ashish Pathak, On convolution for wavelet transform, International Journal for Wavelets, Multiresolution and Information Processing 6 (5) (2008). 739-747. 71. R. S. Pathak and Ashish Pathak, Asymptotic expansion of the wavelet transform with error term (communicated). 72. R. S. Pathak and Ashish Pathak, Asymptotic expansion of the wavelet transform for small a (communicated). 73. R. S. Pathak and S.K. Singh, The wavelet transform on spaces of type S, Proc. Roy. Soc. (Edinburgh) 136A (2006), 837-850. 74. R. S. Pathak and S. K. Singh, The wavelet transform on spaces of type U, Advances in Algebra
174
The Wavelet Transform
and Analysis 1(3) (2006), 1-12. 75. R. S. Pathak and S. K. Singh, Roundedness of the wavelet transform in certain function spaces, J. Inequalities in Pure and Applied Math. 8 (2007). 76. R. S. Pathak and S. K. Singh, lnfraexponential decay of wavelets, Proc. Nat. Acad. Sci. India Sect. A 78 (2008), 155-162. 77. R. S. Pathak and S. Tiwari, Watson convolution of distributions, Int. J. Math. Sci. 3 (I) (2004), 143-155. 78. R. S. Pathak and S. K. Upadhyay, WP-spaces and Fourier transform, Proc. Amer. Math. Soc. 121 ( 1994 ), 733-738. 79. V. Perrier and C. Basdevant, Besov norms in terms of the continuous wavelet transform: Application to structure functions, Math. Models and Methods in Applied Sciences 6 (1991 ), 875-894. 80. E. D. Rainville, Special Functions, The Macmillan Co., New York (1960). 81. A. Rieder, The wavelet transform on Sobolev spaces and its approximation properties, Numer. Math. 58(1991), 875-894. 82. A. Rieder, The high frequency behaviour of continuous wavelet transforms, Applicable Analysis 52 (1994), 125-141. 83. E. Riekstins, Asymptotic representation of certain types of convolution integrals, Laivian Math. Year Book 8 (1970), 223-239. 84. L. Rodino, Linear Partial Differential Operators in Gevrey Spaces, World Scientific Publishing Co. Pte. Ltd., 1993. 85. I. N. Sneddon, The Use oflntegral Transforms, McGraw-Hill Book Co., New York, 1972. 86. R. L. Stens and M. Wehrens, Legendre transform methods and best algebraic approximation, Arbeitsbericht, Lehrstuhl A fiir Mathematik, Rheinisch Westfalische Technische Hochschule Aachen, Germany, 1978. 87. E. C. Titchmarsh, Introduction to Theory of Fourier Integrals, Oxford: Clarendon Press, 1967. 88. K. Trimeche, Generalized Wavelets and Hypergroups, Gordon and Breach Science Publishers, Amsterdam, 1977. 89. R. L. Vandewetering, Variation diminishing Fourier-Jacobi transforms, SIAM J. Math. Anal. 6 (1975), 774-783. 90. M. W. Wong, An Introduction to Pseudodifferential Operators, World Scientific, Singapore, 1991. 91. R. Wong. Explicit error terms for asymptotic expansions of Mellin convolutions, J. Math. Anal. Appl. 72 (1979), 740-756. 92. R. Wong, Error bounds for asymptotic expansions of integrals, SIAM Review 22 (4) (1980), 401-435. 93. R. Wong, Asymptotic Approximations of Integrals, Acad. Press, New York, 1989. 94. R. Wong and J.P. McClure, Generalized Mellin convolutions and their asymptotic expansions, Canad. J Math. 36 (1984), 924-960. 95. S. Zaidman. Distributions and Pseudo-differential Operators, Longman, 1991. 96. A. I. Zayed, Handbook of Function and Generalized Function Transformations, CRC Press, Boca Raton, Florida, USA, 1996. 97. A. H. Zemanian, Distribution Theory and Transform Analysis, McGraw-Hill Book Co., New York, 1965. 98. A. H. Zemanian, Generalized Integral Transformations, lnterscience Publishers, New York, 1968.
Subject Index
Abel limit,
18, 19
Abel summable,
Aerofoil problems,
18 Band limited wavelets,
Abelian theorems for wavelet transform of functions,
Basic generalized function,
59
of distributions,
Basic wavelet,
10, 30, 55, 96,
Bell function,
118, 132
Bjorck space,
6, 26, 35, 43,
44, 136 Approximation theory, Approximation theorem,
94, 103 117
Chebyshev transform,
117
100
Asymptotic expansion,
16, 17, 18, 19,
Confluent hypergeometric fuction, Continuous wavelet transform, Convex inequality,
20, 137, 149, 155 of generalized Stieltjes transform,
142
162
of Mexican hat wavelet transform,
Convolution,
120
12
83
1, 4, 5, 109, 129, 137
of distributions,
8
for Chebyshev transform, 150,
161
for wavelet transform,
117
118, 119
for discrete wavelet transform,
of wavelet transform,
49,
52
99
Asymptotic behaviour,
of Morlet wavelet transform,
117
Compositon of wavelet transforms,
Approximation of wavelet transform,
of Haar wavelet transform,
57
Chebyshev convolution transform,
135, 136
135, 136
Asymptotic order,
81
Chebyshev polynomial,
109
Approximation property of wavelet convolution,
11
Bessel potential,
96
Approximation property,
110, 119,
121
61
Admissibility condition, Analysing wavelet,
80, 81, 82
120
158
149, 158
Dilation parameter,
27, 28, 29
100
Dirac delta function (distribution), 175
2
176
The Wavelet Transform
Distribution,
1, 2, 3, 4
Haar wavelet,
10, 11, 107
Distributional approach,
19
Haar wavelet transform,
162, 167
Distributional derivative,
2, 106
Heaviside unit function,
3
6, 7
Hermite polynomial,
150
Distributional technique for asymptotic
High energy physics,
1
expansion,
Hilbert transform,
Distributional Fourier transform, 19, 141
Discrete wavelet transform,
15, 109, 120 Incomplete Beta integral,
Dispersion relation, Dominated convergence theorem, Dual frame,
102
121
Dual Poisson Laguerre convolution, Transform,
1, 8, 9, 29, 31, 47,48
116
116
138
Inverse Chebychev transform,
117
Inverse Mellin transform,
18, 30
Inverse Hankel transform,
114
Inverse wavelet transform,
15, 29, 32,
34,48 Entire analytic function, Euler's constant,
84
Inversion theorem for Fourier transform,
153
Existence theorems,
5 l,u-transform,
122, 132
Isometry, Fourier transform,
1, 5, 6, 14, 112, 130
Fourier type convolution, Fractional integrals, Frame,
Initial value theorem,
59, 64
137 Kontorovich-Lebedev transform,
29
121
Gaussian function,
115
13, 93
K,u-transform,
115
Laplace convolution,
6
Generalized convolution,
109, Ill
Generalized Mellin transform, Generalized Sobolev space,
155
93, 94, 96,
113, 138, 141
Laplace convolution transform, Laplace type convolution, Laplace transform,
Generalized Stieltjes transform,
137,
141, 142 109,
Legendre transform,
115 115
Localconvergence, Localization operator,
Hankel transform,
12
114, 115 114
Mehler-Fock transform,
Hardy-Littlewood-Sobolev inequality,
Mellin convolution,
124
Mellin transform,
142
103
Local generalized Sobolev space,
Ill Hankel convolution,
138
Lebesgue convergence theorem,
Generalized translation operator,
137
113
Legendre convolution,
135
115
115
113 18, 30, 113
103
177
Subject Index
II, 13, 31, 65
Mexican hat wavelet,
I I, 12
Shan on wavelet,
137
Mellin type convolution,
65, 161,
Mexican hat wavelet transform,
98
Schwarzs inequality,
18
Mellin transform technique,
I, 21
Signal processing,
56, 57, 93,
Sobolev spaces, generalized, 94,96
166 Mother wavelet,
9, II
Sobolev embedding theorem,
Morlet wavelet,
132, 158, 159
Spaces of typeS,
158, 166
Morlet wavelet transform, Mutually dual,
90
67, 68, 69
Spaces of type W,
83, 84, 85
Sturm-Liouville boundary value problem,
85
117 138
Non-commutative convolution, Operation transform formula,
Translation,
155
Oscillatory function, Paley-Wiener theorem,
109, Ill, 121
Transated version,
31, I 07
Ultradistributions,
75
80
Parabolic cy Iinder function,
151, 160
Parseval formula for Fourier transform,
Wavelet,
6,49
Wavelet coefficients, 23, 24, 25
Parseval relation,
4, 6, 7, 8, 19
Tempered distribution,
16
Orthonormal basis,
117 117
Sturm-Liouville transform,
7
I 05
Order of convergence,
Sturm-Liouville convolution,
9, 10, II, 12, 13, 14, 31, 32, 33
Wavelet transform,
Parseval relation for wavelet transform,
16 12, 14, 15, 47, 49,
94,118,129,137,155,158
14,33,34,43
Abelian theorems for,
Plancherel formula for wavelet transform,
Adjoint of,
15, 133
Approximation of,
Potential theory,
59, 61
97 99, 107 I 00
Asymptotic behaviour of,
Product wavelet transform,
49, 53, 55
Composition of,
Pseudodifferential operator,
49
Continuous, 12, 14, 118 Convolution for, 109, 118, 119, 120,
Reconstruction formula,
15, 118
Riesz fractional integral operator, Rodrigues formula, Schwartz space, 57,58,59,61,63
131, 132, 135
17
Remainder term,
49,52
22
150
3, 4, 6, 7, 49, 50, 53, 55,
Discrete,
15, 120 I 03
Local convergence of, Parseval relation for,
14, 33, 34, 43, 133
Plancherrel formula for,
15, 133
of generalized functions,
74
178
The Wavelet Transform
of Gevrey functions of compact support,
order of convergence of,
78
range of,
of tempered distributions, on spaces of type S, on spaces of type W,
50
67
I 05
34, 39, 97
reconstruction formula for, translation for,
109, 119, 121
83
on weighted Sobolev space,
57
Young's inequality,
15, 118
84, 87, 88, 130
