Christian Berg Jens Peter Reus Christensen Paul Ressel
Harmonic Analysis on Semigroups Theory of Positive Definite and Related Functions
Springer-Verlag New York Berlin Heidelberg Tokyo
Christian Berg Jens Peter Reus Christensen Matematisk Institut K0benhavns U niversitet Universitetsparken 5 DK-2100 K0benhavn ~ Denmark
Editorial Board P. R. Halmos Managing Editor Department of Mathematics Indiana University Bloomington, IN 47405 U.S.A.
Paul Ressel Mathematisch-Geographische FakulHit Katholische UniversiHit EichsHitt Residenzplatz 12 D-8078 EichsHitt Federal Republic of Germany
F. W. Gehring
C. C. Moore
Department of Mathematics University of Michigan Ann Arbor, MI 48109 U.S.A.
Department of Mathematics University of California at Berkeley Berkeley, CA 94720 U.S.A.
AMS Classification (1980) Primary: 43-02, 43A35 Secondary: 20M14, 28C15, 43A05, 44AI0, 44A60, 46A55, 52A07, 60E 15 Library of Congress Cataloging in Publication Data Berg, Christian Harmonic analysis on semigroups. (Graduate texts in mathematics; 100) Bibliography: p. Includes index. 1. Harmonic analysis. 2. Semigroups. I. Christensen, Jens Peter Reus. II. Ressel, Paul. III. Title. IV. Series. QA403.B39 1984 515'.2433 83-20122 With 3 Illustrations.
© 1984 by Springer-Verlag New York Inc. All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, U.S.A. Typeset by Composition House Ltd., Salisbury, England. Printed and bound by R. R. Donnelley & Sons, Harrisonburg, Virginia. Printed in the United States of America.
9 87 6 54 32 1
ISBN 0-387-90925-7 Springer-Verlag New York Berlin Heidelberg Tokyo ISBN 3-540-90925-7 Springer-Verlag Berlin Heidelberg New York Tokyo
Preface
The Fourier transform and the Laplace transform of a positive measure share, together with its moment sequence, a positive definiteness property which under certain regularity assumptions is characteristic for such expressions. This is formulated in exact terms in the famous theorems of Bochner, Bernstein-Widder and Hamburger. All three theorems can be viewed as special cases of a general theorem about functions qJ on abelian semigroups with involution (S, +, *) which are positive definite in the sense that the matrix (qJ(sj + Sk)) is positive definite for all finite choices of elements Sl' . . . , Sn from S. The three basic results mentioned above correspond to (IR, +, x* = -x), ([0,00[, +, x* = x) and (No, +, n* = n). The purpose of this book is to provide a treatment of these positive definite functions on abelian semigroups with involution. In doing so we also discuss related topics such as negative definite functions, completely monotone functions and Hoeffding-type inequalities. We view these subjects as important ingredients of harmonic analysis on semigroups. It has been our aim, simultaneously, to write a book which can serve as a textbook for an advanced graduate course, because we feel that the notion of positive definiteness is an important and basic notion which occurs in mathematics as often as the notion of a Hilbert space. The already mentioned Laplace and Fourier transformations, as well as the generating functions for integervalued random variables, belong to the most important analytical tools in probability theory and its applications. Only recently it turned out that positive (resp. negative) definite functions allow a probabilistic characterization in terms of so-called Hoeffding-type inequalities. As prerequisites for the reading of this book we assume the reader to be familiar with the fundamental principles of algebra, analysis and probability, including the basic notions from vector spaces, general topology and abstract
vi
Preface
measure theory and integration. On this basis we have included Chapter 1 about locally convex topological vector spaces with the main objective of proving the Hahn-Banach theorem in different versions which will be used later, in particular, in proving the Krein-Milman theorem. We also present a short introduction to the idea of integral representations in compact convex sets, mainly without proofs because the only version of Choquet's theorem which we use later is derived directly from the Krein-Milman theorem. For later use, however, we need an integration theory for measures on Hausdorff spaces, which are not necessarily locally compact. Chapter 2 contains a treatment of Radon measures, which are inner regular with respect to the family of compact sets on which they are assumed finite. The existence of Radon product measures is based on a general theorem about Radon bimeasures on a product of two Hausdorff spaces being induced by a Radon measure on the product space. Topics like the Riesz representation theorem, adapted spaces, and weak and vague convergence of measures are likewise treated. Many results on positive and negative definite functions are not really dependent on the semigroup structure and are, in fact, true for general positive and negative definite matrices and kernels, and such results are placed in Chapter 3. Chapters 4-8 contain the harmonic analysis on semigroups as well as a study of many concrete examples of semigroups. We will not go into detail with the content here but refer to the Contents for a quick survey. Much work is centered around the representation of positive definite functions on an abelian semigroup (S, +, *) with involution as an integral of semicharacters with respect to a positive measure. It should be emphasized that most of the theory is developed without topology on the semigroup S. The reason for this is simply that a satisfactory general representation theorem for continuous positive definite functions on topological semigroups does not seem to be known. There is, of course, the classical theory of harmonic analysis on locally compact abelian groups, but we have decided not to include this in the exposition in order to keep it within reasonable bounds and because it can be found in many books. As described we have tried to make the book essentially self-contained. However, we have broken this principle in a few places in order to obtain special results, but have never done it if the results were essential for later development. Most of the exercises should be easy to solve, a few are more involved and sometimes require consultations in the literature referred to. At the end of each chapter is a section called Notes and Remarks. Our aim has not been to write an encyclopedia but we hope that the historical comments are fair. Within each chapter sections, propositions, lemmas, definitions, etc. are numbered consecutively as 1.1,1.2,1.3, ... in §1, as 2.1,2.2,2.3, ... in §2, and so on. When making a reference to another chapter we always add the number of that chapter, e.g. 3.1.1.
vii
Preface
We have been fascinated by the present subject since our 1976 paper and have lectured on it on various occasions. Research projects in connection with the material presented have been supported by the Danish Natural Science Research Council, die Thyssen Stiftung, den Deutschen Akademischen Austauschdienst, det Danske U ndervisningsministerium, as well as our home universities. Thanks are due to Flemming Topsq,e for his advice on Chapter 2. We had the good fortune to have Bettina Mann type the manuscript and thank her for the superb typing.
March 1984
CHRISTIAN BERG lENS PETER REUS CHRISTENSEN PAUL RESSEL
Contents
CHAPTER 1
Introduction to Locally Convex Topological Vector Spaces and Dual Pairs §1. Locally Convex Vector Spaces §2. Hahn-Banach Theorems §3. Dual Pairs Notes and Remarks
1 1 5 11 15
CHAPTER 2
Radon Measures and Integral Representations §1. §2. §3. §4. §5.
Introduction to Radon Measures on Hausdorff Spaces The Riesz Representation Theorem Weak Convergence of Finite Radon Measures Vague Convergence of Radon Measures on Locally Compact Spaces Introduction to the Theory of Integral Representations Notes and Remarks
16 16 33 45 50 55 61
CHAPTER 3
General Results on Positive and Negative Definite Matrices and Kernels
66
§1. Definitions and Some Simple Properties of Positive and Negative Definite Kernels §2. Relations Between Positive and Negative Definite Kernels §3. Hilbert Space Representation of Positive and Negative Definite Kernels Notes and Remarks
66 73 81 84
Contents
X
CHAPTER 4
Main Results on Positive and Negative Definite Functions on Semigroups §1. Definitions and Simple Properties §2. Exponentially Bounded Positive Definite Functions on Abelian Semigroups §3. Negative Definite Functions on Abelian Semigroups §4. Examples of Positive and Negative Definite Functions §5. r-Positive Functions §6. Completely Monotone and Alternating Functions Notes and Remarks
86 86 92 98 113 123 129 141
CHAPTER 5
Schoenberg-Type Results for Positive and Negative Definite Functions §1. §2. §3. §4. §5.
Schoenberg Triples Norm Dependent Positive Definite Functions on Banach Spaces Functions Operating on Positive Definite Matrices Schoenberg's Theorem for the Complex Hilbert Sphere The Real Infinite Dimensional Hyperbolic Space Notes and Remarks
144 144 151 155 166 173 176
CHAPTER 6
Positive Definite Functions and Moment Functions §1. §2. §3. §4. §5.
Moment Functions The One-Dimensional Moment Problem The Multi-Dimensional Moment Problem The Two-Sided Moment Problem Perfect Semigroups Notes and Remarks
178 178 185 190 198 203 222
CHAPTER 7
Hoeffding's Inequality and Multivariate Majorization §1. The Discrete Case §2. Extension to Nondiscrete Semigroups §3. Completely Negative Definite Functions and Schur-Monotonicity Notes and Remarks
226 226 235 240 250
CHAPTER 8
Positive and Negative Definite Functions on Abelian Semigroups Without Zero §1. Quasibounded Positive and Negative Definite Functions §2. Completely Monotone and Completely Alternating Functions Notes and Remarks
252 252 263 271
References
273
List of Symbols
281
Index
285
CHAPTER 1
Introduction to Locally Convex Topological Vector Spaces and Dual Pairs
§1. Locally Convex Vector Spaces The purpose of this chapter is to provide a quick introduction to some of the basic aspects of the theory of topological vector spaces. Various versions of the Hahn-Banach theorem will be used later in the book and the exposition therefore centers around a fairly detailed treatment of these fundamental results. Other parts of the theory are only sketched, and we suggest that the reader consult one of the many books on the subject for further information, see e.g. Robertson and Robertson (1964), Rudin (1973) and Schaefer (1971). 1.1. We assume that the reader is familiar with the concept of a vector space E over a field IK, which is always either IK = IR or IK = C, and of a topology (!) on a set X, where (!) means the system of open subsets of X. Generally speaking, whenever a set is equipped with both an algebraic and a topological structure, we will require that the structures match in the sense that the algebraic operations become continuous mappings. To be precise, a vector space E equipped with a topology (!) is called a topological vector space if the mappings (x, y) ~ x + y of E x E into E and (A, x) ~ AX of IK x E into E are continuous. Here it is tacitly assumed that E x E and IK x E are equipped with the product topology and IK = IR or IK = C with its usual topology. A topological vector space E is, in particular, a topological group in the sense that the mappings (x, y) ~ x + y of E x E into E and x ~ - x of E into E are continuous. For each u E E the translation L u : X ~ X + u is a homeomorphism of E, so if fJl is a base for the filter 0/1 of neighbourhoods of zero, then u + fJl is a base for the filter of neighbourhoods of u. Therefore the whole topological structure of E is determined by a base of neighbourhoods of the origin.
2
1. Introduction to Locally Convex Topological Vector Spaces and Dual Pairs
A subset A of a vector space E is called absorbing if for each x E E there exists some M > 0 such that x E AA for all A E lK with IAI ~ M; and it is called balanced, if AA f; A for all A E lK with I AI ~ 1. Finally, A is called absolutely convex, if it is convex and balanced.
1.2. Proposition. Let E be a topological vector space and let 0/1 be the filter of neighbourhoods of zero. Then: (i) every V E 0/1 is absorbing; (ii) for every V E 0/1 there exists V E 0/1 with V + V f; U; (iii) for every V E 0/1, b(V) = nlJlI~ 1 J1V is a balanced neighbourhood of zero
contained in V.
°
PROOF. For a E E the mapping A ~ Aa of lK into E is continuous at A = and this implies (i). Similarly the continuity at (0,0) of the mapping (x, y) ~ x + y implies (ii). Finally, by the continuity of the mapping (A, x) ~ AX at (0, 0) E lK x E we can associate with a given U E 0/1 a number e > and V E 0/1 such that AV f; U for I AI ~ c. Therefore
°
eV
f;
b(V)
~
V
so V contains the balanced set b(V) which is a neighbourhood of zero because e V is so, x ~ ex being a homeomorphism of E. D From Proposition 1.2 it follows that in every topological vector space the filter 0/1 has a base of balanced neighbourhoods. A topological vector space need not have a base for 0/1 consisting of convex sets, but the spaces we will discuss always have such a base.
1.3. Definition. A topological vector space E over lK is called locally convex if the filter of neighbourhoods of zero has a base of convex neighbourhoods. 1.4. Proposition. In a locally convex topological vector space E the filter of neighbourhoods of zero has a base gj with the following properties: (i) Every V E gj is absorbing and absolutely convex. (ii) If V E gj and A 9= 0, then AV E (JI.
Conversely, given a base (JI for a filter on E with the properties (i) and (ii), there is a unique topology on E such that E is a (locally convex) topological vector space with gj as a base for the filter of neighbourhoods of zero. PROOF. If V is a convex neighbourhood of zero then b(V) is absolutely convex. If gjo is a base of convex neighbourhoods, then the family gj = {Ab( V) I V E gj 0' A 9= o} is a base satisfying (i) and (ii). Conversely, suppose that gj is a base for a filter ff on E and satisfies (i) and (ii). Then every set V E ff contains zero. The only possible topology on E which makes E to a topological vector space, and which has ff as the filter of neighbourhoods of zero, has the filter a + ff as filter of neigh-
3
§l. Locally Convex Vector Spaces
bourhoods of a E E. Calling a nonempty subset G s; E "open" if for every a E G there exists U E gj such that a + U S; G., it is easy to see that these "open" sets form a topology with a + f7 as the filter of neighbourhoods of a., and that E is a topological vector space. D In applications of the theory of locally convex vector spaces the topology on a given vector space E is often defined by a family of seminorms.
1.5. Definition. A function p: E following properties:
~
[0,
00 [
is called a seminorm if it has the
(i) homogeneity:p(:1x) = 1:1lp(x)for:1E lK,XEE; (ii) subadditivity: p(x + y) ~ p(x) + p(y) for x, y E E. If, in addition, P- 1( {O}) = {O}, then P is called a norm. If P is a seminorm and (X > 0 then the sets {x EEl p(x) < (X} are absolutely convex and absorbing. For a nonempty set A S; E, we define a mapping PA: E ~ [0, 00] by
PA(X) = inf{:1 > Olx E :1A} (where PA(X) = 00, if the set in question is empty). The following lemma is easy to prove. 1.6. Lemma. If A
S;
E is
(i) absorbing, then PA(X) < 00 for x E E; (ii) convex, then PAis subadditive; (iii) balanced, then PAis homogeneous, and
{x E ElpA(X) < I}
S;
A
S;
{x E ElpA(X)
~
I}.
If A satisfies (i)-(iii) then PA is called the seminorm determined by A. A seminorm P satisfies Ip(x) - p(y) I ~ p(x - y). In particular, if E is a topological vector space then P is continuous if and only if it is continuous at 0 and this is equivalent with {xlp(x) < (X} being a neighbourhood of zero for one (and hence for all) (X > o. We will now see how a family (Pi)iel of seminorms on a vector space E induces a topology on E.
1.7. Proposition. There exists a coarsest topology on E with the properties that E is a topological vector space and each Pi is continuous. Under this topology E is locally convex and the family of sets
is a base for the filter of neighbourhoods of zero.
4
1. Introduction to Locally Convex Topological Vector Spaces and Dual Pairs
PROOF. Let gj denote the above family of sets. Then gj is a base for a filter on E having the properties (i) and (ii) of Proposition 1.4, and the unique topology asserted there is the coarsest topology on E making E to a topological vector space in which each Pi is continuous. D The above topology is called the topology induced by the family (pJiEI of seminorms. Note that in this topology a net (x a ) from E converges to x if and only if lima plx - x a ) = 0 for all i E I. The topology of an arbitrary locally convex topological vector space E is always induced by a family of seminorms, e.g. by the family of all continuous seminorms as is easily seen by 1.4 and 1.6. 1.8. Proposition. Let E be a locally convex topological vector space, where the topology is induced by a family (pJiEI of seminorms. Then E is a Hausdorff space if and only iffor every x E E\ {OJ there exists i E I such that plx) =f O. PROOF. Suppose x =f y and that (pJiEI has the above separation property. Then there exist i E I and G > 0 such that Pi(X - y) = 2c;. The sets
{ulplx - u) < c;}, {ulply - u) < c;} are open disjoint neighbourhoods of x and y. For the converse we prove the apparently stronger statement that the separation property of (pJi E I is a consequence of E being a TI -space (i.e. the one point sets are closed). In fact, if x =f 0 and {x} is closed there exists a neighbourhood U of zero such that x ¢ U. By Proposition 1.7 there exist c; > 0 and finitely many indices iI, , in E I such that
{yIPil(y) < c;,
, Piney) < c;}
so for some i E {iI' ... , in} we have plx) ~ c;.
~
U,
D
1.9. Finest Locally Convex Topology. Let Ebe a vector space over IK. Among the topologies on E., which make E into a locally convex topological vector space., there is a finest one., namely the topology induced by the family of all seminorms on E. This topology is called the finest locally convex topology on E. An alternative way of describing this topology is by saying that the system of all absorbing absolutely convex sets is a base for the filter of neighbourhoods of zero, cf. 1.4. The finest locally convex topology is Hausdorff. In fact, let e E E\ {OJ be given. We choose an algebraic basis for E containing e and let qJ be the linear functional determined by qJ( e) = 1 and qJ being zero on the other vectors of the basis. Then p = IqJ I is a seminorm with pee) = 1, and the result follows from 1.8. Notice that every linear functional is contin uous in the finest locally convex topology. In Chapter 6 the finest locally convex topology will be used on the vector space of polynomials in one or more variables.
§2. Hahn-Banach Theorems
5
1.10. Exercise. Let E be a topological vector space, and let A, B, C, F s; E. (a) Show that A (b) Show that F
+ B is open in E if A is open and B is arbitrary. + C is closed in E if F is closed and C is compact.
1.11. Exercise. Let E be a topological vector space. Show that the interior of a convex set is convex. Show that if U is an absolutely convex neighbourhood of in E then its interior is absolutely convex. It follows that a locally convex topological vector space has a base for the filter of neighbourhoods of consisting of open absolutely convex sets.
°
°
1.12. Exercise. Show that a Hausdorff topological vector space is a regular topological space. (It is actually completely regular, but that is more difficult to prove.) 1.13. Exercise. Let E be a topological vector space and A s; E a nonempty and balanced subset. Then: (i) if A is open, A = {x E ElpA(X) < I}; (ii) if A is closed, A = {x E EI PA(X) ~ I}.
1.14. Exercise. Let P, q be two seminorms on a vector space E. Then if {x E Elp(x) ~ I} = {x E Elq(x) ~ I} it follows that P = q. 1.15. Exercise. Let the topology of the locally convex vector space E be induced by the family (pJiEI of seminorms, and let f be a linear functional on E. Then.f is continuous if and only if there exist C E JO, w[ and some finite subset J s; I such that I f(x) I ~ C • max{pi(x) liE J} for all x E E.
§2. Hahn-Banach Theorems One main result in the theory of locally convex topological vector spaces is the Hahn-Banach theorem about extensions of linear functionals. In the following we treat this and closely related results under the name of HahnBanach theorems. We recall that a hyperplane H in a vector space E over lK is a maximal proper linear subspace of E or, equivalently, a linear subspace of codimension one (i.e. dim E/H = 1). Another equivalent formulation is that a hyperplane is a set of the form qJ - I( {O}) for a linear functional qJ: E ~ lk\ not identically zero. Neither local convexity nor the Hausdorff separation property is needed in our first version of the Hahn-Banach theorem. However the existence of a nonempty open convex set A =F E is a strong implicit assumption on E.
6
1. Introduction to Locally Convex Topological Vector Spaces and Dual Pairs
2.1. Theorem (Geometric Version). Let E be a topological vector space over lK and let A be a nonempty open convex subset of E. If M is a linear subspace of E with A n M = 0, there exists a closed hyperplane H containing M with An H = 0. We first consider the case lK = IR. By Zorn's lemma there exists a maximal linear subspace H of E such that M ~ H and A n H = 0. Let C= H + > 0 AA. The sum of an open set and an arbitrary set is open, hence C is open, cf. Exercise 1.10. We now derive four properties of C and H by contradiction: PROOF.
U;.
(a) C n ( - C) = 0. In fact, if we assume x E C n (-C), we have x = h l + Alal = h 2 - A2a2 with hi E H, ai E A, Ai > 0, i = 1,2. By the convexity of A
which is impossible.
(b) H u C u ( - C) = E. In fact, if there exists x E E\(H u C u ( - C)) we define H = H + IRx, so H is a proper subspace of H. Furthermore A n H = 0 because yEA n H can be written y = h + AX with h E H and A =F (A n H = 0), and then x = (l/A)Y - (l/A)h E C u (-C), which is incompatible with the choice of x. Finally the existence of H is inconsistent with the maximality of H so (b) holds.
°
(c) Hn(Cu(-C))=0. In fact, if x E H n C then x = h + Aa with h E H, a E A and A > 0, but then a = (l/A)(x - h) E A n H, which is a contradiction. From (b) and (c) follows that H is the complement of the open set C u ( - C), hence closed.
(d) H is a hyperplane. If H is not a hyperplane there exists x E E\H such that H = H + IRx =F E. Without loss of generality we may assume x E C and we can choose y E ( - C)\H. The function f: [0, 1] ~ E defined by f(A) = (1 - A)X + Ay is continuous, sO.r - l(C) and.r - l( - C) are disjoint open subsets of [0, 1] and 1. Since [0" 1] is connected there exists containing" respectively" ct E ]0, 1[such thatf(ct) E H. But this implies y = (l/ct)(f(ct) - (1 - ct)x) E H, which is a contradiction. This finishes the proof of the real case. A complex vector space can be considered as a real vector space, and if H denotes a real closed hyperplane containing M and such that A n H = 0, then H n (iH) is a complex hyperplane with the desired properties. D
°
7
§2. Hahn-Banach Theorems
The following important criterion for continuity of a linear functional will be used several times. 2.2. Proposition. Let E be a topological vector space over IK, let qJ: E -+ IK be a nonzero linear functional and let H = qJ - 1( {O}) be the corresponding hyperplane. Then precisely one of the following two statements is true: (i) qJ is continuous and H is closed; (ii) qJ is discontinuous and H is dense.
PROOF. The closure H is a linear subspace of E. By the maximality of H we therefore have either H = H or H = E. If qJ is continuous then H = qJ - 1( {O}) is closed. Suppose next that H is closed. Let a E E\H be chosen such that qJ(a) = 1. By Proposition 1.2 there exists a balanced neighbourhood V of zero such that (a + V) n H = 0, and therefore qJ(V) is a balanced subset of lK such that 0 fll + qJ(V), hence qJ(V) f; {x E IK Ilx I < l}. It follows that IqJ(X) I < G for all x E GV, G > 0, so qJ is continuous at zero, and hence continuous. D 2.3. Theorem of Separation. Let E be a locally convex topological vector space over IK. Suppose F and C are disjoint nonempty convex subsets of E such that F is closed and C is compact. Then there exists a continuous linear functional qJ: E -+ IK such that sup Re qJ(x) < inf Re qJ(x). xeC
xeF
PROOF. Let us first suppose IK = ~, and consider the set B = F - C. Obviously B is convex, and using the compactness of C it may be seen that B is closed, cf. Exercise 1.10. Since F n C = 0 we have 0 €I B, so by 1.4 there exists an absolutely convex neighbourhood U of 0 such that U n B = 0. The interior V of U is an open absolutely convex neighbourhood (cf. Exercise 1.11) so A = B + V = B - V is a nonempty open convex set (1.10) such that 0 €I A. Since {O} is a linear subspace not intersecting A, there exists by Theorem 2.1 a closed hyperplane H with A n H = 0. Let qJ be a linear functional on E with H = qJ - 1( {O}). By 2.2, qJ is continuous. Now qJ(A) is a convex subset of ~, hence an interval, and since 0 €I qJ(A) we may assume qJ(A) f; ]0, 00 [. (If this is not the case we replace qJ by - qJ). We next claim inf qJ(x) > 0, xeB
which is equivalent to the assertion. If the contrary was true there exists a sequence (x n ) from B such that qJ(x n ) -+ O. Since V is absorbing there exists u E V with qJ(u) < 0, but X n + u E A so that qJ(x n) + qJ(u) > 0 for all n, which is in contradiction with qJ(x n) -+ O. In the case IK = C we consider E as a real vector space and find a ~-linear functional qJ: E -+ ~ as above. To finish the proof we notice that there exists precisely one C-linear functional t/J: E -+ C with Re t/J = qJ namely t/J(x) = qJ(x) - iqJ(ix), which is continuous since qJ is so. D
8
1. Introduction to Locally Convex Topological Vector Spaces and Dual Pairs
Applying the theorem to two one-point sets we find
2.4. Corollary. Let E be a locally convex Hausdorff topological vector space. For a, bEE, a =f= b, there exists a continuous linear functional f on E such that f(a) =f= f(b). We shall now treat the versions of the Hahn-Banach theorem which are called extension theorems. Although they may be derived from the geometric version, we give a direct proof using Zorn's lemma. The first extension theorem is purely algebraic and very useful in the theory of integral representations. It uses the following weakened form of the concept of a seminorm.
2.5. Definition. Let E be a vector space. A function p: E -+ IR is called sublinear if it has the following properties: (i) positive homogeneity: p(AX) = Ap(X) for A ~ 0, X E E; (ii) subadditivity: p(x + y) ~ p(x) + p(y) for x, y E E. A functionf: E
-+
IR is called dominated by p iff(x)
~
p(x) for all x E E.
2.6. Theorem (Extension Version). Let M be a linear subspace ofa real vector space E and let p: E -+ IR be a sublinear function. If f: M -+ IR is linear and dominated by p on M, there exists a linear extension 1: E -+ IR off, which is dominated by p. PROOF. We first show that it is always possible to perform one-dimensional extensions assuming M =f= E. Let e E E\M and define M' = span(M u {e}). Every element x' E M' has a unique representation as x' = x + te with x E M, r E IR. For every rx E IR the functional f~: M' -+ IR defined by f~(x + te) = f(x) + trx is a linear extension off. We shall see that rx may be chosen such thatf~ is dominated by p. By the subadditivity of p we get for all x, y E M
f(x)
+
f(y) = f(x
+ y)
~
p(x
+ y)
~
p(x - e)
or
f(x) - p(x - e)
~
p(e
+ y)
- f(y).
Defining
k = sup{f(x) - p(x - e)lx EM}, K = inf{p(e
+ y)
- f(y)ly EM},
we have - 00
< k
~
K <
00.
+ p(e + y),
9
§2. Hahn-Banach Theorems
It is easily seen that a necessary condition for f~ to be dominated by p on M' is that rx E [k, K]. This condition is also sufficient. In fact, for rx E [k, K], x, Y E M and t > 0, we have
Multiplying by t > 0 and rearranging yields
.l(x) - trx
~
p(x - te),
fey)
+ trx
~
p(y
+ te)
and shows that f~ is dominated by p on M'. We next consider the set §' of pairs (M', f'), where M' ~ M is a linear subspace of E andf' is a linear p-dominated extension off to M'. For (M',f'), (M", f") E §' we define (M', f') -< (M", f") if and only if M' ~ M" and f" is an extension of f'. Under this relation §' is inductively ordered, so by Zorn's lemma there exists a maximal element CM,!). The preceding discusD sion shows that Nt = E, which finishes the proof. The following corollary was established by Choquet (1962) in his treatment of the moment problem. 2.7. Corollary. Let M be a linear subspace of a real vector space E, and let P be a convex cone in E such that M + P = E. Then every linear functional f: M ~ IR, which is nonnegative on M n P, can be extended to a linear functional!: E ~ IR which is nonnegative on P. PROOF. On E we define the order relation x ~ Y by Y - X there exist Yl' Y2 E M such that Yl ~ x ~ Y2 because x, -x implies that the expression
p(x) = inf{f(Y)IY E M, Y
~
x},
X E
E E
P. For x E E M + P. This
E
satisfies - 00 < p(x) < 00, and it is clear that p is sublinear andf(x) = p(x) for x E M. Let!: E ~ IR be a linear extension off which is dominated by p. We shall see that!(x) ~ 0 for all x E P. Indeed, for x E P we have -x ~ 0 and hence l( -x) ~ p( -x) ~ f(O) = o. D 2.8. Theorem. Let M be a linear subspace of a vector space E over IK and let p: E ~ [0, oo[ be a seminorm. Iff: M ~ IK is linear and satisfies I f(x)1 ~ p(x)for all x E M, there exists a linear extensionl: E ~ IK offwhich satisfies II(x)1 ~ p(x)for all x E E. PROOF.
The real case follows immediately from Theorem 2.6 since a seminorm
p is sublinear and satisfies p( - x) = p(x). In the complex case, we consider E as a real vector space and extend g = Re(f) to a IR-linear functional g: E ~ IR satisfying Ig(x) I ~ p(x) for x E E. Let finally!: E ~ C be the unique C-linear functional with Re(!) = g, i.e.l(x) = g(x) - ig(ix) for x E E. Since Re(IIM) = glM = g = Re(f) we
10
1. Introduction to Locally Convex Topological Vector Spaces and Dual Pairs
necessarily have] I!vI = f. For x rx](x) = 1](x)l, and find
E
E we choose rx
I ](x) I = ](rxx) = Re .!(rxx) = g(rxx)
E
~ p(rxx)
C with Irx I = 1 such that
= Irx Ip(x) = p(x).
D
2.9. Corollary. Let E be a locally convex topological vector space and M a linear subspace. A continuous linear functional on M can be extended to a continuous linear functional on E. PROOF. There exists an absolutely convex neighbourhood V of 0 in E such that the linear functional f on M sati~fies I f(x) I ~ 1 for x E V n M. Let x E M and let A > 0 be such that x E AV. Then A-1 X E V n M and hence I f(x) I ~ A. This shows that the seminorm Pu determined by V (cf. 1.6) satisfies If(x)1 ~ Pu(x) for x EM. Let]be a linear extension offsatisfying 1](x)1 ~ Pu(x) for x E E. Then 1](x)1 ~ G for x E GV, which shows that]is continuous. D
2.10. If E denotes a topological vector space we denote by E' the vector space of continuous linear functionals on E, and E' is called the topological dual space, which is a linear subspace of the algebraic dual space E* of all linear functionals on E.
2.11. Exercise. Let E be a real vector space and p a sublinear function on E. Show that
p(x) = sup{f(x)lfEE*,f
~
p}.
2.12. Exercise. Let P1' ... , Pn: E -+ IR be sublinear functions on a real vector space E and let.f: E -+ IR be linear and satisfyingf(x) ~ P1(X) + ... + Pn(x) for x E E. Show that there exist linear functions f1' ... , fn: E -+ IR such that f = f1 + ... + fn and such that h is dominated by Pi for i = 1, ... , n. Hint: Consider the product space En. 2.13. Exercise. With the notation as in Theorem 2.6 we denote by A(f, E) the set of linear extensions]: E -+ IR offwhich are dominated by p. Clearly A (f, E) is convex. Show by a Zorn's lemma argument that A(f, E) has extreme points. Let Atf, E) such that
Xo E
E. Show that there exists an extreme point]o in
(For the notion of an extreme point see 2.5.1. The result of the exercise is due to Vincent-Smith (1966, private communication). For a generalization see Andenaes (1970).)
11
§3. Dual Pairs
§3. Dual Pairs Let r\J o = {O, 1,2, ... }, let E = ~~o be the vector space of real sequences s = (Sk)k~O and let F be the vector space of polynomials p(x) = Lk=O CkX" with real coefficients. Note that F can be identified with the subspace of sequences SEE with only finitely many nonzero terms. For sEE and p E F we can define 00
(s, p> =
L SkCk k=O
and (., . > is a bilinear mapping of E x F into ~, which clearly satisfies the axioms in the following definition, so E and F form a dual pair under ( ., .
>.
3.1. Definition. Let E and F be vector spaces over IK and (., . >: E x F -+ IK a bilinear form, i.e. separately linear. We say that E and F form a dual pair under (., . >if the following conditions hold: (i) For every e E E\{O} there existsf E F such that (e,f> =f= (ii) For every f E F\ {O} there exists e E E such that (e,f> =f=
o. o.
3.2. A locally convex Hausdorff topological vector space E and its topological dual space E' form a dual pair under the bilinear form (x, qJ> = qJ(x) for x E E, qJ E E'. The condition (ii) is clearly true and (i) follows from Corollary 2.4. A vector space E and its algebraic dual space E* form a dual pair under the bilinear form (x, qJ) = qJ(x). This example is a special case of the above example if E is equipped with the finest locally convex topology, cf. 1.9. We see below that every dual pair (E, F, (., . arises in the above way in the sense that there exist a topology 1J on E, such that E is a locally convex Hausdorff topological vector space, and an isomorphism j: F -+ E' such thatj(f)(e) = (e,f> for e E E,f E F. Such a topology 1J is called compatible with the duality between E and F. In general there exist many different topologies on E of this kind, and we will now define one, which turns out to be the coarsest compatible with the duality and therefore is called the weak topology.
»
3.3. Definition. Let E and F be a dual pair under (., . >. The weak topology a(E, F) on E is the topology induced by the family (Pf)feF of seminorms, where Pf(e) = I(e,f> I· Condition (i) of 3.1 implies that a(E, F) is Hausdorff, cf. 1.8. By reasons of symmetry there is also a weak topology a(F, E) on F.
3.4. Proposition. The topology a(E, F) is the coarsest of the topologies compatible with the duality between E and F. PROOF. If 1J is a topology compatible with the duality then e ~ (e,f> is 1J-continuous for all f E F, and so are the seminorms (Pf)feF. By 1.7 it
12
1. Introduction to Locally Convex Topological Vector Spaces and Dual Pairs
follows that aCE, F) is coarser than fJ. If E is equipped with the weak topology then e 1--+ <e,f> is a continuous linear functional on E for each f E F, and the mapping j: F -+ E' given by j(f)(e) = <e,f> is linear and one-to-one (condition (ii) of 3.1). To see thatj is onto we consider a aCE, F)-continuous linear functional qJ on E. By 1.7 there exists £ > 0 and .fl' ... ,In E F such that pflx) < £, i = 1, ... , n, implies IqJ(x) I ~ 1. This gives at once that
{x EEl <x,.h > = 0, i = 1, ... , n} ~ qJ Let us consider the linear mapping tjJ: E
tjJ(x) = «X,fl >,
-+
1(
{O} ).
(1)
/Kn defined by
... , <x,fn»'
XE
E.
The image tjJ(E) is a linear subspace of /Kn and the inclusion (1) implies that ip: tjJ(E) -+ /K is well defined by ijJ(tjJ(x)) = qJ(x), x E E. But a linear functional on a subspace of /Kn may be written n
L Ai Yb
ijJ(y) =
Y E tjJ(E)
~
/Kn,
i= 1
for a not necessarily unique vector (A l , ... , An) E /Kn, and this shows that qJ(x) = <x,f> withf = 1 Ai.h E F, hence j(f) = qJ. D
L?=
It is only slightly more difficult to show that there is also a finest topology on E compatible with the duality. This topology is called the Mackey topology and is denoted T(E, F), cf. Exercise 3.13. We now associate with each subset of one of the two vector spaces of a dual pair a subset of the other space of the pair, called the polar subset.
3.5. Definition. Let E and F be a dual pair under the polar subset A ° is given by
AO = {f
E
FIRe<e,f>
~
<., .>. For a subset A ~ E
1 for all e E A}.
For e E E the set {e}O = {f E FI Re<e,f> ~ I} is convex and closed in any topology, on F compatible with the duality. Therefore also
is ,-closed and convex. Furthermore 0 E AO.
3.6. The Bipolar Theorem. Let fJ be any topology on E compatible with the duality between E and F and let A ~ E. The bipolar set AOO = (AO)O is the smallest fJ-closed and convex subset of E" containing A u {O}. PROOF. From the above remark it follows that A is an fJ-closed and convex set containing A u {O}. To finish the proof we show that the existence of an fJ-closed convex set B containing A u {O} and a point e E AOO\B will lead to a contradiction. In fact, by the separation theorem (2.3) there exists an 00
13
§3. Dual Pairs
1]-continuous linear functional qJ: E
-+
IK and a number A E IR such that
Re qJ(e) < A < inf Re qJ(b). beB
Since 0 E B we have A < O. Ifj~'E F is such that qJ(x) = (x,f> for x E E we find
~~~ Re(b,~ f) < 1 < Re(e, ~ f)' The first inequality shows that (1/ A)f E A ° and the last inequality is then incompatible with e E A 00. D 3.7. Remark. If A is balanced we have
AO= {fEF/I(e,f>1 ~ IforalleEA}. This is often used as a definition of the (absolute) polar set. If A is a cone (i.e. AA £; A for all A ~ 0) we have
AO = {f E FI Re(e,f>
~
0 for all e E A},
which is a convex cone. With A £; E we also associate another convex cone Ai- £; F, which is closed in any topology on F compatible with the duality between E and F, namely Ai- = {fEFI(e,f> ~ Ofor alleE A}. Clearly Ai-£; - A ° and if E and F are real vector spaces and if A is a cone then Ai- = -AO. For a set A containing 0 the bipolar theorem states that A 00 is the 1]-closed convex hull of A. Using translations we therefore have the following consequence of the bipolar theorem: 3.8. Proposition. The closed convex hull of a subset of E is the same for all topologies on E compatible with a given duality. If E is a finite dimensional vector space over IK, hence isomorphic with IK n where n is the dimension of E, there is exactly one topology on E compatible with the duality between E and E*. More generally there is exactly one Hausdorff topology on E such that E is a topological vector space. We will refer to this topology as the canonical topology of E. These assertions are contained in the following result. 3.9. Proposition. Let E be a finite dimensional subspace of a Hausdorff topological vector space F. Then E is closed in F, and any algebraic isomorphism qJ: IK n -+ E (n = dim(E)) is a homeomorphism, when IK n is equipped with the topology generated by the euclidean norm.
14
1. Introduction to Locally Convex Topological Vector Spaces and Dual Pairs
We first show by induction that any isomorphism qJ: IK n -+ E is a homeomorphism. For n = 1 we put qJ(l) = e. The continuity of scalar multiplication implies that qJ: A 1--+ Ae is continuous from IK to E. The inverse qJ - 1 is a linear functional on E, and its kernel is the hyperplane {O}, which is closed since E is Hausdorff. By 2.2 it follows that qJ - 1 is continuous. Let us assume that the above statement is true for all dimensions less than n and let qJ: IK n -+ E be an algebraic isomorphism. As before the continuity of the algebraic operations shows that qJ is continuous. To see that n qJ - 1 : E -+ IK is continuous it suffices to prove that each linear functional on E is continuous. To get a contradiction let us assume that t/J: E -+ IK is a discontinuous linear functional and put H = t/J - 1({O}). Then H is a (n - 1)dimensional hyperplane, which is dense in E by 2.2. Let 11·11 be the euclidean norm (or any norm) on H. By the induction hypothesis the norm topology on H coincides with the topology induced from E, so there exists an open set V in E such that V n H = {xEHlllxll < I}. PROOF.
Since H is dense in E and V is open, we have V n H = V, where the closures are in E. But the set {x EHlllxli ~ I} is compact in H, hence in E and in particular closed in E, so we get V ~
V = V n H ~ {x EHlllxli ~ I}. Since V is absorbing in E we get E = H. By this absurdity
qJ is indeed a homeomorphism. We finally show that E is closed in F. If this is not true there exists x E E\E. Then E = span(E u {x}) is a (n + I)-dimensional space. If et, ... , en is an algebraic basis for E, then qJ: IK n + 1 -+ E given by qJ(A t , .•. , An' A) = 1 Ai ei + AX is an algebraic isomorphism, hence a homeomorphism. It follows that E is closed in E, hence x E EnE = E, which is a contradiction.
Li=
D
.>.
3.10. Exercise. Let E and F be a dual pair under (., Then the weak topology (J(E, F) is the coarsest topology on E for which the mappings e 1--+ (e,f> are continuous when .f ranges over F. 3.11. Exercise (Theorem of Alaoglu-Bourbaki). Let E be a locally convex Hausdorff topological vector space with topological dual space E' and let V be a neighbourhood of zero in E. Show that VO is (J(E', E)-compact. Hint: Show that for x E E there exists A > 0 such that I(x, .f> I ~ A for all
f
E
vo.
3.12. Exercise. Let E, F be a dual pair under (., . >and let t1 be a topology on E compatible with the duality. Let V be a closed, absolutely convex neighbourhood of zero in E and let Pu be the seminorm determined by V, cf. 1.6. Show that XE E. Pu(x) = sup{ I(x,f> f E Va},
II
15
Notes and Remarks
3.13. Exercise (Theorem of Mackey-Arens). Let E, F be a dual pair under (., .), and let d be the family of all absolutely convex and u(F, E)-compact subsets of F. For A E t91 we define
IlellA
= sup{ I(e,f)
II f
E
A},
e E E.
Show that II·IIA is a seminorm on E. Use 3.11 and 3.12 to show that if fJ is a topology on E compatible with the duality then fJ is induced by some subfamily of (11·11 A)A E.>4· Show finally that the topology induced by the family (11·IIA)AEd is the Mackey topology, i.e. the finest topology on E compatible with the duality.
Notes and Remarks In the period up to the 1940's most results in functional analysis were about normed spaces. The development of the theory of distributions of Schwartz was one main motivation for a study of general spaces, since the basic spaces of test functions and distributions are nonnormable in their natural topology. Today locally convex Hausdorff topological vector spaces are a natural frame for many theories and problems in functional analysis, e.g. the theory of integral representations, which we shall discuss in the next chapter. For historical information on the theory of topological vector spaces we refer the reader to the book by Dieudonne (1981).
CHAPTER 2
Radon Measures and Integral Representations
§1. Introduction to Radon Measures on Hausdorff Spaces It is well known that the pure set-theoretical theory of measure and integration has its limitations, and many interesting results need a topological frame because measure spaces without an underlying "nice" topological structure may be very pathological. In classical analysis this difficulty was overcome by introducing the theory of Radon measures on locally compact spaces. On these spaces there is a particularly important one-to-one relationship between Radon measures and certain linear functionals (see below) which in many treatments on analysis leads to the definition, that a Radon measure is a linear functional with certain properties. Another branch of mathematics with a need for a highly developed measure theory is probability theory. Here the class of locally compact spaces turned out to be far too narrow, partly due to the fact that an infinite dimensional topological vector space never can be locally compact. For example, it was found that the class of polish spaces (i.e. separable and completely metrizable spaces) was much more appropriate for probabilistic purposes. Later on it became clear that a very satisfactory theory of Radon measures can be developed on arbitrary Hausdorff spaces. This has been done, for example, in L. Schwartz's monograph (1973). We shall follow an approach to Radon measure theory which has been initiated by Kisynski and developed by Tops~e. It deviates, for example, from the Schwartz-Bourbaki theory in working only with inner approximation, but we hope to show that it gives an easy and elegant access to the main results.
§1. Introduction to Radon Measures on Hausdorff Spaces
17
In the following let X denote an arbitrary Hausdorff space. The natural a-algebra on which the measures considered will be defined will always be the a-algebra f!4 = f!4(X) of all Borel subsets of X, i.e. the a-algebra generated by the open subsets of X. In our terminology a measure will always be nonnegative; a measure defined on f!4(X) will be called a Borel measure on X. Later on we also need to consider a-additive functions on f!4(X) which may assume negative values, these functions will be called signed measures.
1.1. Definition. A Radon measure J1 on the Hausdorff space X is a Borel measure on X satisfying (i) Jl(C) < 00 for each compact subset C £ X, (ii) Jl(B) = sup{Jl(C) I C £ B, C compact} for each B E f!4(X). The set of all Radon measures on X is denoted M +(X).
Remark. Many authors require a Radon measure to be locally finite, i.e. each point has an open neighbourhood with finite measure. There are good reasons for not having this condition as part of the definition, see Notes and Remarks at the end of this chapter. A finite Radon measure Jl (i.e. Jl(X) < (0) satisfies for B E f!4(X) Jl(B) = inf{Jl(G)IB £ G, G open} as is easily seen by considering the property (ii) for Be. However for arbitrary Radon measures this need not be true as is shown by Exercise 1.30 below. Let .ff = .ff(X) denote the family of all compact subsets of X. Clearly the restriction to .ff of a Radon measure Jl is a set function
A: .ff
-+
[0, oo[
satisfying the axioms of a Radon content below.
1.2. Definition. A Radon content is a set function A:.ff -+ [0, oo[ which satisfies (1) A(C 2 ) - A(C 1 ) = sup{A(C)IC £ C 2 \C 1 , CE.ff} for all C 1 , C 2 E.ff with C 1 £ C 2 • The key result in our approach to Radon measure theory is the extension theorem (1.4) below, the proof of which will need the following lemma.
1.3. Lemma. A Radon content A has the following properties: (i) A(C t ) ~ A(C 2 ),for all Cf, C 2 E ~ C t £ C 2 , i.e. A is monotone. (ii) A(C t U C 2 ) + A(C t n C 2 ) = A(C 1 ) + A(C 2 )'1 i.e. A is modular. (iii) lj'a net (Ca)(J. E A in.ff is decreasing with C = na C a then A( C) = lima A(Ca) = infa A( C a). In particular for a decreasing sequence C 1 ;2 C 2 ;2 . . . of compact sets we have lim n _ oo A(C n) = A(n:=l Cn).
18
2. Radon Measures and Integral Representations
(i) as well as A(0) = 0 is obvious. (ii) We have (C 1 u C 2 )\C 2 = C 1 \(C 1 n C 2 ) and therefore
PROOF.
A(C 1 u C 2 )
A(C 2 ) = A(C 1)
-
A(C 1 n C 2 )
-
as an immediate consequence of (1). (iii) Assume that b := in(iA(C a) - A(C» > O. We choose a fixed set C ao and C' £ Cao \ C, C' E .ff such that
A(Cao) - A( C) - A(C') < b. Now na~ao(C' n C a) = 0 and therefore C' n Cal = 0 for some Cal £ Cao since C' is compact and (Ca)aE A is decreasing. From (ii) we get
A(C' u Ca) = A(C') < A(C')
+ A(Ca) ~ A(Cao ) + A(C) + b
o
implying A(C a) - A(C) < J, a contradiction.
1.4. Theorem. Any Radon content on a Hausdorffspace has a unique extension to a Radon measure. PROOF. Let A be a Radon content on X. We define for any subset A £ X the inner measure by A*(A):= sup{A(C)IC £ A, C E.ff}
and have to show that J1 := A* I~ is a measure. Of course A* is an extension of A, but it may assume the value + 00, if Ais unbounded. In a certain analogy with Caratheodory's famous abstract measure extension theorem we consider the set system
.91:= {A
£
X/A*(C n A)
+
A*(C n A C ) = A*(C) for all C
E
.ff},
and we will show that d is a a-algebra containing ~, on which the restriction of A* is a-additive. From the very definition d is closed under complements and contains the empty set. The defining property (1) of a Radon content shows that .91 even contains all open subsets of X. Let AI' A 2 E d be disjoint and let C 1 £ A b C 2 £ A 2 be compact. Then the modularity of A gives
A(C 1 )
+
A(C 2 ) = A(C 1 u C 2 ) ~ A*(A 1 u A 2 )
and hence
A*(A 1 )
+
A*(A 2 ) ~ A*(A 1 u A 2 ),
i.e. A* is "superadditive". As a consequence d may also be written as
d = {A £ X/A*(C n A)
+
A*(C n A C ) ~ A*(C) for all C
E
.ff}.
Now let a sequence AI' A 2 , ••• E.91 be given and fix C E.ff as well as o. Then there exist compact sets K j £ C n A j and L j £ C n Aj such that
G
>
j
= 1,2, ....
(2)
19
§1. Introduction to Radon Measures on Hausdorff Spaces
From the modularity of A we get
A(V: K
i)
+ A(K n + 1n
/)1 Ki )
n
~Cn
=
A(VI K
i)
+ A(K n +
I)
(3)
as well as
We have
and
I n := L n + hence
K
n
and
1 U
j-= 1
Li
(n
A~+ I)'
Aj u
j= 1
In are disjoint compact subsets of C, so that A(K n ) + A(I n ) ~ A(C).
(5)
Adding the equalities (3) and (4) and inserting (2) and (5) give
ACV: K
i)
+ AC6: Li ) =
A(~I K i ) - A(K n )
+
-
A(OI L
i)
I)
+ A(K n + + A(Ln +
A(I n )
(6)
~ A() Ki ) + A(.n Li ) - 2n~ )=1 )=1 Ifwe add (6) over n
I)
l'
= 1,2, ... , N - 1 and use (2) for j = 1 we get
A(~I K i )
+ A(6 Li )
~ A(C) -
/;-
JI ~.
(7)
Put A:= Uf=1 A j ; then A(U7=1 K j ) ~ A*(C n A) for all Nand
A(C\ Li ) =
!~n:, A(b Li ) ~ A*(C n
A
C )
(8)
by Lemma 1.3(iii), hence letting N tend to infinity in (7) gives
A*(C n A)
+ A*(C n
A
C )
~
A(C) -
B,
and since this holds for all B > 0, we have in fact shown A E or", hence d is a a-algebra containing the open sets and therefore the Borel sets. Let us now furthermore assume that the sets A l' A 2' ... E d are pairwise disjoint and that C £ A. Then lim N _ oo A(n7=1 L j ) = 0 by (8), and taking again the limit in (7) gives
Jc(C) -
B
~ ~~~ A(~I Ki)= ~~ itl A(K) = JI A(K) ~ i~1 A*(A)
20
for all
2. Radon Measures and Integral Representations
G
> O. Letting G -+ 0 we find 00
A*(A) = sup{A(C)IC £ A, C
E
L A*(A
$'} ~
j ),
j= 1
and since the reverse inequality is obvious by the superadditivity of A* we have that A* I.rtf is a measure, thus finishing our proof. D The result we are now going to prove is a kind of monotone convergence theorem for Radon measures. The usual form of this theorem on general measure spaces deals with an increasing sequence of nonnegative measurable functions; however, if the underlying measure is a Radon measure and if the functions to be integrated are lower semicontinuous (i.e. {.f > t} is open for all t E [R), then the sequence may be replaced by an arbitrary increasing net of functions, as we shall see. In the sequel we shall make repeated use of the obvious inequality 1
00
o ~ J.:= 2" i~l l u >i/2"j ~ f
(9)
being valid for arbitrary functions f with values in [0, 00]. Iff is finite the infinite series in (9) reduces to a finite sum (pointwise) and f - .in ~ 1/2n • Note thatfn increases tofalso iff assumes the value 00. Let us mention that the family of all lower semicontinuous functions is closed under finite sums, multiplication with a nonnegative constant, and that the supremum of an arbitrary subfamily of these functions is still lower semicontinuous. Noting finally that an indicator functionf = I G is lower semicontinuous if and only if G is open, we see that the functionsfn defined in (9) are lower semicontinuous if fis.
1.5. Theorem. Let J1 be a Radon measure on the Hausdorffspace X. Then the .following holds: (i) If a net (Ga)aEA of open subsets of X is increasing with
Ua Ga=
G then
J1(G) = sup J1(G a) = lim J1(G a). (ii) Ifa net (fa)(J.EA oflower semicontinuousfunctions X with sUPa j~ = f then
f
f dJl =
f
s~p f~ dJl = li~
ffa
-+
[0, 00] is increasing
dJlo
PROOF. (i) Let C £ G be compact. Then finitely many Gal' ... , Gak cover C and by assumption there is some CXo such that Gal U ••. U Gak £ Gao' implying J1(C) ~ J1(G ao ) ~ SUPa J1(G a) and therefore
J1(G) = sup{J1(C)I C £ G, C The reverse inequality is trivial.
E
$'}
~
sup J1(G a).
21
§l. Introduction to Radon Measures on Hausdorff Spaces
(ii) For every t E IR the open sets {fa> t} increase to {f > t}. Using the functions.in and the correspondingfa,n as defined in (9) we find
If~d~
=
;n ~~({f > ;n}) = ;n ~li~~({fa > ;n})
=
li~ ;n ~ ~ ({fa > ;n}) = li~ If~. nd~,
where the interchange of limits is justified, both limits being suprema, and using this device once more we get
I
f d~ = s~p I fn d~ = s~p s~p I f~.
n
d~
= sup sup Ifa,n dfl = sup I fa dfl, a
n
a
D
applying, of course, the usual monotone convergence theorem.
1.6. Remark. Theorem 1.5 can be applied to an upwards filtering family A of sets or functions by defining an increasing net in the following way: The index set and the mapping of the net will be A and the identical mapping. A Borel measure satisfying property (i) of the above theorem is usually called a r-smooth measure. The class of these measures is in general larger than the class of Radon measures, however, for finite Borel measures on locally compact spaces the two notions coincide. The generalized monotone convergence theorem expressed as property (ii) of the above theorem uses only the r-smoothness of the underlying Radon measure and therefore remains valid for r-smooth measures as well, see Top;¢>e (1970) and Varadarajan (1965). We shall need in the following the notion of restriction ofa Radon measure to a Borel subset. If X is a Hausdorff space and B E 81(X), then B is again a Hausdorff space with respect to the trace topology {B n GIG open in X} and it is easy to see that the Borel subsets of B are given by
81(B) = {B n AlA
E
81(X)} = {D
so that in fact 81(B) £ 81(X). For fl
E
E
81(X)ID £ B}
M +(X) we now define
fll B: 81(B)
~
[0, 00]
as the restriction of fl to 81(B)'l i.e. (fll B)(A) := fl(A) for A mediately seen that fll B is again a Radon measure.
E
81(B). It is im-
1.7. Proposition. Let J1 be a Radon measure on X. lfthefunctionf: X is Borel measurable, then
If d~
= sup Ke,*,"
r f d~,
JK
~
[0,00]
(10)
22 and
2. Radon Measures and Integral Representations
if f:
X
-+ [o~
oo[ is continuous then v: &I(X) v(B):=
-+ [o~
00] defined by
{! d,u
is again a Radon measure. The measure v is often denoted fJ1 or f dJ1. PROOF. If I = I B for some B E &I(X), then (10) follows from the definition of a Radon measure. It is obvious that (10) remains true iff'is an elementary measurable nonnegative function, i.e. f = = I CXi 1Bi with pairwise disjoint Borel sets Bf, ... , Bn and CXI' ••• , CXn ~ o. But it is well known that an arbitrary Borel measurable I ~ 0 is the pointwise limit of some increasing sequence of elementary functions, so that the usual monotone convergence theorem and the possibility of interchanging two suprema give (10) in the general case also. Let now f: X -+ IR+ be continuous and v(B) = SBI dj1, BE &I(X). Obviously, v is finite on compact sets. Applying (10) to the restrictions J1IB and.flB we find
Ii
{!
{{!
d,u = su p
d,uIK
E
Jf; K
~ B}
0
1.8. Let j1 be a Radon measure on X and consider the family ~ of all open J1-zero sets in X. The system of all finite unions of sets in ~ filters upwards to the union G of all sets in ~ and J1(G) = 0 by Theorem 1.5. The open set G is therefore maximal in ~ and its complement is called the support of j1 or abbreviated supp(J1). It is immediate that supp(j1) is closed and that sUPP(J1) = {x
E
X IJ1(U) > 0 for each open set U such that x
E
U}.
Particularly simple examples of Radon measures are those with a finite support which we will call molecular measures, and among these are the one-point or Dirac measures Gx defined by Gx ( {x}) = 1 and Gx ( {x }C) = o. Of course supp(Gx ) = {x} and if J1 = I CXiG Xi is a molecular measure with Xi Xj for i j, then sUPP(j1) = {XdCXi > O}. The set of molecular measures is denoted Mol+(X).
+
+
I?=
In the usual set-theoretical measure theory, as well as in the theory of Radon measures, the notion of a product measure is of central importance. In the latter case we are immediately confronted with the following problem: Let X and Y be two Hausdorff spaces; then the product of the two u-algebras of Borel sets~ usually denoted &I(X) ® &I(Y)~ is by definition the smallest u-algebra on X x Y rendering the two canonical projections 1Cx: X x Y -+ X and1Cy: X x Y -+ Ymeasurable,i.e.&I(X) ® &I(Y)istheu-algebragenerated by 1Cx I (&I(X» U 1Cf 1(&I( Y». By definition of the product topology these two projections are continuous on X x Y and therefore Borel measurable, so that always
&I(X) ® &I(Y) £ &I(X x Y).
23
§l. Introduction to Radon Measures on Hausdorff Spaces
On "nice" spaces we even have equality of these two a-algebras on X x Y, but this need not always hold, see the exercises below. Our next goal will be to show existence and uniqueness of the product of two arbitrary Radon measures. This stands in some contrast to set-theoretical measure theory where usually a-finiteness of the measures is required in order to guarantee a uniquely determined product measure. We begin with a lemma.
1.9. Lemma. Let Z be a Hausdorffspace and let stl be an algebra ofsubsets of Z containing a base for the topology. If A: .91 --+ [0, 00 [ is finitely additive then A: %(Z) --+ [0, oo[ defined by ,1(C):= inf{A(G)I C £ G, G open, G E d} is a Radon content on Z. PROOF. Let C £ Z be compact, then every point x E C has an open neighbourhood Gx E d. Finitely many of these neighbourhoods cover C and their union is still in d. Hence ,1( C) is certainly finite. Now let two compact sets C 1 £ C 2 be given. For B > 0 there is an open set G 1 ;2 C b G 1 Ed such that A(G 1 ) - ,1(C 1 ) < B. The set C:= C 2 n G1 is compact, too, allowing us to choose a further open set G E .91, G ;2 C with A(G) - ,1(C) < B. Of course, C 2 £ G U G 1 E .91 so that ,1(C 2 ) ~ A(G) + A(G 1) and therefore ,1(C 2 ) - ,1(C 1) ~ A(G) + A(G 1 ) + B - A(G 1 ) < ,1(C) + 2B. Hence
,1(C 2 )
-
,1(C 1 )
~
sup{,1(C) I C £ C 2 \C 1 , C
E
%}.
The reverse inequality will follow immediately if we can show that A is additive on disjoint compact sets. Therefore let K, L E % with K n L = 0 be given. One direction, namely
,1(K u L)
~
,1(K)
+ ,1(L)
is obvious, so it remains to be shown that for arbitrary
,1(K)
+ ,1(L)
~
,1(K u L)
B
> 0
+ B.
By definition there is an open set WEd containing K u L such that A(W) - ,1(K u L) < B. The assumption made on the algebra .91 implies that K and L may be separated by open sets G, H belonging to .91, i.e. we have K £G,
L £H,
Gn H = 0.
Hence
,1(K)
+ ,1(L)
~
A(G n W)
+ A(H n
W)
= A«G u H) n W) ~
thus finishing the proof.
A(W) < ,1(K u L)
+ B, D
24
2. Radon Measures and Integral Representations
Later on we shall need existence and unicity of certain Radon measures on the product of two Hausdorff spaces X and Y not only for the product of two measures, but also for so-called Radon bimeasures. If(X, d) and (Y,!!l) are just two measurable spaces (without an underlying topological structure) then a bimeasure <1> is by definition a function <1>: .91 x !!l
-+
[0,
00]
such that for fixed A E .91 the partial function B ~ <1>(A, B) is a measure on !!l and for fixed B E !!l the function A ~<1>(A, B) is a measure on d. Obviously, if K is a measure on .91 @ !!l, then (A, B) ~ K(A x B) is a bimeasure, but in general not even a bounded bimeasure is induced in this way, cf. Exercise 1.31. Our next result will, however, show that for Radon bimeasures such pathologies do not exist, where by definition <1> is a Radon bimeasure if <1> is a bimeasure defined on !!l(X) x !!l( Y) such that <1>(K, L) < 00 for all compact sets K, Land <1>(A, B) = sup{<1>(K, L)IA ;2 K E g(X), B;2 L E g(y)} for all Borel sets A and B. 1.10. Theorem. Let X and Y be two Hausdorff spaces and let <1>: !!l(X) x !!l(Y) -+ [0, 00] denote a Radon bimeasure. Then there is a uniquely determined Radon measure K on X x Y with the property
<1>(K, L) = K(K x L)
.for all
K
E
g(X),
L
E
g(y).
Furthermore, the equality <1>(A, B) = K(A x B) holds for all Borel sets A
E
!!l(X), B
E
!!l( Y).
PROOF. Denote Z:= X X Y and let .91 be the algebra generated by the "measurable rectangles" A x B, where A E !!l(X) and BE !!l(Y). This algebra contains, of course, the products of open sets in X (resp. Y) and therefore a base for the topology on Z. It is easy to see that there is a uniquely determined finitely additive set function A on .91 fulfilling
A(A x B) = <1>(A, B)
for all
A
E
!!l(X)
and
B E !!l( Y).
Let us now first assume that <1>(X, Y) < 00. Then we may apply Lemma 1.9 which, combined with the extension theorem 1.4, shows the existence of a Radon measure K on Z such that K(C)
= inf{A(G)IC
£
G E .91, G open}
for each compact set C £ Z. If C = K x L is the product of two compact sets K £ X, L £ Y, then C E .91 and K(K x L)
~
A(K x L) = <1>(K, L)
by monotonicity of A. On the other hand, we may use the two finite Radon measures ,u(A):= <1>(A, Y) on X and v(B):= <1>(X, B) on Y to provide us
25
§ 1. Introduction to Radon Measures on Hausdorff Spaces
with open sets G
;2
K, H
;2
L such that J.1(G\K) <
8
and v(H\L) <
+ A(X x + v(H\L) < 28,
A(G x H\K x L) ~ A«G\K) x Y) = J.1(G\K)
8.
Then
(H\L»
and thus K(K x L)
~
A(K xL),
i.e. we have the desired equality. If A E 81(X), B E 81(Y) and C is a compact subset of Ax B, then the projections K:= nx(C) and L:= ny(C) are still compact and C £ K x L £ A x B, implying K(A x B)
= sup{K(C)1 C
£ A x B, C E ff(Z)}
= sup{K(K x L)IK £ A, L £ B, K
E
ff(X), L
= sup{
(K, L)IK £ A, L £ B,KE ff(X), L
E
E
ff(Y)}
ff(Y)}
= (A, B), using in the last equality once more that is a Radon bimeasure. We also see from the preceding argument that K is indeed uniquely determined from its values on products of compact sets (still assuming (X, Y) < (0). In the second step we abandon the finiteness restriction on <1>. For two compact sets K £ X, L £ Y we know that there is a uniquely determined KK,L E M +(K x L) such that KK, L(A x B)
=
(A, B)
for all Borel sets A £ K, B £ L. Of course these measures patible in the sense that K 1 £ K 2 , L 1 £ L 2 implies KK2,L 2I K
l
X
K K, L
are com-
L 1 = KKt,Lt.
Ifnow C £ Z is compact, then C £ K x L for suitable compact sets K £ X, L £ Y, and irrespective of the choice of K and L the value K(C):= KK,L(C) is well defined; furthermore, we see immediately that K is even a Radon content on Z whose extension to a Radon measure on Z we still denote byK. Repeating the argument already used we see that also in this case K(A x B)
= (A, B)
for all
A
E
81(X),
B
E
81( Y).
Since the values K (C) for compact subsets C £ Z are uniquely determined by the values K (K x L) for K E ff(X), L E ff(Y), so is finally K itself, thereby finishing the proof. D A particularly important special case is the following: let J1 E M + (X) and v E M +(Y) denote two Radon measures, then (A, B) := ,u(A)· v(B) is of course a Radon bimeasure leading to
26
2. Radon Measures and Integral Representations
1.11. Corollary. rf J1 and v are two Radon measures on the Hausdorff spaces X and Y" then there is a uniquely determined Radon measure on X x Y" called the product of J1 and v and denoted J1 ® v, with the property
J1 ® v(K x L) = J1(K) . v(L)
For all Borel sets A
£;
X and B
£;
for all
K
E
$'(X),
L
E
$'(Y).
Y we have
J1 ® v(A x B) = J1(A) . v(B), so that, in particular, the restriction ofJ1 ® v to the product a-algebra 81(X)
@
81( Y) is a product measure of J1 and v in the usual sense.
Later on we shall also need an amended version of the Fubini theorem, being more general in allowing the interchange of the order of integration for some Borel measurable functions on the product X x Y which are not necessarily measurable with respect to 81(X) ® 81(Y). In particular" this interchange will be possible for all nonnegative continuous functions on X x Y.
1.12. Theorem. Let J1 and v be two Radon measures on the Hausdorff spaces X and Y and let f: X x Y -+ [0" 00 ] be lower semicontinuous. Then the two functions X 1-+
J
f(x, y) dv(y)
and
Jf(x, y) dJl(x)
(11)
Jyr fx .f(x, y) dJ1(x) dv(y).
(12)
Y 1-+
are again lower semicontinuous and
f
x
x
yf
d(/l ® v) =
ff X
Y
f(x, y) dv(y) dJl(x) =
If J1 and v are a-finite Radon measures and f: X x Y -+ [0, 00 ] is Borel measurable, then the two functions in (11) are again Borel functions and (12) continues to hold. PROOF. We know from the preceding corollary that the restriction of J1 @ v to 81(X) ® 81(Y) is a product measure in the usual sense. Let us first consider the simple case where f = 1A x B for Borel sets A £; X and B £; Y. Then S f(x, y) dv(y) = v(B)· l A (x), S f(x, y) dJ1(x) = J1(A) . I B (y) are certainly measurable on X (resp. Y) and (12) obviously holds. This result extends immediately to the case where f is the indicator function of a set in the algebra spanned by the" measurable rectangles" A x B" A E gB(X) and BE8I(Y),sothatitholds,inparticular,forf = l u whereU = U7=1(G i x Hi) and Gi f; X, Hi £; Yare open sets. In this case, however,.f is also lower semicontinuous and we have asserted that the partial integrations in (11) yield again lower semicontinuous functions. To show this we have to make
27
§1. Introduction to Radon Measures on Hausdorff Spaces
use of the sections of a subset of X x Y, defined for an arbitrary V by Vx :={y
E
YI(x, y)
E
V}
VY:= {x
E
XI(x, y)
E
V}.
£;
X x Y
and
All the properties of sections which we shall need follow from the simple fact that Vx is the preimage of V under the continuous mapping y ~ (x, y) and the corresponding fact for VY. If all the sections of V are Borel sets in X (resp. Y) (which certainly is true for V E 84(X x Y», then the two functions in (11) are well defined for .f = 1v and
f
f(x, y) dv(y)
=
f
v(Vx),
f(x, y) df.1(x)
Now let us continue to assumej' = Iv with U Hi being open. For given t E IR+ let
=
=
f.1(vY).
U?=I(G i x HJ, Gi and
then {xEXlv(V x ) > t} =
U nG
i
CXED t
iECX
is an open set, hence v(V x) is lower semicontinuous as a function of x and, of course, y ~ /l(VY) is also lower semicontinuous. If V £; X x Y is an arbitrary open set, then V is the union of an upwards filtering family of open sets V y of the above simple type, i.e. each U y is a finite union of open rectangles. In this case and are again lower semicontinuous, and then Theorem 1.5 shows that (12) remains valid for.f = 1v. The extension to an arbitrary nonnegative lower semicontinuous function f is now easily obtained using the approximating functions ,fn as defined in (9) and using once more Theorem 1.5. Let us now assume that /l and v both are finite measures and put Z := X X Y. Then the set system ~
:=
{V
E
84(Z) I v(Vx ) and /l(VY) are Borel measurable and
(12) is valid for.f
= Iv}
has the following three properties: (i) Z E ~ (ii) A E ~ ~ A C E ~ (iii) AI, A 2 , ... E ~ pairwise disjoint ~ U~ I Ai E ~.
28
2. Radon Measures and Integral Representations
This means that ~ is a so-called Dynkin class and the main theorem about these classes is as follows (cf. Bauer 1978, Satz 2.4): If Q is a nonempty set and ~ is a family of subsets of Q closed under finite intersections, then the smallest Dynkin class containing ~ equals the a-algebra generated by ~. Applying this result in our special situation where Q = Z and where ~ is the family of all open subsets of Z, we may conclude that ~ = PJ(Z), so that (12) is indeed valid for all f = lv, V E PJ(Z), and then, by the usual extension, for all Borel measurable.f: Z --. [0, 00]. The extension to the case where Il and v are a-finite is completely straightforward and therefore omitted. D
1.13. It is, of course, a natural question to ask if equality in (12) holds for more general functions than just nonnegative lower semicontinuous ones. The following example shows that one cannot, in general, hope for too much. Let X be the unit interval with usual topology and with Lebesgue measure Il, and let Y be the unit interval with discrete topology (i.e. every subset is open in Y). On Y, we consider the counting measure v, i.e. v(B) = card(B) for all B £; Y. Both measures Il and v are Radon measures, so that Theorem 1.12 may be applied. The diagonal L\:= {(x, x)IO ~ x ~ I} is closed in X x Y, hencef := 1A is a bounded nonnegative upper semicontinuous function, in particular fis Borel measurable. But
II
lA(x, y) dJ1(x) dv(y)
=0
and
1.14. Another important method of generating new Radon measures from given ones is the formation of image measures. Let X and Y be two Hausdorff spaces, let Il be a Radon measure on X and suppose that the mapping f: X --. Y is continuous. Then a set function III may be defined on the Borel sets of Y by
B E PJ(Y) and it is immediate that III is a-additive, i.e. III is a Borel measure on Y, called the image of Il under f. The simple example of Lebesgue measure on the real line and a constant function shows that the image of a Radon measure need not again be of this type. We have, however, the following positive result which will be sufficient in many cases of interest.
§1. Introduction to Radon Measures on Hausdorff Spaces
29
1.15. Proposition. Let X and Y be Hausdorffspaces, let /l be a Radon measure on X and suppose that f: X --. Y is continuous. ~f /l(f - 1 (K» < 00 for each compact set K £; Y then III is a Radon measure. This condition holds if either fleX) < 00 or if f is proper, i.e. f - I(K) is compact for each compact set K £; Y. PROOF. We have only to verify condition (ii) of Definition 1.1 for /lI, since condition (i) is part of the assumptions. Let B E 8l(Y) be given. For any a < /lI(B) = /l(f - l(B» there exists a compact set K £; f -1(B) such that a < /l(K). Now C := f(K) is a compact subset of Band
/lI(C) = /l(j' -1(C» ~ /l(K) > a
D
which shows condition (ii).
1.16. Later on in this book we will work repeatedly with the so-called convolution of finite Radon measures on a Hausdorff topological semigroup or group. We are now going to give the precise definition of this notion. Let S denote a Hausdorff topological abelian semigroup, i.e. S is a Hausdorff space and there is a composition law + : S x S --. S which is assumed to be associative, commutative and continuous. For a detailed discussion of this subject see Chapter 4. Let /l and v be two finite Radon measures on S. Then their convolution /l * v is defined by
/l
* v := (/l ® v) + ,
i.e. as the image of the product measure /l ® v under the composition law. By the preceding proposition /l * v is again a finite Radon measure on Sand it is not difficult to see that
/l* v = V*/l and
(/l
* v) * K
=
/l
* (v * K)
hold for all /l, v, K E M~ (S), the set of finite Radon measures on S; another way to express this is, ofcourse, that (M~ (S), *) is again an abelian semigroup. We shall see later (cf. 3.4) that M~(S) is even a topological semigroup in a naturally chosen topology. A special case deserves mention. Suppose S is an abstract abelian semigroup, i.e. no topology is given on S in advance. If we then declare every subset of S as open, i.e. if we equip S with the so-called discrete topology, S becomes a topological semigroup in which the compact subsets are just the finite ones. Every molecular measure It
~ /l = L...
CX'G l St
i= 1
(where {Sb ... ' Sn} £; S, {CXI' ... ' CXn} £; IR+) is, of course, a finite Radon measure on S, so that the convolution of molecular measures is well defined.
30
2. Radon Measures and Integral Representations
In fact, if v =
LJ= 1 [JjG tj is a second molecular measure on S then n
J1
*v=
m
L1 L1 r:J.i[JjGSi+tj·
i=
j=
We finish this section by proving the so-called "localization principle" for Radon measures which will turn out later to be very important for the proofs of several integral representation theorems. The following lemma will be needed in the proof of the localization principle.
1.17. Lemma. Let X be a Hausdorffspace and let C £; X be a compact subset covered by finitely many open sets G}, ... , Gn, i.e. C £; GI U ... u Gn • Then there are compact subsets C i £; Gi , 1 ~ i ~ n, such that C = C I u··· U Cn. We use induction on n. For n = 2 we have C £; G I U G 2 , hence the disjoint compact sets C n G~, C n G~ can be separated by open sets VI' V 2 , i.e. PROOF.
C n G~
£;
VI'
Cn
G~ £;
V2
and
But then C I := C n V~ £; G b C 2 := C n V~ £; G 2 and, of course, C = CI U C2 . Assuming the assertion for n, let now C £; G I U U Gn + 1 = (G I u··· u Gn ) U Gn+ I. ThenC = K u Cn + I whereK £; G I u u Gnand C n + I £; Gn+ I are compact. By assumption K = C I u··· U C n for compact sets C i £; G i , i ~ n, thus finishing the proof. D
1.18. Theorem. Let (Gcx)cxED be an open covering of the Hausdorffspace X and on each Gcx let a Radon measure J1cx be given such that J1cx(B) = J1p(B)for each pair of indices r:J., [J E D and for each Borel set B £; Gcx n Gp. Then there is a uniquely determined Radon measure J1 on X such that J1(B) = J1cx(B) if B is a Borel set contained in GCX.
Ui=
PROOF. Let C £; X be compact. We say that C = 1 Ai is a decomposition of C if (Ai) is a finite family of pairwise disjoint Borel sets such that for each i = 1, ... , n the (compact) closure Ai is contained in some GCXi ' r:J.i E D. Decompositions always exist, because by compactness C £; 1 GCXi for suitable r:J.}, ... , r:J. n E D, and by Lemma 1.17 there exist compact sets C i £; GCXi with C = 1 C i · Finally we put Al := C b Ai := Ci\(C 1 U ... u C i - I ) for i = 2, ... , n. If we have two decompositions of a compact set C, C = 1 Ai = UJ=I Bj with Ai ~ GCXi ' Bj £; Gpj , then
Ui=
Ui=
Ui=
it,ualAi) = J/a.(9! Ai n Bj) = it! jt/alAi n B
j)
JI it/fJiAi
n
Bj) = jt/fJj(~1 Ai n Bj) = J/fJJ(B),
31
§ 1. Introduction to Radon Measures on Hausdorff Spaces
so that n
A(C):= L /lexlAi) i= 1
is a well-defined function A: .% -.. [0, 00[. We show that Ais a Radon content. Indeed,ifCf,C 2 andLarecompactsubsetsofXwithC I £; C 2 ,L £; C 2 \Cf, and if C 2 = Al U ... U An is a decomposition, then C I = (AI n C I ) u ... U (An n C 1) and L = (A In L) u ... U (An n L) are decompositions, too, and n
A(C 2 )
-
A(C l )
= L [/lexlAi) - /lex.(A i n
C I )]
i=l n
n
= L/lex.(Ai\C I ) ~ L/lexlAi n L) i= I
= A(L).
i=l
On the other hand, given e > 0, there exist compact subsets K i £; Ai\C I such that /lexlAi\CI) - /lexlKi) < eln, and K = I K i is a decomposition of the compact set K £; C 2 \ C I ; hence
Ui=
n
A(K)
=
n
L/lexlKi) > L/lexlAi\CI) - e i=l
= A(C 2 )
i=l -
A(C I )
-
e.
Let /l be the unique extension of A to a Radon measure on X. Then if B is a Borel subset of some Gex , we have /l(C) = /lex(C) for each compact subset C £; B and therefore /l(B) = /lex(B). If v is another Radon measure on X such that v(B) = /lex(B) for B E 8l(X), B £; Gex , a E D, and if C = I Ai is a decomposition of the compact set C £; X, then
Ui=
n
v( C)
n
= L v(A i) = L /lexlAi) = A( C) = /l( C) i=l
i=l
D
implying, of course, equality of /l and v.
1.19. Exercise. Let X be a Hausdorff space with a countable base ~ (i.e. each open set in X is the union of some subfamily of ~), then 8l(X) equals the a-algebra generated by ~. If Y is a further Hausdorff space with a countable base, then 8l(X x Y) = 8l(X) ® 8l(Y). 1.20. Exercise. Let IRs be the real line equipped with the Sorgenfrey topology (i.e. a neighbourhood base of x E IR is given by {[x, a[ Ix < a < oo}). Then IRs is a Hausdorff space, 8l(lRs) = 8l(IR), but 8l(lRs) ® 8l(lRs) 8l(lRs x IRs). Hint: The topology induced by IR; on the second diagonal
*
Li:= {(x, is discrete.
-x)lx
E
IR}
32
2. Radon Measures and Integral Representations
1.21. Exercise. Show that any Borel measure fl on IR n which is finite on compact sets, is already a Radon measure.
1.22. Exercise. There are a-finite measures on IR which are not Radon measures. E 84(X), then flA: 84(X) ~ fl(A n B) is again a Radon measure.
1.23. Exercise. If fl is a Radon measure on X and A [0,
00
J defined
by flA(B)
:=
1.24. Exercise. If fl and v are Radon measures on X and Y, then SUPP(fl ® v) = SUPP(fl) x supp(v). If fl is finite andj': X ~ Y is continuous, then SUPP(flf) = .f(SUPP(fl)). If X = Y is an abelian topological semigroup and fl, v are both finite, then SUPP(fl * v) = SUPP(fl) + supp(v).
1.25. Exercise. Let X be a Hausdorff space and let the set function fl: 84(X) ~ [0, 00 J be finitely additive, finite on compact sets and inner regular, i.e. fl(B) = SUP{fl(K) I B ;2 K E $'(X)} for all B E 84(X). Then fl is already a-additive and hence a Radon measure. 1.26. Exercise. If fl is a Radon measure on X, v is a Radon measure on Y, and f: X x Y ~ Y x X is defined by f(x, y) = (y, x) then (fl ® v)f = v ® fl.
1.27. Exercise. Let fl and v be two finite Radon measures on a completely
J
J
regular Hausdorff space X. If f dfl ~ f dv for each bounded nonnegative continuous function then fl ~ v, i.e. fl(B) ~ v(B) for all Borel sets B ~ X.
1.28. Exercise. Let X be a Hausdorff space, let (fln) be a sequence of Radon measures on X and let the set function fl: 84(X) ~ [0, 00 J be defined by 00
fl(B) =
L fln(B)
for
B E 84( X).
n=l
Show that if fl(K) < measure on X and
00
for all compact sets K ~ X then fl is a Radon
for all Borel measurable functions f: X functions,f: X ~ c.
~
[0,
00
J
and all fl-integrable
1.29. Exercise. Let X be a Hausdorff space, let (flaJx. E A be an increasing net of Radon measures on X (i.e.
rJ.,
f3 E A,
BE 84(X)), and let the set function fl: 84(X) fl(B)
= sup flex(B) = lim flex(B) exEA
exEA
f3 => flex(B) ~ flp(B) for all [0, ooJ be defined by
rJ. ~
~
for
B
E
84(X).
33
§2. The Riesz Representation Theorem
Show that if Ji(K) < measure on X and
If
00
dJi
for all compact sets K
= sup (lEA
If
dJi(l
If
= lim (lEA
for all Borel measurable functions f: X --. [0,
£;
00
X then Ji is a Radon
dJi(l
J. Furthermore,
for all Ji-integrable functions f: X --. C. (Note that 1.28 is a special case of 1.29.)
1.30. Exercise. Let X = ~ U {oo} be the Appert-Varadarajan space, i.e. all sets {n} £; ~ are open and a subset G £; X containing 00 is open if and only if its "density" lim n _ oo (ljn)IGn {1, ... ,n}1 equals one. Show that X is a normal Hausdorff space in which only finite sets are compact. Therefore the counting measure on X is a Radon measure which is not locally finite. 1.31. Exercise. Let m denote Lebesgue measure on [0, IJ and let X b X 2 be disjoint nonmeasurable subsets of [0, 1] both with outer Lebesgue measure 1. (For the existence of such sets see Oxtoby (197L pp. 23-24).) On Xi we consider the Borel a-algebra 81(XJ = {Xi n BIB E 81([0, I])}, i = 1, 2. Show that
is independent of the choice of B i E 81([0, 1]) such that Ai = Xi n B b i = 1,2, and that
=
for
Ai
E
81(XJ,
i = 1,2.
Hint: Consider the decreasing sequence of sets in 81(X 1) ® 81(X 2)
§2. The Riesz Representation Theorem In the introduction to §1 we have mentioned the close connection between certain linear functionals and Radon measures on locally compact spaces, a connection made precise in the famous Riesz representation theorem. There is, however, a much more general integral representation theorem due to Pollard and Tops~e (1975) which implies not only numerous topological representation theorems but also, for example, the abstract Daniell extension
34
2. Radon Measures and Integral Representations
theorem. We shall not prove this result in full generality, but instead confine our presentation to the topological setting. Let X be a Hausdorff space and let r.c be a convex cone of continuous IR + -valued functions on X, separating the points in X (i.e. if x =f= y then f(x) =f= fey) for some! E r.c). We consider a mapping
T:
r.c ~ [0, 00],
r.c and T fulfil the following conditions: Iff, g E r.c thenf 1\ g E r.c, (f - g) + E r.c and f 1\ 1 E r.c. T(f + g) = T(f) + T(g) for allf, g E r.c. For each.f E r.c we have T(f) = sup{T(g)lf ~ g E r.c, g bounded, T(g) < oo}. For each compact set K £; X there is some.f E r.c with lK
and we shall assume that (i) (ii) (iii)
(iv)
T(f) <
~
f and
00.
(v) Givenf E r.c such that T(f) < 00 and given c > 0 there is a compact set K £; X such that g E r.c, g ~ f and g I K = 0 implies T(g) < c. Note that these conditions are obviously all fulfilled in the "classical" setting where r.c = CC+ (X) is the set of all nonnegative continuous functions with compact support on a locally compact space X, and where T is the restriction to r.c of some positive linear functional on CC(X). An immediate consequence of (i) and (ii) is the monotonicity of T, i.e. we have T(f) ~ T(g) if f ~ g, and this again, together with the additivity of T, implies
T(af) = aT(f)
for all .f
E
r.c
and all
a
E
IR+ .
One might consider condition (v) as the "crucial" one among (i) to (v); it follows from Proposition 1.7 that (v) necessarily holds if T(f) = f dJi for some Radon measure Ji on X. For the proof of the announced representation theorem we need the lemma below.
J
2.1. Lemma. Assuming the above conditions on hold:
r.c
the following properties
(a) For all x =f= y in X there is some.f E r.c such that.f(x) = 1 and.f(y) = o. (b) For all x =t= y in X there are disjoint neighbourhoods U of x and V of y such that j'l U = 1 and f I V = O.for some j' E r.c. (c) If K and L are disjoint compact subsets of X then j'l K = 1 and f IL = 0 for some f E r.c. (d) If K and L are disjoint compact subsets o.f X then there are two functions f, g E r.c such that l K ~ f ~ 1, l L ~ g ~ 1 and.f 1\ g = O. PROOF. (a) With the cone r.c being point separating we find some h E r.c with hex) =f= hey) and without restriction we may assume max{h(x), hey)} = 1.
35
§2. The Riesz Representation Theorem
Using property (iv) there is some g then
.f := 1 _ belongs to put
r.c and.f(x) =
E
r.c such that l{x, y}
1 h(x) (g
1\
1 - h)
~
g. Now if hex) < hey)
+
l,f(y) = 0; if on the other hand hex) > hey) then
h'
:=
1 _lh(Y) (g
1\
1 - h)+,
and in this case.f := (g 1\ 1 - h') + has the desired properties. (b) Using (a) we find two functions g, h E r.c such that g(x) = 1, g(y) = 0, hex) = 0 and hey) = 1. The two sets U
:=
{g >
i, h < i},
V
:=
{g <
i, h > i}
are open disjoint neighbourhoods of x (resp. y), and for the functions g' := t(g 1\ i), h' := t(h 1\ i) we see that g'l U = 1 = h'l V whereas g'l V < and also h'l U < t. Now it is clear that
t
f:= [g' - (h' - g')+J+ is one on U and zero on V. (c) For each pair x E K, y E L there are disjoint neighbourhoods U x of x, Vy of y and functionsfx,y E r.c such thatfx,yl U x = O,fx,yl ~ = 1. Let us first fix a point y E L, then there is a finite subset {Xl' ... ,xn } ~ K such that K ~ U Xl U ... U U x n • The function
fy:= min{fxI,y, ... , fxn,y} belongs to r.c, equals one in some neighbourhood ~ of y and is zero on all of K. Using again (iv) we choose some g E r.c with l KuL ~ g; then
hy := (g
1\
1 - fy) +
is zero on ~ and one on K. Now by compactness of L we find {YI' ... ,Ym} ~ L with L ~ ~l U ... U ~m and finally f := min{h yl ,· .. , h ym } has the desired properties. (d) By (c) and (i) there is some h E r.c such that h ~ 1, hlK = 1 and hlL = 0; again we choose qJ E r.c with l KuL ~ qJ ~ 1. Putting
.f:= [h - (qJ - h)+J+ and
g:= [(qJ - h)+ - hJ+, we have indeedflK = 1, giL = 1 andf
1\
g
=
O.
D
2.2. Theorem. Let X be a Hausdorff space, let r.c denote a point separating convex cone of continuous functions f: X ~ [0, 00 [, and let T: r.c ~ [0, 00 J together with r.c .fulfil the above conditions (i) - (v). Then there is a uniquely
36
2. Radon Measures and Integral Representations
determined Radon measure Ji on X such that
f f dJl
T(f) =
for all f
E
re.
The measure Ji is furthermore locally finite. PROOF. We define
A: $'(X) --. [0, w[ by A(K)
:=
inf {T(f) 11 K ~
f
E
re}
and we shall show that A is a Radon content. It is very easy to see that A is subadditive, i.e. A(K u L) ~ A(K)
+ A(L)
for all K, L E $'(X). If, furthermore, K n L = 0, then Lemma 2.1 ensures existence off, g E rc such that IK ~ f, IL ~ g and.f 1\ g = 0. For any function h E rc with IKuL ~ h we then have A(K)
+ A(L)
~
T(h
1\
= T(h
1\
~
f) + T(h 1\ g) j' + h 1\ g)
T(h)
and therefore A(K)
+ A(L) =
A(K u L),
i.e. A is additive on disjoint compact sets, and this implies for compact sets C l £; C 2 that sup{A(K)IK
£;
C 2 \C l , K
E
$'(X)} ~ A(C 2 )
-
A(C l ).
To show the other direction we choose, given e > 0, some.f E re such that leI ~ f and T(f) ~ A(C l ) + e. We also fix some number a E JO, 1[ and define K ex := C 2 n {f ~ a}. Certainly K ex is a compact subset of C 2 \C l , and if 1K(J( ~ g E rc then
implying A(C 2 ) ~ A(K ex )
I T(f) a
+-
~ A(K ex )
1 [A(C l ) a
+-
+ eJ
°
Taking now on the right-hand side the limit for a --. I and e --. we see that A indeed is a Radon content. Let Ji denote the Radon extension of A. We have to show that Ji represents T, i.e. that T(f) = f dJi for all f E re. To see that dJi ~ T(f) it is certainly enough to prove that
J
J.f
n
h = IailKi ~f i=l
37
§2, The Riesz Representation Theorem
implies n
L QiP(KJ ~
T(f),
i= 1
whenever K l' ... , K n are pairwise disjoint compact sets and Qi > 0 for all i = 1, ... , n. Again we use Lemma 2.1 providing us for each 1 ~ i < j ~ n with functions hj, gij E such that
rc
l K, ~ hj ~ 1,
l Kj ~ gij ~ 1
and hj
1\
gij = O.
For convenience puthi = gii = 1 and let
.f; := (min gki) k~i
Then hi K i = 1 and.f;
1\
(min hj) j~i
1\
(~i f),
i = 1, ... , n.
Q
.f} = 0 for i =t= j. Furthermore,
1\
n
h ~ LQih ~f, i= 1
whence
itl aiJ1(KJ
~ itlaiT(f;) = T(tlaiJ;) ~ T(f), J
and therefore, as already remarked, f dp ~ T(f). The reverse inequality remains to be shown, i.e. T(f) ~ j' dp for each f E C(j. By (iii) we may assume 0 ~ f ~ 1 and T(f) < 00. Given e > 0 we choose a compact set K as indicated in (v). There also exists a function h E C(j with l K ~ h ~ 1 and T(h) < 00, and then for n suitably chosen we have (l/n)T(h) < e. Consider now the compact sets K·:= K n J
J
{r L} '
j
?:. n '
= 1, ... , n
leading to the inequality
(f - -n1) l
K
~
L -1l Kj ~ f n
j= 1
n
and thus to
J.r
1 n { dp ~ - L p(K j) ~ inf T(g)lg n j= 1
E C(j,
g~
Ln -1 l Kj } , j= 1
n
where the last inequality is an immediate consequence of the definition of p on %(X), i.e. the definition of the Radon content A. (In view of what has already been shown, this last inequality is in fact an equality.)
38
2. Radon Measures and Integral Representations
Let us now choose agE
C(j,
g ~ (lin) 1
LJ= 1 1Kj , such that
n
T(g) ~ - L J1(K j ) n j= 1 and without restriction g
f.
Then
(I - g - ~) ~ I,
.f' := and therefore T(f') <
~
+£
f'IK
+
£
=
0
by choice of K. But
I-g~.f'+~, n and so
T(f - g)
= T(f) - T(g)
~
2£,
and finally
T(.f) ~ T(g)
+ 28
n ~ ;;1 j~/(K)
+ 38
~
II
dJ1
+ 38.
We now show uniqueness of the representing measure. Let v be some Radon measure on X which also represents T, i.e. S f dv = T(f) = S f dJ1 for allf E C(j. Given K E %(X) and 1K ~ f E C(j we have
v(K)
~ II dv =
T(.f)
and therefore v(K) ~ J1(K) by construction of J1. For the other direction we choose some h E C(j such that 1K ~ h ~ 1 and T(h) < 00. By 1.7 the set function B ~ SB h dv is a finite Radon measure on X, hence given £ > 0 there is a compact set L disjoint with K such that
r
h dv <
£.
J(KUL)C
Once more by Lemma 2.1 there is some f E C(j with f IK = 1,.f IL = 0 and without restriction f ~ h, implying
J1(K)
~
T(f) =
~
v(K)
+
II dv =
r
r I dv + r I dv + r
JK
h dv < v(K)
JL
I dv
J(KUL(
+ 8.
J(KULf
Hence J1 and v agree on compact sets and are therefore equal. The local finiteness of J1 results as an immediate consequence of property (iv). D
39
§2. The Riesz Representation Theorem
If X is a Hausdorff space we let C(X) denote the vector space of all realvalued continuous functions on X. Three subspaces of C(X) deserve particular interest: the space Co(X) of continuous functions" tending to zero at infinity" in the sense that {I f I ~ e} is compact for each e > 0, the space CC(X) of continuous functions with compact support, where the support of f E C(X) is defined by supp(f)
:=
tl
=+=
O},
and the space Cb(X) of bounded continuous functions, being a Banach space with respect to the supremum norm. Clearly, CC(X)
£;
Co(X)
£;
Cb(X)
£;
C(X)
and the four spaces coincide in case X is compact. If X is locally compact, then it follows from Urysohn's lemma that Co(X) is the uniform closure of CC(X). As a corollary we now get the classical Riesz representation theorem. 2.3. Corollary. Let X be locally compact. Then there is a bijection between all positive linear functionals T on CC(X) and all Radon measures J1 on X
given by T(.f)
= ff
.f
dj1,
E
CC(X).
In the following we also have to consider a-additive set-functions which may assume negative values. If J1l and J12 are two finite measures on a measurable space (X, (JI) then J1 := J1l - J12 is of this type and will be called a finite signed measure. Conversely, the Jordan-Hahn decomposition theorem tells that any a-additive set function J1: (JI ~ IR is representable as the difference of two finite measures J1l and J12' even in such a way that J1l and J12 are concentrated on disjoint measurable sets. It follows in particular that a a-additive real-valued function defined on a a-algebra is bounded. Any difference of two Radon measures will be called a signed Radon measure. This is perfectly well-defined when dealing with finite Radon measures. If infinite Radon measures are involved, the difference is as a set function only well-defined on the family of all relatively compact Borel sets. In the first chapter we have defined for any topological vector space E the topological dual E' consisting of all scalar-valued continuous linear functions on E. We shall now identify E' in some cases where E is a certain space of continuous functions. At first we shall treat the case where X is compact and C(X) denotes the space of all real-valued continuous functions on X. The space C(X) will be given the sup-norm; it is well known that C(X) in this norm is a Banach space, and that (C(X))' is a Banach space, too, if we define the norm of T E (C(X))' by II
TII:= sup 1I!11
~ 1
I
TC.f) I·
40
2. Radon Measures and Integral Representations
The set M(X) of all signed Radon measures on X will be given the total variation norm; if J1 E M(X) then 11J111
:=
sup{ I J1(A) I + I J1(A C )
II A E &leX)}.
2.4. Theorem. If X is compact then (C(X))' = M(X) in the sense that there is a bijective linear isometry J1 ~ TJl from M(X) onto (C(X))', given by the natural mapping TIl(f)
where Jf dJ1:= Jf dJ11 such that J1 = J11 - J12·
Jf
= f f dJ1,
dJ12 is independent ofthe choice o.fJ11' J12 EM + (X)
TJl E (C(X))' for J1 E M(X). For let J1 = J11 - J12 with (positive) Radon measures J11' J12 being concentrated on disjoint Borel sets; then PROOF. At first we remark that
IJ fdJ1
1
~ flfldJ11 + f'f'dJ12 ~ Ilfll·IIJ111
showing also II TJlII ~ 11J11l· Let now T E (C(X))' be given. We then define T+: C+(X) ~ IR by
T+(f)
It follows that
°
~
:=
sup{T(h) I f ~ h E C +(X)},
T + (f)
II TIl II.f II <
~
T+C.f
+ g) ~
Let.f, g, h E C+(X) such that h
~
f
T+(f)
+ g.
h(x)f(x) h'(x):=
{
~(x)
+ g(x)
h(x)g(x) ~(x)
h"(x):=
+ g(x)
{ then h', h"
E
C+(X), h'
~
00,
f
E
and for f, g
+
C +(X). E
C +(X) we have
T+(g).
We put
if.f(x)
+ g(x) >
if f(x)
= g(x) = 0,
if f(x)
+ g(x) >
0,
0,
if.f(x) = g(x) = 0,
f, h"
~
g and h'
T(h)
~
T+(f)
+ h" = h, implying + T+(g)
so that finally T+ is additive on C +(X). We put T- := T+ - T which also is additive, nonnegative and positively homogeneous on C + (X). By Corollary 2.3 there are two Radon measures J1b J12 on X such that
41
§2. The Riesz Representation Theorem
implying for f E C(X) TC{) = T(f+) - T(f-) = T+(f+) - T-(f+) - [T+(f-) - T-(f-)]
= I f+ d(J.11 - J.12) - I f- d(J.11 - J.12) J1
= II dJ.1,
:=
J1l - J12
so that T = ~. By Corollary 2.3 we also get immediately that there is only one signed Radon measure J1 with this property. Let J1 = J1+ - J1- denote the Jordan-Hahn decomposition of J1, i.e. J1 +(B) = J1(B n D) and J1- (B) = J1(B n DC) where D E &leX) is chosen in such a way that both J1 + and J1- are nonnegative (see, for example, Billingsley (1979, p. 373)). Then J1+ and J1- are Radon measures and furthermore
I I dJ.11
=
sup{Ih
~
I f dJ.1+
dJ.1+ - I h dJ.1-IO
~ h ~ f, h E C(X)}
for all.{EC+(X), hence J1l ~ J1+ by Exercise 1.27 and similarly J12 ~ J1-. Consequently we get J1l (DC) = 0 = J12(D) and therefore J1l = J1 +, J12 = J1as well as 1IJ111 = J1l (D) + J12(D C). We still have to show the reverse inequality II TJlII ~ 1IJ111. Given e > 0 we choose compact sets K+ £; D, K- £; DC such that
By Urysohn's lemma there is a continuous functionf: X thatflK+ == 1 andflK- == -1, implying
If
f dJ.11
~ J.11(K+) + J.1iK-) -
2/:
~ 11J.111 -
~
[ -1,1] such
4/:.
This shows II TJlII ~ 1IJ111 and finishes the proof of the theorem.
D
We shall now consider the case where X is locally compact. For a given compact subset K £; X we denote by C K(X) the vector space of all continuous functions on X whose support is contained in K; of course, CK(X) is a Banach space with respect to the supremum norm and CC(X)
=
U
CK(X).
K E Jf"(X)
A linear functional T on CC(X) is called continuous if all the restrictions TIC K(X) are continuous in the usual sense.
2.5. Theorem. For a locally compact Hausdorff space X the continuous linear jiinctionals T on CC(X) are in a bijective linear relation with the signed
42
2. Radon Measures and Integral Representations
Radon measures J1 on X via the natural formula T(.f) =
f f dJ1.
If J1 is a signed Radon measure, i.e. J1 = J1l - J12 for Radon measures J1l' J12' then TCf) = f dJ1 := f dJ1l .f dJ12 is well defined and continuous in the above sense; the unicity of the representing measure J1 is obvious by Corollary 2.3. Now let a continuous linear functional T: CC(X) ---+ IR be given. Imitating the proof of Theorem 2.4 we define for f E CC+ (X) PROOF.
J
J
J
T+(f):= sup{T(h)l.f
~
hE CC+(X)}.
Any function h occurring here belongs to Csupp(flX), implying that for some constant a > 0 IT(h) I ~ allhll ~ all.!'II, hence
o~
T+(j) <
00.
The same arguments used already in the preceding proof show that T+ and also T- := T+ - T are both additive. Again an application of Corollary 2.3 finishes the proof. D Let X be a locally compart space and V £; C(X) a linear subspace of continuous functions. Motivated by the classical moment problem Choquet introduced a simple sufficient condition on V in order that every positive linear functional L: V ---+ IR can be represented by a Radon measure on X as in the Riesz representation theorem where V = CC(X), cf. Choquet (1962, 1969). For functions .f, g: X ---+ IR we write.f E o(g) if formally f jg vanishes at infinity on X, but due to possible zeros of g the precise meaning off E o(g) is the following:
For every e > 0 there exists a compact subset K elg(x)l.for x E X\K.
£;
X such that If(x)1
~
2.6. Definition. A convex cone C £; C +(X) of nonnegative continuous functions is called an adapted cone if: (i) For every x (ii) For everY.f
E E
X there exists.f E C such that.f(x) > O. C there exists gEe such that.f E o(g).
A linear subspace V
£;
C(X) is called an adapted space if:
(iii) V = V+ - V+, where V+ = V n C +(X). (iv) V+ is an adapted cone.
If.f E CC+(X) then.f E o(f) and for.f E C1(X) we have.f E o(J]), which shows that CC(X) and CO(X) are adapted spaces. If p and q are two real
43
§2. The Riesz Representation Theorem
polynomials in one variable then p E o(q) if and only if deg(p) < deg(q). It follows easily that the space of polynomials in one variable is an adapted space of continuous functions on X = IR. Similarly the space of polynomials in k variables is adapted on X = IRk. The main property of adapted spaces is given in the following: 2.7. Theorem. Let V be an adapted space of continuous functions on a locally compact space X. For every positive linear functional L: V ~ IR there exists a Radon measure J1 on X such that V £; ,pl(J1) and
PROOF.
L(f)
= ff
{f
C(X)llfl ~ g for some g E V+}.
for all .f E V.
dp
We define
v=
E
Then V is a subspace of C(X) containing V, and a simple compactness argument combined with (i) shows that CC(X) £; V. We claim that V= V+ + Jt: In fact, iff E Ii and g E V+ is such that If I ~ g, thenf = (g + f) + (-g) shows the assertion. By Corollary 1.2.7 it follows that L can be extended to a positive linear functional L: V ~ IR, and by the Riesz representation theorem there exists a Radon measure J1 on X representing [ ICC(X). For g E V+ and qJ E CC(X) satisfying 0 ~ qJ ~ g we have
f qJ dp.
L(g) = L(g) ;;;; L(qJ) = By U rysohn's lemma the family ff to g, so by Theorem 1.5
f
g dp
=
:=
{qJ
CC(X) I0 ~ qJ ~ g} filters upwards
E
sup{f qJ dpl qJ E.9F} ~ L(g) < 00, J
hence g E !fJl(J1). To see that g dJ1 = L(g) we choose h E V+ such that g E o(h). Let e > 0 be given. There exists a compact set K £; X such that ~
g(x) We choose qJ
E
eh(x)
CC(X) such that Ix
x
for ~
o~ g -
qJ
~
gqJ
E
X\K.
1 and find ~
eh,
hence
o ~ L(g) -
l(gqJ) ~ eL(h),
or
L(g)
~
f
gqJ dp
+ eL(h) ~
f
g dp
+ eL(h),
44
2. Radon Measures and Integral Representations
which suffices since e > 0 is arbitrary and independent of h. The equality
L(g) =
f
g d/l
D
extends clearly from V+ to V.
2.8. Exercise. Let X be locally compact. Show that the dual (Co(X))' of the Banach space CO(X) equals Mb(X), the space of all finite signed Radon measures on X (which is again a Banach space with respect to the total variation norm) in the sense, that there exists a linear bijective isometry from Mb(X) onto (CO(X))'. Hint: Use the one-point compactification of X. 2.9. Exercise. Show that the following conditions are equivalent for a locally compact space X: (i) X is a-compact, i.e. X is a countable union of compact sets. (ii) There is a strictly positive function.r E CO(X). (iii) There is a function f E C(X) with f(x) ---+ 00 for x ---+ 00, i.e. such that tr ~ a} is compact for all a E IR. (iv) C(X) is an adapted space.
2.10. Exercise. Try to find a finite signed Radon measure J1 on IR such that supp(J1+) = sUPP(J1-) composition of J1.
= IR~ where J1 = J1+ - J1- is the Jordan-Hahn de-
2.11. Exercise. Let X be locally compact and consider on CC(X) the following four families (Pi)ielj,j
XI0 =f
II
=
12
= f(X),
{A
£;
= 1, ... ,4, of seminorms: PAC.{):= max 1.{(x)l,
A finite},
xeA
PK(f):= max If(x)l, xeK
13 = {oo},
Poo(f)
:=
sup I f(x) I, xeX
{pili
E
I 4 } = {pip is a seminorm on CC(X) whose restriction to CK(X) is continuous for all K
E
f(X)},
where CK(X) := {f E CC(X) Isupp(f) £; K} is considered as a Banach space with respect to the supremum norm. Let (!) l' ... , (!) 4 denote the corresponding locally convex topologies on CC(X) (which are all Hausdorff). Show that the topological duals of CC(X) with respect to these four topologies are given (by natural identification) in the following way: (CC(X), (D.)' = {J1
E
M(X)lsupp(J1) is finite},
(CC(X), (D2)' = {J1
E
M(X)lsupp(J1) is compact},
(CC(X), (D3)' = {J1
E
M(X) 111J111 < oo},
(CC(X), (D4)'
=
M(X),
~3.
45
Weak Convergence of Finite Radon Measures
where supp(J1):= supp(,u+) decomposition of ,u.
U
supp(J1-) if J1 = J1+ - J1- is the Jordan-Hahn
2.12. Exercise. Let V be an adapted space of continuous functions on a locally compact space X, let F
X be a closed subset and put
£;
V~ = {f E V I f(x) ~ 0 for all x E F}.
Show that any linear functional L: V be represented as
L(f) =
ff
---+
IR which is nonnegative on V~ can for
dp
f
E
V,
where ,u E M +(X) is supported by F.
2.13. Exercise. Let X, Y be locally compact spaces and f: X ---+ Y a continuous surjective mapping such that f-l(K) is compact in X for each compact subset K £; Y. Show that each v E M +(Y) is of the form J11 for some ,u E M +(X). Hint: Use Corollary 1.2.7 and the Riesz representation theorem. 2.14. Exercise. Let X and Y be locally compact spaces and T: CC(X) x CC(Y) ---+ IR a bilinear mapping which is positive in the sense that T(f, g) ~ 0 if.f E CC+ (X) and g E CC+ (Y). Show that
T(f, g) = for some uniquely determined J1
E
ff
® 9 dp
M + (X x Y).
§3. Weak Convergence of Finite Radon Measures The theory of weak convergence of finite Radon measures is a well-developed theory which is of great importance in probability theory, in particular when dealing with stochastic processes. We will later need only a very few basic facts which we are going to develop in this section. Let X be a Hausdorff space and denote by M~(X) the set of all finite Radon measures on X, i.e. all Radon measures J1 with J1(X) < 00. The weak topology on M~ (X) is the coarsest topology such that the functions J1 ~ f dJ1 become lower semicontinuous for every bounded lower semicontinuous .f: X ---+ IR. The family of sets
J
Gf,I:=
{p
E
M:(X)! ff
dp >
t}
is a subbase for the weak topology when f ranges over the bounded lower semicontinuous functions on X and t E IR. The following result is part of the so-called portmanteau theorem (cf. Tops~e 1970, Theorem 8.1).
46
2. Radon Measures and Integral Representations
3.1. Theorem. For J1 E M~(X) and a net (J1(l)(lEA in M~(X) the following properties are equivalent: (i) (ii) (iii) (iv) (v)
J1(l
~
lim lim lim lim
J1 weakly, i.e. in the weak topology; inf J1(l(G) ~ J1(G)for all open G £; X and lim J1(l(X) = J1(X); sup J1(l(F) ~ J1(F)for all closed F £; X and lim J1(l(X) = J1(X); inf Jf dJ1(l ~ Jf dJ1for all bounded lower semicontinuousf: X ~ ~; sup Jf dJ1(l ~ Jf dJ1for all bounded upper semicontinuousf: X -+ IR.
If (i)-(v) are fulfilled, then lim Jf dJ1(l = Jf dJ1 for each bounded continuous f: X ~ IR, and this property implies (i)-(v) if X is in addition a completely regular space. PROOF. By definition (i) is equivalent with (iv), and the equivalence of (ii) and (iii) (resp. (iv) and (v)) is immediate. "(ii) => (iv)" Let f: X -+ IR be lower semicontinuous and assume without restriction 0 ~ f ~ 1. From the obvious inequality
1)+ ~ -I L n
(f - -
n
-
l
1{!>i/n}
ni=l
~
f
we get
lim inf
f
f
d~a ~ ~ ~ lim inf ~a({f > ~}) ~ ~ ~ ~({f > ~}). J
where the last expression converges to f dJ1 as n tends to 00. The implication "(iv) => (ii)" is obvious. If f: X ~ lR is bounded and continuous and J1(l ~ J1 weakly, then by (iv) and (v) f dJ1(l ~ f dJ1. Now suppose that X is completely regular and lim f dJ1(l = f dJ1 for all continuous bounded f: X ~ IR. To show (ii) let G ~ X be open and let K £; G be compact. As an immediate consequence of the very definition of complete regularity we fi~d a continuous function f: X ~ [0,1] such thatflK == 1 andflG c == o. Then
J
J
J
J
~a
~
f
f
d~a -+
f
f
d~ ~ ~(K),
J1(K) and finally lim inf J1(l(G)
~
J1(G), which had to
D
3.2. Proposition. The space M~(X) offinite Radon measures is a Hausdorff space in the weak topology. PROOF. Let (J1(l) be a net in M~(X) converging to J11 as well as to J12 weakly. Then J11 (X) = J12(X) and, of course, to show J11 = J12 it is enough to prove that J11 (B) ~ J12(B) for all Borel subsets B £; X. If A £; X, then the above theorem implies
J11 (A) ~ lim inf J1(l(A) ~ lim sup J1(l(A) ~ J12(A).
47
§3. Weak Convergence of Finite Radon Measures
Now let B E 8l(X) and e > 0 be given. There exist compact sets K 1 ~ B, K 2 ~ BC such that J.l1 (B\K 1) < e and J.l2(B C\K2) < e. A simple compactness argument shows the existence of two open sets G l' G 2 ~ X separating K 1 and K 2 , i.e. K 1 ~ G 1, K 2 ~ G2 and G 1 n G 2 = 0. Then
K 1 ~ G1 ~ G1 ~ G~ ~ K~ and hence G ~
J.l1(B) This holds for all
G
J.l1(K 1) ~ J.l2(K~) ~ J.l2(B)
> 0 so that J.l1 (B)
~
+ e. D
J.l2(B).
In §1 it was shown that given two Radon measures J.l, v on spaces X and ~ there is a unique product Radon measure J.l ® v on X x Ycharacterized by
J.l ® v(C x D) = J.l(C)v(D)
for all compact
C
~
X,
D
~
Y,
and giving for all measurable rectangles the" right" value. In particular the product of two finite Radon measures is finite again (as it should be), and it is only natural to guess that J.l ® v depends continuously (in the weak topology) on both of its arguments.
3.3. Theorem. Let X and Y be two Hausdorff spaces. Then the mapping (J.l, v) ~ J.l ® v from M~(X) x M~(Y) to M~(X x Y) is weakly continuous. PROOF.
In a first step we show that the mapping
X x
M~(Y) ~ M~(X x
Y)
(x, v)~Gx ® v
is continuous. Assume that XiX ~ x and Vix ~ v. Let G 1 , ••• , Gn ~ X and H l' ... , H n ~ Y be open and put U := 1 (G i x Hi). We show first that
u?=
lim inf e xex ® vix(U) ~ This holds trivially if x ~
u?=
1
® v(U).
ex
G i • Suppose now that
1:= {i ~ nix Then there exists some ~o such that we get
E
GJ
+0.
XiX E niEI
G i for all ~ ~ ~o and for those
~
hence lim inf Gxex ® vix(U)
~ lim inf vix(.U Hi) lEI
~ v(.U Hi) lEI
=
ex
® v(U).
48
2. Radon Measures and Integral Representations
Every open set U ~ X x Y has the form U = U;'EA (G;. x H;.) for suitable open sets G;. ~ X and H;. ~ Y. By Theorem 1.5 we can find, given G > 0, finitely many At, ... , An E A such that
ex ®
V(Vl(G A, x RAJ) > ex ® v(U) -
e.
This implies lim inf ex « Q9 v,,(U)
~ lim inf ex
«
Q9
V{Vl (G A, x RAJ)
~ ex Q9 V(Vl (G A, x RAJ) > ex Q9 v(U) -
e.
Hence lim inf Gx ", (8) v(X(U) ~ Gx ® v(U) and 3.1 implies Gx ", ® V(X ~ Gx ® v. The second step will now be an easy consequence of the first one. Let f: X x Y ~ [0, 00] be lower semicontinuous. From the continuity of (x, v) ........ Gx ® v we get that
X x
M~(Y) ~
(x, v) 1-+
[0, 00]
r f(x, y) dv(y) = sup Jyr (n
Jy
1\
f(x, y» dv(y)
n
is also lower semicontinuous, and using the fact that M~(Y) is again a Hausdorff space as well as the Fubini theorem for lower semicontinuous functions (1.12), we may repeat this argument and conclude that M~(X) x M~(Y) ~
(Ji, v) ........
[0, 00],
fxJr
f(x, y) dv(y) dJi(x) =
y
f
f d(Ji (8) v)
Xxy
is lower semicontinuous, too. If now f: X x Y ~ IR is lower semicontinuous and bounded, then f + c ~ for some C E IR, and
°
(j1, v) 1-+
f f d(j1 Q9 v) = f (f + c) d(j1 Q9 v) -
Cj1(X)v(Y)
is again lower semicontinuous, the second term on the right being a continuous function of Ji and v. This finishes our proof. D
3.4. Corollary. If S is a Hausdorff topological semigroup, then so is with respect to convolution.
M~(S)
PROOF. The mapping (Ji, v) ........ Ji * v is continuous as composition of (Ji, v) ........ Ji (8) v and the mapping <1>: M~(S x S) ~ M~(S) defined by (K):= K+, where + denotes the semigroup operation and K+ is the image of K under + ~ cf. 1.15~ 1.16 and Exercise 3.7 below. D
49
§3. Weak Convergence of Finite Radon Measures
It is a general principle in mathematics to approximate complicated functions (or other objects) by simpler ones. Among the Radon measures the so-called molecular measures (i.e. finite positive linear combinations of one-point (or "atomic ") measures) are considered to be "simple" objects. They are dense in the set of all finite Radon measures even in a stronger sense than with respect to the weak topology.
3.5. Proposition. For every Hausdorffspace X the set of molecular measures is a dense subset of M~ (X) with respect to the pointwise convergence on 86(X). PROOF. As the directed set we choose the family A of all finite Borel partitions of X, i.e. the set of all ri = {B 1 , ••• , B n } ~ 86(X) such that B i =F 0, B i n Bj = 0 for i =F j and U7= 1 B i = X, ordered by refinement. For such ri we put, given any f.1 E M~(X), f.1rJ.:= 2:7=1 f.1(BJe xi , where Xi E B i is chosen arbitrarily. Now let B E 86(X)\ {0, X} be given; then for all ri E A finer than rio = {B, Be} we have f.1rJ.(B) = f.1(B); and f.1rJ.(B) = f.1(B) for all ri E A when BE {0, X}. D 3.6. Exercise. Let (f.1rJ.) and f.1 be finite Radon measures on the Hausdorff space X and let them all be concentrated on the Borel subset Y ~ X. Then f.1rJ. -+ f.1 weakly in M~ (X) if and only if f.1rJ.1 Y -+ f.11 Y weakly in M~ (Y). 3.7. Exercise. Let X and Y be two Hausdorff spaces and letf: X -+ Y be a continuous mapping. Then for any f.1 E M~(X) the image measure f.1 f belongs to M~(Y), cf. Proposition 1.15. Show that the transformation f.1 ~ f.1 f from M~(X) to M~(Y) is continuous. 3.8. Exercise. Let X be a Hausdorff space and M~(X) the set of Radon probability measures on X, i.e. M~(X) = {f.1 E M~(X)If.1(X) = I}. Let E := {ex IX E X} be the set of all one-point measures. Show that every {O, 1}valued measure f.1 E M~(X) already belongs to E. Show further that E is a weakly closed subset of M~(X) homeomorphic to X and that E = ex(M~(X». (For the notion of an extreme point see 5.1.) 3.9. Exercise. Let f.1 be a Radon measure on the Hausdorff space X. For K E %(X) define f.1K(B):= f.1(B n K) and f.1K(B):= f.1(B n K)/f.1(K) (if f.1(K) > 0). Show that the net (f.1K) converges to f.1 pointwise on 86(X) and that
li~
f
g d,uK =
f
g d,u
for every f.1-integrable function g: X -+ C. Show that if f.1(X) < (f.1K) (resp. (f.1K» converges weakly to f.1 (resp. f.1/f.1(X».
00
then
3.10. Exercise. Let (prJ.) denote a net of probability Radon measures on the product X x Y of two Hausdorff spaces, and denote by JlrJ. (resp. vrJ.) the
50
2. Radon Measures and Integral Representations
marginal distribution of Pa. on X (resp. Y) (i.e. the image measures under the two canonical projections). If (/la.) converges to /l and ( va.) converges to some one-point measure ey, then (Pa.) tends to /l (8) eye
3.11. Exercise. Let (S, +, 0) denote a Hausdorff topological abelian semigroup with neutral element 0 (cf. 1.16) and let Ji E M~(S). Then 00
Ji*n
n=O
n!
L- =
lim (eo
+ Ji/n)*n
n-+oo
holds in the sense that both limits exist and agree for all Borel subsets of S. Their common value is often called the exponential of Ji and abbreviated exp(Ji). In particular both limits exist and agree with respect to the weak topology.
3.12. Exercise. Let Sand T be two Hausdorff topological semigroups and let h: S ~ T be a continuous homomorphism. Then for Ji, v E M~(S) we have
§4. Vague Convergence of Radon Measures on Locally Compact Spaces In this section X denotes a locally compact Hausdorff space. The vector space CC(X) of continuous functions j: X ~ IR with compact support and the vector space M(X) of signed Radon measures on X form a dual pair under the bilinear form Ji
E
M(X),
4.1. Definition. The vague topology on M(X) is the weak topology a(M(X), CC(X», i.e. the coarsest topology in which the mappings /l ~ (Ji,j> are continuous, when j ranges over CC(X), cf. 1.3.10. In particular the vague topology is a Hausdorff topology. We first remark that for any lower semicontinuous functionj: X the function J11---+
~
[0,00]
f f dJ1
is lower semicontinuous on M + (X) with the vague topology. In fact, from Urysohn's lemma it follows that any suchjis the supremum of the upward filtering family of functions qJ E CC(X) satisfying 0 ~ qJ ~ j,
§4. Vague Convergence of Radon Measures on Locally Compact Spaces
51
so by Theorem 1.5
f f dll sup{f cp dill cp =
E
CC(X), 0
~ cp ~ f}-
A combination of this remark and Theorem 3.1 immediately gives the following relationship between the weak topology on M~(X) and the restriction of the vague topology to M~ (X):
4.2. Proposition. Let (/l(l)(l E A be a net on M ~ (X) and let /l E M ~ (X). Then (/l(l) converges weakly to /l if and only if (/l(l) converges vaguely to /l and lim /l(l(X) = /leX). 4.3. Corollary. The vague topology and the weak topology coincide on the set of Radon probability measures.
M~(X)
Under some extra assumptions on a net (/l(l)(lEA in M +(X) the vague convergence implies the convergence of f d/l(l)(lEA for certain functions f E C(X)\CC(X). As an important example we have
(J
4.4. Proposition. Let (/l(l)(lEA be a net on /l E M~(X). If C :=
then /leX) ~ c, andfor allf
E
M~ (X)
sup /l(l(X) <
converging vaguely to
00,
Co(X) we have (1)
PROOF.
For any qJ
E
CC(X) satisfying 0
~ qJ ~
1 we have
hence
/leX) = sup{
II f -
E
Co(X). For any
I
qJ> 0 ~ qJ ~ 1, qJ E E
CC(X)} ~ c.
> 0 there exists
qJ E
CC(X) such that
qJ II 00 ~ E, and therefore we find
and now it is easy to see that (1) holds.
D
The following result characterizes the relatively compact subsets M ~ M +(X) in the vague topology, i.e. the subsets M for which the vague closure M is vaguely compact.
52
2. Radon Measures and Integral Representations
~
4.5. Theorem. A subset M only if
M +(X) is relatively vaguely compact if and
sup {
for each
qJ E
>III EM}
<
00
CC+(X).
PROOF. The space of functions of CC(X) into IR can be considered as the CC product space n := IR (X) which will be equipped with the product topology. The subset A of positive linear functionals of CC(X) into IR is closed in n, and because of the Riesz representation theorem (2.3) there is a bijection of M +(X) onto A which is a homeomorphism when M +(X) carries the vague topology and A carries the topology inherited from n. Therefore, M ~ M + (X) is relatively compact in the vague topology if and only if the corresponding set of positive linear functionals is relatively compact in n. By the Tychonoff theorem this is the case if and only if { III E M} is relatively compact in IR for each qJ E CC(X), which gives the conditions of the theorem since a subset of IR is relatively compact precisely if it is bounded. D
4.6. Proposition. Let c > O. The set {II
E M~(X)IIl(X) ~
c}
is vaguely compact. PROOF. The set in question is closed by Proposition 4.4 and relatively compact by Theorem 4.5. D
4.7. Corollary. Suppose X is a compact space. Then weak (or vague) topology.
M~(X)
is compact in the
PROOF. By Proposition 4.6 we have that M~(X) is relatively compact in the vague topology, but also closed since 1 E CC(X). The proof is finished by the observation in Corollary 4.3. D The following result was established by Choquet (1962) in his treatment of the moment problem.
4.8. Proposition. Let V be an adapted space of continuous .functions on a locally compact space X and let L: V -+ IR be a positive linear functional. The set of representing measures, i.e. the set for all
f
E
V}
is convex and compact in the vague topology. PROOF. It is clear that C is convex. For qJ E CC+(X) there existsf that qJ ~ f (cf. the proof of Theorem 2.7), hence
<11.
qJ>
;£
ff
dll = LU)
for
11
E
C,
E
V+ such
§4. Vague Convergence of Radon Measures on Locally Compact Spaces
53
so C is relatively compact by Theorem 4.5. Let (1l(l)(lEA be a net from C converging vaguely to 11 E M + (X) and let.f E V+ . By the remark following 4.1 we get
If
I
~ lim inf f
dll
dll a
= L(f).
We choose g E V+ such thatf E o(g), and let e > 0 be given. There exists a compact set K ~ X such thatf(x) ~ eg(x) for x E X\K, and if qJ E C~(X) is such that 1K ~ qJ ~ 1 we havef(1 - qJ) ~ ego From
I
f d(1l - Ila) =
I
f(1 - cp) d(1l - Ila)
+
I
fcp d(1l - Ila)
we therefore get
and sincefqJ
E
CC(X) the last term tends to zero, so we have
IIf Since
f
E
f,
dll - L(f)1
~ 2t:L(g).
> 0 is arbitrary we get Jf dll = L(f) for all.f E V+ and then for all 'I
D
V, thus proving that 11 E C.
Let 11 be a signed Radon measure on X and let G ~ X be an open subset. Then G is itself locally compact and if.f E CC(G) is extended to X by f(x) = {f(X),
0,
X E
G,
x E X\G,
then .f E CC(X). The mapping f 1--+ Jf dll is a continuous linear functional on CC(G), hence by Theorem 2.5 represented by a signed Radon measure on G, denoted III G, and called the restriction of 11 to G. In case of a (nonnegative) Radon measure 11 on X the restriction III G is ofcourse the usual set-theoretical restriction, introduced already after Remark 1.6. Vague convergence of Radon measures is a local concept as the following result shows: 4.9. Theorem. Let (G(l)(lED be an open covering of a locally compact space X. A net (lli)iEI of signed Radon measures on X converges vaguely to 11 E M(X) if and only ~f (Ili IGCX)iE I converges vaguely to (Ill G(l) for each tYw E D. PROOF. The" only if" part is obvious, so suppose that (Ili IG(l)i E I converges to (Ill G(l) for each tYw E D. Letf E CC(X) have the compact support C, and choose tYw 1 , . . . , tYw n ED such that C ~ G(ll U G(l2 U ... u G(ln. By Lemma 1.17 there exist compact sets C k ~ GCXk k = 1, ... , n such that C = C 1 U ... U Cn. By Urysohn's lemma there exist functions qJk E CC(X) such that 1Ck ~ qJk ~ 1Gock
54
2. Radon Measures and Integral Representations
and SUPP(qJk) ~ G(lk' k
= 1, ... , n. The functions f(x)
!k(x)
nqJk(X)
,
XE
C,
X E
X\C,
L qJj(x)
:=
j= 1
0,
belong to CC(X), sUPP(h) ~ G(lk andf
= Lk= 1 fk' so we have for
and the assertion follows.
i E I,
D
The following result will be used occasionally. For the proof, see e.g. Bauer (1978, p. 233).
4.10. Proposition. Suppose that the locally compact space X has a countable base for the topology. Then the vague topology on M + (X) is metrizable. 4.11. Exercise. Let X be locally compact. For a net (Ji(l) on M +(X) and Ji E M + (X) the following conditions are equivalent: (i) Ji(l ~ Ji vaguely; (ii) lim sup Ji(l(K) ~ Ji(K) for each compact K ~ X and lim inf Ji(l(G) ~ Ji(G) for all relatively compact open sets G ~ X; (iii) lim Ji(l(B) = Ji(B) for all relatively compact Borel sets B ~ X such that Ji(oB) = O.
4.12. Exercise. Show that the set of all Radon measures on a locally compact space taking only values in No = {O, 1, 2, ... , oo} is vaguely closed. 4.13. Exercise. Let the Radon measures Ji(l tend vaguely to Ji and assume that all the Ji(l are concentrated on the closed subset Y ~ X. Then Ji is concentrated on Y, too, and Ji(l tends vaguely to Ji also on the locally compact space Y. 4.14. Exercise. Show that (Ji, v) 1---+ Ji ® v is vaguely continuous as a mapping from M +(X) x M +(Y) to M +(X x Y) for two locally compact spaces X and Y. Hint: Use the Stone-Weierstrass theorem. 4.15. Exercise. Property (ii) in Exercise 4.11 above makes sense on any Hausdorff space X and hence can be used to define vague convergence of Radon measures in this generality. Show however that if M +(X) with respect to vague convergence is a Hausdorff space, then X is necessarily locally compact. Hint: If X o E X has no relatively compact neighbourhood, then for a certain net (X(l) in X we have X(l ~ XO , GXat ~ GXQ and GXat ~ o.
§5. Introduction to the Theory of Integral Representations
55
§5. Introduction to the Theory of Integral Representations Throughout this section E denotes a locally convex Hausdorff topological vector space over IR. The theory below can be applied to complex vector spaces by restricting the multiplication with scalars to real scalars. The basic observation, which should be kept in mind when reading the following general theorems, is that a convex polyhedron is the convex hull of its corners, or equivalently, that any point in the convex polyhedron is the centre of gravity of a molecular probability measure on the corners. The basic notion is that of an extreme point of a set, which is a generalization of a corner of a convex polyhedron and defined below. We shall limit ourselves here to those parts of the theory of integral representations which will be needed in the sequel. A detailed exposition can be found in Alfsen (1971) or Phelps (1966).
5.1. Definition. Let A ~ B ~ E be subsets of E. Then A is called an extreme subset of B if for all x, y E B and A E ]0, 1[: AX
+ (1
- A)Y
E
A ~ x, YEA.
A point a E A is called an extreme point of A if {a} is an extreme subset of A. It is easy to see that if A is convex then a only if for all x, yEA: a = !(X
+ y) ~ x
E
A is an extreme point if and
= y = a.
The set of extreme points of A is denoted ex(A) and in some literature called the extreme boundary of A. Notice that" extreme subset of" is a transitive relation in the set of subsets of E. The above definitions of course make sense for an arbitrary real vector space without topology. Let Xl' .•• , X n be points in E and let ,1.1' ••. , An be numbers E [0, 1] with with 1 Ai = 1. The corresponding convex combination of Xl' ..• , X n is the point
L?=
n
b
=
L Ai x i·
i= 1
If J.1 denotes the molecular measure on E with mass Ai at Xi' i.e. n
J.1
=
L AiGxi'
i= 1
56
2. Radon Measures and Integral Representations
then the point b is called the barycentre of JL For any (continuous) linear functional.f on E we have
f(b) =
Jl
AJ(X i ) =
f
f dJ1,
and this is the motivation for the following:
5.2. Definition. Let X be a compact subset of E and let J1 E M~(X) be a Radon probability measure on X. A point bEE is called the barycentre of J1 if and only if
f(b) =
Ix f dJ1
for all j'
E
E'.
Remark. There exists at most one barycentre of J1 E M~(X). In fact, if b l , b 2 are barycentres of J1 we have .f(b l ) = j'(b 2 ) for all.f E E', and since E' separates the points of E we find b l = b2 . As well as barycentre one also encounters the words centre o.f gravity and resultant. The barycentre is "the value of" the vector integral
Ix x dJ1(x), but such a vector integral need not always converge in E, so a barycentre need not exist. We have, however, the following result:
5.3. Proposition. Let X be a compact subset of E such that K = conv(X) is compact. Then.for every J1 E M~(X) the barycentre exists and belongs to K. Conversely, every point x E K is barycentre of some J1 E M ~ (X). PROOF.
For fEE' and J1
E M~(X)
HI =
{x
E
we put
Elf(x) =
ff
dJ1}
The intersection of the H f's for fEE' is the set of barycentres of J1, hence either empty or a singleton. We shall show
nH
feE'
f
n K =t=
0·
The set H f n K is a closed subset of K, so by a result from general topology it suffices to prove that (1)
for an arbitrary finite subset {ft, ... ,fn} ~ E'. For such a subset we define a continuous linear mapping T: E ~ lR n by
T(x) = Cft(x), ... ,fn(x»,
~5.
57
Introduction to the Theory of Integral Representations
and claim that p:=
(5
fl djl, ... ,
5
fn djl)
E
T(K),
which shows (1). If the contrary is true, there exists by the separation theorem 1.2.3 a linear form qJ: IR n ~ IR such that sup qJ(T(K)) < qJ(p).
(2)
The linear form qJ is given as qJ(x) = for some a and defining g = = 1 a i /; E E', (2) can be expressed
I?
= (at, . .. ,an) E ~n,
sup g(K) < { g djl, which is impossible, J1 being a probability. We next have to prove that any x E conv(X) is the barycentre of some J1 E M~(X). This is clear if x E conv(X), in fact such a point is the barycentre of a molecular measure as remarked earlier. For x E conv(X) there exist nets (x~)~ E A of points from con v(X) con verging to x and (J1~)~ E A of molecular measures from M~(X) such that x~ is the barycentre of J1~ for each ~ E A. By Corollary 4.7 there exist J1 E M~(X) and a subnet (J1~p) converging weakly to J1. For fEE' we then have
5
f djl
= li~
5
f djlap
= li~
f(x ap )
which shows that x is the barycentre of J1.
= f(x), D
5.4. Remark. If E is complete then conv(X) is compact for every compact X ~ E, so every J1 E M~(X) has a barycentre in this case. This applies in particular to Frechet spaces and Banach spaces. For details, see, e.g. Robertson and Robertson (1964). Already Minkowski proved that a compact convex set K in ~n is the convex hull of ex(K). In 1940 Krein and Milman found a far-reaching generalization of Minkowski's result:
5.5. Theorem. Every compact convex set K in E is the closed convex hull of its extreme points, i.e. K
= conv(ex(K)).
We first show that any nonempty compact set C has extreme points. We form the family ~ of nonempty, closed extreme subsets of C. Notice that C E~. A Zorn's lemma argument shows that ~ contains a minimal element M with respect to inclusion. To see that M has only one point, which is then an extreme point of C, we assume the existence of x, y E M, PROOF.
58
x
2. Radon Measures and Integral Representations
+y, and choosef E E' such thatf(x) > f(y). Then M o = {zEMlf(z) = supf(M)}
is easily seen to be an extreme subset of M, hence MoE ~ Since M 0 is a proper subset of M we are led to a contradiction. Clearly conv(ex(K» is a compact convex subset of K. If they are not equat there exists by the separation theorem 1.2.3 an fEE' such that
supf(K) > supf(conv(ex(K»).
(3)
The set
M = {x
E
Klf(x) = supf(K)}
is a nonempty compact subset of K, and by the first part of the proof ex(M) 0. Since M is an extreme subset of K we have ex(M) ~ ex(K), but this is impossible due to (3). D
+
For every subset A ~ E we have conv(A) = conv(A). An equivalent formulation of the Krein-Milman theorem is therefore that K = conv(ex(K». Using Proposition 5.3 we can reformulate the theorem in the following way:
5.6. Theorem. Let K be a compact convex set in E. Every x centre of a measure Jl E M~(ex(K».
E
K is the bary-
A natural and important question in connection with the above theorem is whether Jl can be chosen such that Jl is concentrated on the set of extreme points, i.e. such that Jl(K\ex(K» = O. The answer is yes if K is metrizable, and this is the content of Choquet's theorem. Tbe answer is no in general if K is nonmetrizable, simply because ex(K) can be a nonmeasurable subset of K in this case. There is however a satisfactory solution to the question in the nonmetrizable case also, related to the notio.n of a boundary measure. For a compact convex subset K of E we define a partial ordering <J on M~(K) by Jl <] v
f
~ f
dJl
for all continuous convex functionsf: K
f
~ f
dv
-+ ~.
5.7. Definition. A measure Jl E M~(K) is called a boundary measure if it is maximal with respect to the ordering <J. A boundary measure Jl is pseudo-concentrated on ex(K) in the sense that Jl(G) = 0 for all G~-sets G ~ K such that G n ex(K) = 0. (We recall that a subset of a topological space is called a G~-set if it is the intersection of countably many open sets.)
59
§5. Introduction to the Theory of Integral Representations
In the metrizable case the set ex(K) of extreme points is a Borel set, in fact even a G~-set. Furthermore, any compact subset of K is a Gd-set. It follows that a boundary measure Jl on K is concentrated on ex(K) in the ordinary sense. We can now formulate the generalization of Theorem 5.6 going back to Choquet in the metrizable case and to Bishop and de Leeuw in the general case. 5.8. Theorem. Let K be a compact convex set in E. Every x centre of a boundary measure Jl
E
E
K is the bary-
M~(K).
Remark. In our applications of the present theory the compact convex sets K which we consider always have the property that ex(K) is closed, so we may apply Theorem 5.6 instead of Theorem 5.8. We will mention briefly the very important modern notion of a simplex.
5.9. Definition. A compact convex set K ~ E is called a simplex if every x E K is the barycentre of precisely one boundary measure" and it is called a Bauer simplex if furthermore ex(K) is closed. 5.10. Example. Let X be a compact Hausdorff space and let E = M(X) be the vector space of signed Radon measures with the vague topology. The set K = M~(X) is a compact convex set, and by Exercise 3.8 ex(K) = {Gx Ix E X} is a compact set. It is easily seen that K is a Bauer simplex. In applications we often want to give an integral representation of the elements in a convex cone C. This is possible if C has a compact base B, i.e. a compact convex subset B ~ C\{O} such that for any x E C\{O} there exists a unique number A. > 0 with Ax E B. The following general result was proved by Neumann (1983) and will be applied later. Let X be a nonempty set, let E = ex be the vector space of functions f: X -+ e with the topology of pointwise convergence and let K ~ E be a compact convex set. We assume that y: X -+ X is a mapping and define the following subsets of K:
II f(x) r = {f E K II f(x)
Ky
=
{f E K
2
~ f(y(x» for all x EX},
2
= f(y(x» for all x EX}.
1
1
5.11. Proposition. With the above notation K y is a compact convex set and
r
~
ex(K y ).
The function Z 1---+ I Z 12 is convex from e to IR and therefore K y is a convex set which is clearly closed, hence compact. Let fEr and suppose PROOF.
60
2. Radon Measures and Integral Representations
j' = t(g
+ h) with g, h E Ky.
t(
I
g(X) 12
+
I
For x
E
X we then have
h(x) 12 ) ~ !(g(y(X))
= If(x)1
2
+ h(y(x))) = f(y(x)) = ilg(x) + h(x) 12
whichimplieslg(x) - h(x)1 2 ~ O,henceg = h.Thisshowsthatfisanextreme point of Ky. D
5.12. Corollary. If ex(K) PROOF.
hence
r
~
r
then ex(K) =
r.
If ex(K) ~ r, we get by the Krein-Milman theorem that K = K y , s; ex(K) by 5.11. D
5.13. Exercise. Let K be a metrizable compact convex set in E and let --+ [0, oo[ be a metric defining the topology of K. For each n E N
d: K x K define
Fn
=
{x
E
Klx = t(y + z), y, Z E K, d(y,
z) ~~}
Show that
nK\F 00
ex(K)
=
n
n=1
and deduce that ex(K) is a
G~-set.
5.14. Exercise. Let K be a compact convex set in E and let b:
M~(K) --+ E
be the mapping which to fl E M~(K) associates the barycentre b(fl) of fl. Show that b is continuous when M~(K) carries the weak topology and that b is affine.
5.15. Exercise. Let K be the set of stochastic n x n matrices, i.e. A = (aij) E K if and only if aij ~ 0 and Lj= 1 aij = 1 for i = 1, ... , n. Show that K is a compact convex set in the vector space of all real n x n matrices with the canonical topology. Show that A is an extreme point of K if and only if each row has n - 1 zero entries. Show that the set On of doubly stochastic n x n matrices is a compact convex subset of K"l where a stochastic matrix A is called doubly stochastic if also the sum of each column is one. Show that the extreme points of Q n are the permutation matrices arising from the unit matrix by permutations of the columns. (This result is due to G. Birkhoff.) Hint: Consider the matrix as a chessboard. If a doubly stochastic matrix A is not a permutation matrix there exists a closed circuit for a rook with each move starting on a position ij with 0 < aij < 1. Adding successively G"l -G"l G"l ..• and -G"l G"l -G"l .•. at each position of the circuit with G > 0 sufficiently smalt we get two matrices A+ and A_ in On such that A = t(A+ + A_).
61
Notes and Remarks
5.16. Exercise. Let K be the unit ball {x E Hlllxli ~ I} ofa Hilbert space H. Show that ex(K) is the topological boundary of K, i.e.
ex(K)
= {x E HI IIxll = I}.
5.17. Exercise. Let (X, d) be a measurable space and let E be the Banach space over [K of bounded .91 -measurable functions f: X -+- [K with the uniform norm IIf'II = sup{lf(x)llx E X}. Show that the set of extreme points of the unit ball {f E E III f I ~ I} of E is the set of functionsf E E with If(x)1 = 1 for all x EX. Formulate and prove a similar result when X is a compact Hausdorff space and E = C(X). 5.18. Exercise. Let K denote the set of con vex functions f: ]0, 2IJ [ -+- [0, 1] considered as a subset of IRl O, oo[ with the topology of pointwise convergence. Show that K is a metrizable compact convex set. Show that every f E K is decreasing, continuous and differentiable from the left and the right. For E K and t > 0 we define
.r
!rex) = {f(t) + (x - t)f'-(t) f(x)
for 0 < x ~ t, for x > t.
Show that!r, f - !r E K and deduce that the extreme points of K are the following functions qJo == 0, qJoo == 1, qJt(x) = (1 - x/t) + , 0 < t < 00. Show finally that K is a Bauer simplex. (This way of finding the extreme points of K is taken from Johansen (1967).) 5.19. Exercise. Show that the Riesz representation theorem on a compact Hausdorff space is a special case of the Krein-Milman theorem. 5.20. Exercise. (Douglas 1964). Let V be an adapted space of continuous functions on a locally compact space X and let C be the convex set of representing measures for a positive linear functional L: V -+- IR. Show that J1 E C is an extreme point of C if and only if V is dense in 2 1(X, J1). 5.21. Exercise. Let K and L be compact convex subsets of locally convex spaces~ and letf: K -+- Lbe continuous~ affine and onto. Show that ex(L) ~ f(ex(K))~ and use this result to determine the set of extreme points of
L:= {( ft df.1(t), ft df.1(t),···, ft" df.1(t)) If.1 E 2
M~([O, I])}.
Notes and Remarks We have not assumed a Radon measure to be locally finite. The reason for this is simply that we do not need this condition to derive any of the main results and, on the other hand, on many spaces a Radon measure is automatically locally finite. Certainly this is the case for locally compact spaces
62
2, Radon Measures and Integral Representations
but it holds, for example, also for metric spaces: Let fl be a Radon measure on the metric space X and suppose there is some x E X such that fl(G) = 00 for each open set G containing x, then in particular fl(B n ) = 00 if B n is the open ball of radius lin around x, and hence fl(K n ) ~ n for a suitable compact set K n ~ B n , where without restriction x E K n • Now the point is that K:= U:= 1 K n is again compact, because if K ~ UAEA GA for a family of open sets (G A)' then for some Ao we have x E GAo' and then for some no x E B no ~ GAo implying U:=no K n ~ GAO' The fact that fl(K) = 00 shows that our assumption was wrong. Exercise 1.30 shows that non locally finite Radon measures may occur, and this depends on the fact that each compact subset is finite. As another example where this is the case we mention the fine topology of potential theory, for instance on [R3, cf. Helms (1969). The fine topology is by definition the coarsest topology on 1R 3 in which all superharmonic functions are continuous. The fine topology is completely regular (Brelot 1971, p. 5), and the fact that every finely compact set is finite is proved in Helms (1969,
p.208). Next we want to relate our approach to Radon measures with the "classical" one as developed, for example, in Bourbaki (1965-1969). There, as already mentioned in the introduction, the "functional point of view" is prevalent, a (Radon) measure fl being by definition a positive linear form on CC(X), if X is locally compact. Two set functions are then considered, fa mesure exterieure fl* defined by fl*(G) = sup{l.f
E
CC(X), 0 ~
f
~
IG}
for open
G
~
X
and /l*(A)
= inf{fl*(G)IA
~
G, G open}
for
A ~ X,
and fl' derived from l'integrale superieure essentielle and given by /l'(A)
=
SUP{fl*(A n K) IKE %(X)}
for
A ~ X.
Both fl* and fl' are Borel measures, i.e. a-additive when restricted to 8l(X), and fl' is a Radon measure in our sense whereas /l* is not so in general. One has fl' ~ fl* and they agree on open sets and on Borel sets B with /l*(B) < 00, and in particular on compact sets. It follows that for locally compact and a-compact spaces one has /l'(B) = /l*(B) for all BE 8l(X), so for these spaces Bourbaki's notion of a Radon measure is equivalent to ours. This holds in particular for compact spaces. Bourbaki (1965-1969, Ch. IV, §L Ex. 5) gives an example where fl'(r') = 0 and fl*(r') = 00 for a certain closed subset r' in some locally compact space, and this shows that fl* is not a Radon measure in our sense. Bourbaki defines a (Radon) premeasure on a Hausdorff space X as a mapping W which to every compact subset K ~ X associates a Radon measure WK on K such that WK IL = WL if L is a compact subset of K. Then the following set function W' is considered W'(A) = sup{(WK)'(A n K)IK
E
%(X)}
for
A
~
X.
63
Notes and Remarks
The restriction of W· to f4(X) is a Radon measure in our sense. Bourbaki calls W a (Radon) measure if W· is in addition locally finite. It follows that Bourbaki's notion of (Radon) measures is equivalent with locally finite Radon measures in our sense. Let X be a Hausdorff space and consider Borel measures v: f4(X) --+ [0, 00 ] satisfying (a) (b) (c)
v(K) < 00 for K E f(X); v(G) = sup{v(K)IK s; G,KEf(X)} foropen G s; X; v(B) = inf{ v(G) I B s; G, G open} for B E fJ6(X).
There is a one-to-one correspondence between locally finite Radon measures J.l on X and Borel measures v satisfying (a), (b) and (c). In fact, if J.l is a locally finite Radon measure then J.l*(B) = inf{J.l(G)IB s; G, G open}
for
B
E
fJ6(X)
is a Borel measure satisfying (a), (b) and (c), and if v has these properties then v·(B)
= sup{ v(K) I K
s; B, K E f(X)}
for
B
E
fJ6( X)
is a locally finite Radon measure. Furthermore (J.l*r = J.l and (v·)* = v. For the proof of these assertions see Schwartz (1973), where a third equivalent definition of a locally finite Radon measure is given, namely as a pair (m, M) of Borel measures satisfying certain conditions realized by (J.l, J.l*) and (v·, v) in the above notation. Notice that a Borel measure v satisfying (b) and (c) is locally finite if and only if (a) holds. In the locally compact case the exterior measure J.l* satisfies (a), (b) and (c) and (J.l*r = J.l., (J.l.)* = J.l*. The generalized monotone convergence theorem (1.5) involves only the values of the underlying Radon measure J.l on open sets, so it holds for any Borel measure v which agrees with J.l on open sets. In particular, we have
°
for each lower semicontinuous function f ~ on X. For a continuous real-valued functionj'integrability with respect to J.l and v are equivalent and Sf dJ.l = Sf dv in case of integrability. Bourbaki (Ch. IX, §3) also considers the possibility of "extending" a set function A: f(X) --+ [0, oo[ to a Radon measure, however the crucial property (1) of our §1, the defining property of a Radon content, which goes back to Kisynski (1968), is not discussed there. Theoreme 1 of §3 in Bourbaki should be compared with our Lemma 1.3. Theorem 1.4 is due to Kisynski. Replacing the Hausdorff space X by an abstract set and the family f(X) of compact subsets of X by a suitable set system called "compact paving", Tops¢>e (1978) proved an abstract measure extension theorem which not only contains Theorem 1.4 but also, for example, Caratheodory's classical result.
64
2. Radon Measures and Integral Representations
It should be mentioned that on many "nice" spaces a finite Borel measure is automatically a Radon measure. This holds in particular on Polish spaces, i.e. separable and completely metrizable spaces, cf. for example, Bauer (1978, Satz 41.3), but it can even be shown for so-called analytic spaces, i.e. Hausdorff spaces which are the continuous image of some Polish space; see Dellacherie and Meyer (1978, Chap. III) for a proof. The last-mentioned book also contains a proof of the bimeasure theorem for finite Radon bimeasures on separable metric spaces. A slightly more general version may be found in Morando (1969), but both results are in fact a special case of a theorem of Marczewski and Ryll-Nardzewski (1953) about nondirect products of measures. The Riesz representation theorem is certainly a cornerstone of functional analysis. Our proof is based on Pollard and Tops¢e (1975) and we refer to the references given there, in particular to Batt (1973) for further information on this important topic. The theory of adapted spaces has had important applications in potential theory, see Sibony (1967-1968). The theory of weak convergence of finite (or probability) Radon measures is mainly motivated by its applications in probability theory and mathematical statistics. For a thorough treatment on metric spaces we refer to Billingsley (1968) and Parthasarathy (1967). Later on Tops¢>e (1970) discovered that a satisfactory theory of weak convergence can be developed on arbitrary Hausdorff spaces. For a completely regular space X the weak topology on M~(X) is induced by the weak topology a(Mb(X), Cb(X)). However, for Hausdorff spaces in general it is not possible to extend the weak topology from M~(X) to Mb(X) in such a way that Mb(X) is a Hausdorff topological vector space. In fact, if such an extension was possible then Mb(X) would be completely regular and so would {£xlx EX}, which is homeomorphic to X by Exercise 3.8. A particularly important topic for probabilistic applications is the characterization of relatively compact subsets of M~(X) in the weak topology, and we should mention the striking result due to Prohorov: For Polish spaces X a subset M ~ M~(X) is weakly relatively compact if and only if for each £ > 0 there is a compact set K ~ X such that SUPjiEM J1(X\K) < £, a condition on M called uniform tightness, see Billingsley (1968, Theorems 6.1 and 6.2). Theorem 3.3 may be found in Ressel (1977); Exercise 3.10 is a generalization of Slutsky's theorem, cf. Ressel (1982b). For a locally compact space X the space CC(X) is often equipped with the inductive limit topology of the Banach spaces CK(X), K E f(X), appearing before Theorem 2.5. With this topology, which is equal to the topology gi ven in Exercise 2.11 by the family 14 , CC( X) is a barrelled space, and the topological dual space is M(X) with the vague topology. This approach is the starting point in Bourbaki (1965-1969), who also seems to be the first who has systematically studied the vague topology. Theorem 4.5 is a special case of the Alaoglu- Bourbaki theorem, cf. Exercise 1.3.11. In the special case of X = IR Theorem 4.5 is sometimes called Helly's selection theorem. In
Notes and Remarks
65
fact it follows from 4.5 and 4.10 that any sequence of measures in M~(IR) with bounded total mass has a vaguely convergent subsequence. The result in Exercise 4.15 is due to TopscPe. The importance of the theory of integral representations lies undoubtedly in the fact that it gives a unified approach to a great number of classical formulas and theorems, cf. Phelps (1966). Let usjust mention here Herglotz' formula for nonnegative harmonic functions in a ball, the far more general Martin representation, and the theorems of Bernstein and Bochner. In Chapter 4 we shall use the theory to prove integral representation theorems for positive definite functions on abelian semigroups. The idea of considering a point in a metrizable compact convex set K as the barycentre of a probability measure concentrated on ex(K) is due to Choquet, and the whole theory is often called Choquet theory. In our applications of the theory we use only the special case where ex(K) is closed, in which case the representation theorem is equivalent with the Krein-Milman theorem. Therefore we have given a complete proof of the latter and only indicated the general results, which can be found in many books, for example, Alfsen (1971) and Phelps (1966).
CHAPTER 3
General Results on Positive and Negative Definite Matrices and Kernels
§1. Definitions and Some Simple Properties of
Positive and Negative Definite Kernels When dealing with positive and negative definite kernels a certain amount of confusion often arises concerning terminology. A positive definite kernel defined on a finite set is usually called a positive semidefinite matrix. Sometimes it is only called "positive", which may be misleading. When working on groups, the name positive definite function is used traditionally. In our previous papers on abelian semigroups we also followed this tradition. Instead of calling a kernel t/J negative definite, some authors call the kernel - t/J "conditionally positive definite" or "almost positive." In this book we use mainly the larger class of "semidefinite" kernels of all kinds and therefore prefer to avoid the prefix" semi" which otherwise would appear several hundred times. Adapting the above point of view, an n x n matrix A = (ajk) of complex numbers is called positive definite if and only if n
L
cjcka jk
~ 0
j,k= 1
for all {c 1 , ••• , Cn} ~ c. It is well known that this is the case if and only if A is hermitian (i.e. ajk = akj for j, k = 1, ... , n) and the eigenvalues of A are all ~ o. Similarly A is called negative definite if and only if A is hermitian and n
L
j,k= 1
cjckajk
~0
§I. Definitions and Some Simple Properties of Positive and Negative Definite Kernels
67
for all {c l , ... , Cn} ~ C with the extra condition Lj= I Cj = O. (This definition requires n ~ 2. Any 1 x 1 matrix A = (all) with real all is called negative definite.)
1.1. Definition. Let X be a nonempty set. A function qJ: X x X called a positive definite kernel if and only if
--+
C is
n
L
j,k= I
CjCkqJ(Xj, Xk) ~ 0
for all n E N, {Xl' ... ' X n } ~ X and {C l , ... , Cn} ~ C. We call the function qJ a negative definite kernel if and only if it is hermitian (i.e. qJ(Y, x) = qJ(x, y) for all x, Y E X) and n
L
j,k= I
CjCkqJ(xj, Xk) ~ 0
for all n ~ 2, {Xl' ... , x n} ~ X and {c l , ... , cn} ~ C with Lj= I Cj = O. If the above inequalities are strict whenever X I' ... 'Xn are different and at least one of the C I' ... , Cn does not vanish, then the kernel qJ is called strictly positive (resp. strictly negative) definite.
1.2. Remark. In the above definitions it is enough to consider mutually different elements Xl' ... , Xn E X. In fact, if Xl' ... , Xn E X are arbitrary and x a1 ' • • • , x ap are the mutually different elements among the x/s, then n
L
p
j,k= I
cjCkqJ(x j , Xk) =
L
j,k= I
djdkqJ(x aj , x ak ),
where dk
:=
L
Ci ,
k = 1, ... ,p.
{iIXi=X czk }
Furthermore, if (1: X --+ X is a bijection, then qJ is a positive (resp. negative) definite kernel if and only if qJ «(1 x (1) is a positive (resp. negative) definite kernel. If X is a finite set, say X = {Xl' ... , x n }, then plainly qJ is positive (resp. negative) definite if and only if the n x n matrix 0
(qJ(Xj' Xk»l ~j,k~n is positive (resp. negative) definite. We now list some simple properties and examples of positive and negative definite kernels.
1.3. A kernel qJ on X x X is positive (resp. negative) definite if and only if for every finite subset X 0 ~ X the restriction of qJ to X 0 x X 0 is positive (resp. negative) definite.
68
3. General Results on Positive and Negative Definite Matrices and Kernels
1.4. If qJ is positive definite, then qJ(x, x) ~ 0 for all x on the diagonal L\:= {(x, x)1 x EX}. 1.5. Let (:
E
X, i.e. qJ is nonnegative
:) be a positive definite 2 x 2 matrix. Then
o~
(1,1)(: :)C) =
a
+b+c+d
= - 1m c. Further,
implying 1m b
o~
0,0(: :)( _~) =
implying Re b = Re c, i.e. b definite kernel is hermitian.
= c.
a - ib
+ ic + d
It follows immediately that any positive
qJ on X x X is positive (resp. negative) definite if and only if qJ is symmetric (i.e. qJ(x, y) = qJ(Y, x) for all x, Y E X) and
1.6. A real-valued kernel n
L
~ 0
CjCkqJ(X j , x k )
(resp.
~O)
j,k= I
for all n E N, {Xl' ... , X n} £ X and {c l , ... , en} £ IR (resp. addition). For, if Cj = Qj + ib j , Qj and b j being real, then n
L
LJ= 1 Cj
=
0 in
n
Cj0c qJ(Xj' x k )
=
j,k= I
L
(QjQ k
+ bjbk)qJ(x j , x k)
j,k= I n
+i L
(bjQ k -
Qjbk)qJ(X j , X k ),
j, k= I
and the last sum is zero if qJ is symmetric.
1.7. A 2 x 2 matrix (:
~) is negative definite if and only if a, d E IR, b = c
and
and this inequality is equivalent with a
+d
~
2 Re b.
Therefore, we have for any negative definite kernell/J the inequality
l/J(X, x)
+
l/J(y, y) ~ 2 Re l/J(x, y).
69
§l. Definitions and Some Simple Properties of Positive and Negative Definite Kernels
1.8. Let (:
(w,
~) be a hermitian 2 x
Z)(: ~)(~) = alwI
2
2 matrix. Then for z,
iC we have
+ 2 Re(bzw) + dlzl 2 2
b 1 a w + ~z
=
WE
2
Izl + -;;(ad -lbI 2 )
(for a =F 0).
I
The matrix is therefore positive definite if and only if a det(:
~) = ad -
Ibl 2
~
0, d
~
0 and
~ O.
Hence for any positive definite kernel qJ we have I
qJ(x, y) 12 ~ qJ(X, x) . qJ(Y, y).
1.9. Iff: X -+ C is an arbitrary function, then qJ(x, y) definite, because
j.~lCjCkqJ(Xj'
Xk)
=
IJl cJ
(X)12
:=
f(x)f(y) is positive
~ O.
The kernel ljJ(x, y) := f(x) + f(y) is negative definite, for if IJ= 1 Cj = 0, then even IJ,k= 1 cjCkljJ(x j , x k) = O. In particular, a constant kernel (x, Y)H C is positive definite if and only if c ~ 0 and negative definite if and only if C E IR. 1.10. The kernelljJ(x, y) = (x - y)2 on IR x IR is negative definite, + Cn = 0 implying (for real numbers Cj' see 1.6)
In j,k=l
CjCk(X j - Xk)2 = -2
(
In
CjX j
C1
+ ...
)2 ~ o.
j=l
1.11. If X is a nonempty set, then the family of all positive (resp. negative) definite kernels on X x X is a convex cone, closed in the topology of pointwise convergence. A very important property of positive definite kernels is their closure under pointwise multiplication which was proved by Schur (1911) (in the case of matrices): 1.12. Theorem. Let qJl' qJ2: X x X -+ C be positive definite kernels. Then qJ2: X x X -+ C is positive definite, too.
qJl .
PROOF. It suffices to prove that if A = (a jk ) and B = (b jk ) are positive definite n x n matrices, then C := (ajkbjk) is positive definite.
70
3. General Results on Positive and Negative Definite Matrices and Kernels
Now it is well known from linear algebra (and also follows from 3.1 below) that there are n functions .fI' ... ,.fn: {I, ... , n} -+ e such that n
ajk = Let
C l' ... , Cn E
L fp(j)fp(k),
for
j, k
p=1
= 1, ... , n.
e be arbitrary, then n
n n cjckajkbjk = L L cjfp(j)ckfp(k)bjk ~ j,k= 1 p= 1 j,k= 1
L
o.
D
1.13. Corollary. Let qJt: X x X -+ e and qJ2: Y x Y -+ e be positive definite kernels. Then their tensor product qJt ® qJ2: (X x Y) x (X x Y) -+ e defined by qJt ® qJ2(X t , Yt, x 2, Y2) = qJt(X t , x 2)· qJ2(Yt, Y2) is also positive definite. PROOF. qJt(X t ,
If (PI: (X x Y) x (X x Y) -+ e is defined by (PI(X t , Yt' X 2, Y2) = x 2) and analogously (P2(X t , Yt, X 2' Y2) = qJ2(Yt, Y2)' then qJt ® qJ2 =
D
(PI . (P2 and is therefore positive definite.
1.14. Corollary. Let qJ: X x X -+ e be positive definite such that IqJ(x, y) I < p for all (x, y) E X X X. Then if f(z) = L:=o anz n is holomorphic in {z E e II z I < p} and an ~ 0 for all n ~ 0, the composed kernel f qJ is again positive definite. In particular if qJ is positive definite, then so is exp( qJ). 0
PROOF. By Theorem 1.12 for each n E ~ the kernel qJn is positive definite, therefore L~= 0 an qJn is positive definite for all N E ~ and so is its pointwise limit f 0 qJ. D
1.15. Remark. In contrast to Theorem 1.12 above the ordinary matrix product of two positive definite matrices is positive definite if and only if the two matrices commute. This follows from the simultaneous diagonalization of these matrices. In particular the matrix exponential of any positive definite matrix again has this property. By using the Jordan decomposition one can show that the matrix exponential of every symmetric real matrix is positive definite (even strictly). The following remarkable criterion for strict positive definiteness is often useful.
1.16. Theorem. Let A strictly positive definite
=
(a jk ) be some hermitian n x n matrix. Then A is
if and only if
det«ajk)j,k~p)
> 0
for p = 1, ... , n. Suppose first A to be strictly positive definite. As in the proof of 1.12 we choose n vectors z t, ... , Zn E en such that
PROOF.
j, k
= 1, ... , n,
§l. Definitions and Some Simple Properties of Positive and Negative Definite Kernels
71
which of course can also be written as A = BB*, where B is the n x n matrix with rows Zl' ..• ' Zn. This implies det A = Idet BI 2 ~ 0 and certainly A cannot be singular so that in fact det A > o. Obviously the same reasoning can be applied to the submatrices (ajk)j,k~P for p = 1, ... , n. For the other direction, we proceed by induction on n. The case n = 1 being trivially true, let us suppose that the theorem holds for n - 1. By assumption all> 0 and we subtract a1k/a11 times the first column from the kth column, k = 2, ... , n. The new matrix (a}k)j,k~n (where the first column remained unchanged whereas for k ~ 2 we have a}k = ajk - (a 1k/a 11 )aj1) has the same principal minors as (ajk), i.e. det«ajk)j,k~p)
=
det«ajk)j,k~p),
p
= 1, ... , n
and if we now change the matrix (a}k) to B, where all
0
0
B=
then still det«bjk)j,k~p) = det«ajk)j,k~p) for all p, and B is furthermore hermitian. Now
for p = 2, 3, ... , n implying by assumption that the (n - 1) x (n - 1) matrix
n
+
L C 1c;a 1 k + k=2
2
I C1 1
all
72
3. General Results on Positive and Negative Definite Matrices and Kernels
If (c 2 , •.. , cn) =t= 0 then the first sum is >0, and for (c 2 , ... , cn) == 0 but CI 0 the second term is strictly positive. Hence A is a strictly positive definite matrix. D
*
One might expect that a corresponding result for positive definite matrices holds if the determinants in the above theorem are only supposed to be nonnegative. The simple counterexample given by the 2 x 2 matrix
(0o 0)
-1 destroys this hope. However, the following result seems to be rather satisfactory, at least theoretically.
1.17. Theorem. Let definite
qJ:
X x X
-+
C be a hermitian kernel. Then qJ is positive
if and only if det«qJ(x j , Xk»j,k~n) ~ 0
for all n
E ~
and all
{Xl' ... ,
~
xn}
X.
PROOF. If qJ is positive definite, then as in the beginning of the proof of the above theorem we see that all determinants in question are nonnegative. Let us on the other hand assume this condition. We define a slightly perturbed kernel qJe:== qJ + e· 1.1' where e > 0 and L\ is the diagonal in X x X. For mutually different elements Xl' ... , X n E X it is easily seen that n
det«qJe(x j , Xk»j,k~n) ==
Ld
p
P e ,
p=o
where dn == 1 and dp ==
L
det«qJ(x j ,
Xk»j,keA)
~ 0
AS{I, ... ,n}
IAI=n-p
for p == 0, 1, ... , n - 1. Therefore det«qJe(x j , Xk»j.k~n) ~ en > 0 implying that qJe is a strictly positive definite kernel. Hence the pointwise limit qJ == lim e _ o qJe is positive definite. D The special case n == 2 has already been derived in 1.8.
1.18. Exercise. If qJ is a positive definite kernel, then also Re qJ, iP and IqJ 12 are positive definite, but not necessarily IqJ I. If t/J is negative definite, then so are Re t/J and If.
1.19. Exercise. For
ZE
C define M z :==
(~ ~ Z
Z
:). Then M z is positive
1
definite if and only if I z I ;£ 1 and [3 - 2 Re(z)] 1 z 12 ;£ 1. For -1 ;£ z < the matrix M z is not positive definite.
-1
73
§2. Relations Between Positive and Negative Definite Kernels
1.20. Exercise. Let H be a complex (pre-) Hilbert space. Then its scalar product > is a positive definite kernel. The squared distance l/J(x, y) := Ilx - YII 2 is negative definite.
<., .
1.21. Exercise. Let qJ: X x X definite if (and only if)
-+
IR be a symmetric kernel. Then
qJ
is positive
L CjCk qJ(X j , X k) ~ 0 for all n
E
N,
{Xl' ... , x n} ~
X and {c l ,
1.22. Exercise. Show that for each a on IR x IR fulfils the inequalities
E
... ,
cn}
~
7L.
IR the kernell/Ja(x, y):= (a
+x
_ y)2
n
L j,k=
cjckl/Ja(x j , x k)
~ 0
I
for all n ~ 1, all Xl' ... , x n E IR and all C I , ... theless l/J a is negative definite only for a = O.
, Cn E
IR with
Lc
j
= O. Never-
1.23. Exercise. Let X be a nonempty set and let T ~ X x X contain the diagonal. Then the kernel IT is positive definite if and only if T is an equivalence relation. 1.24. Exercise. Let X = [a, b] be a compact interval and let qJ: [a, b] x [a, b] -+ C be continuous. Then qJ is positive definite if and only if
ff
c(x)c(y)cp(x, y) dx dy
for each continuous function c: X
-+
~0
C.
1.25. Exercise. An invertible square matrix is positive definite if and only if its inverse has this property (and in this case both matrices are strictly positive definite). 1.26. Exercise. Let A = (a jk ) be a real negative definite
2
tr A ~ - n I
L
n
x
n
matrix. Then
ajk·
j
§2. Relations Between Positive and Negative Definite Kernels There are many interesting and important relations between positive and negative definite kernels some of which were first known in special cases only, say for positive (resp. negative) definite functions on groups. Later on
74
3. General Results on Positive and Negative Definite Matrices and Kernels
they turned out to hold for more general kernels, too. Of course, it is trivially true that - qJ is negative definite whenever qJ is positive definite. The remarkable part in the following lemma is therefore, that it is an "if and only if" statement.
2.1. Lemma. Let X be a nonempty set, X o E X, and let t/J: X x X -+ C be a hermitian kernel. Put qJ(x, y) := t/J(x, x o) + t/J(y, x o) - t/J(x, y) - t/J(x o , x o)· Then qJ is positive definite if and only ift/J is negative definite. Ift/J(x o , x o) ~ 0 and qJo(x, y) := t/J(x, x o) + t/J(y, x o) - t/J(x, y), then qJo is positive definite if and only if t/J is negative definite.
n
n
L Cj~ qJ(Xj' x k) = L cjc kqJo(xj, x k) j,k= 1 j,k= 1 n
= -
L j,k=
cjckt/J(Xj' x k)· 1
Hence positive definiteness of qJ or of qJo implies negative definiteness of t/J. Suppose on the other hand that t/J is negative definite. Let Xl' ... , Xn E X and C l , •.• , Cn E C be given and put Co := - LJ= 1 cj • Then n
n
O~
L j,k=O
Cj~t/J(Xj' x k)
=
L CjCkt/J(X j, Xk) j,k= 1
n
+
n
LCjcot/J(Xj'x o) j=l
+
LCo~t/J(XO,Xk)+ k=l
ICo I2 t/J(x O 'x o)
n
L
j,k= 1
Cj~[t/J(Xj' x k) - t/J(xj, x o) - t/J(x o , x k)
+ t/J(x o , x o)]
n
=
-
L Cj~qJ(Xj' Xk), j,k= 1
thus showing that qJ is positive definite. Now if t/J(x o , x o) ~ 0 then qJo = qJ + t/J(x o , x o) is positive definite.
D
The following result, due mainly to Schoenberg, is of central importance.
2.2. Theorem. Let X be a nonempty set and let t/J: X x X -+ C be a kernel. Then t/J is negative definite if and only if exp( - tt/J) is positive definite for all t > O. If exp( - tt/J) is positive definite, then 1 - exp( - tt/J) is, of course, negative definite and so is therefore the pointwise limit
PROOF.
t/J =
lim O
~ (I t
- exp( -tt/J».
75
*2. Relations Between Positive and Negative Definite Kernels
Now suppose that t/J is negative definite. For obvious reasons we need only show that exp( - tt/J) is positive definite for t = 1. We choose X o E X and with qJ as in the above lemma we have
- t/J(x, y) = qJ(x, y) - t/J(x, x o) - t/J(y, x o)
+ t/J(x o , x o),
where qJ is positive definite. Hence exp( - t/J(x, y))
= exp( qJ(x, y)) . exp( - t/J(x, x o)) . exp( - t/J(y, x o)) . exp( t/J(x o , x o))
and from 1.14, 1.9 and 1.12 we conclude that exp( -
t/J) is positive definite. D
In the following let C+ = {z E C IRe z ~ O}. It is known that a kernel x X -+ C+ is negative definite if and only if (t + t/J)-l is positive definite for all t > O. Instead of proving this we show the following more general result.
t/J: X
2.3. Theorem. Let Jl be a probability measure on the half-line IR+ such that J~ s dJl(s) < 00, and let !RJl denote its Laplace transform, i.e. !RJl(z) = J~ e- sz dJl(s), Z E C+. Then t/J: X x X -+ C+ is negative definite if and only if !RJl(tt/J) is positive definite for all t > O.
o<
PROOF.
If t/J is negative definite then for t > 0 we have !l'j1(tljJ)
=
roo exp( -tsljJ) dj1(s)
pointwise on X x X, which is positive definite, being a mixture of the positive definite kernels exp( - tst/J). If on the other hand !RJl(tt/J) is positive definite for all t > 0, then for each (x, Y) E X X X we get
f.oo
~ [1 _ !l'j1(tljJ(x, Y))] = 1 - exp[ -tsIjJ(x, y)] dj1(s) t o t -+
IjJ(x, y)
roo s dj1(s)
for
t
-+
0,
where we could apply Lebesgue's theorem because of
11 - exp[ -tsljJ(x, Y)]I ~ t
'IjJ(x, y)ls.
-
Being a pointwise limit of negative definite kernels, definite, too.
t/J
itself is negative
0
Choosing Jl = 8 1 in the above theorem, we get back Theorem 2.2 for C+-valued t/J, and the choice of Jl = e- t dt shows, as already mentioned, that t/J: X x X -+ C+ is negative definite if and only if (t + t/J)-l is positive definite for all t > O.
76
3. General Results on Positive and Negative Definite Matrices and Kernels
2.4. Remark. If a probability measure J1 on ~ + has infinite first moment and t/J: X x X -+ C + is negative definite, then 2 J1(tt/J) is still positive definite for all t > 0, but, in general, the converse does not hold. It follows from later results (cf. 2.10 and 4.4.5) that exp( -~) is the Laplace transform of some probability measure J1 on ~+. Now a matrix (a jk ) of the form ajk = (Sj + Sk)2, where S b ..• , Sn ~ 0, is not negative definite in general, C 1 + ... + Cn = 0 cjckajk = 2(I Cj Sj)2, but nevertheless for all t > 0 implying
L
2J1(ta jk )
= exp( -~) = exp( -JiSj) exp( -JiSk)
is positive definite by 1.9. qJ is positive definite and qJ I~ == C for some C E ~ + then obviously C - qJ is negative definite, bounded and vanishes on the diagonal ~. A similar statement in the other direction (which may be found in the literature) is not generally true; see, however, 4.3.15. For Xl = -1, x 2 = 0, X3 = + 1, the 3 x 3 matrix
2.5. Remark. If
is negative definite, vanishes on the diagonal, and is bounded by 4, but for no real number t the matrix (t - ajk) is positive definite, because t
det( t - a jk)
= t- 1 t-4
t -
1
t t-l
t -
4
t -
1 - -8
for all
t E ~.
Negative definite kernels are intimately related to so-called "infinitely divisible" positive definite kernels. 2.6. Definition. A positive definite kernel qJ is called infinitely divisible if for each n E N there exists a positive definite kernel qJn such that qJ = (qJn)n. If t/J is negative definite then qJ = e - '" is infinitely divisible since qJn = exp( -(I/n)t/J) is positive definite and (qJn)n = qJ. Furthermore, qJ has no zeros. Proposition 2.7 below shows, in particular, that every strictly positive infinitely divisible kernel has this form. Let qJ be infinitely divisible. Then
k, n
~
1,
so that the nonnegative kernel IqJ I is infinitely divisible inside the family of all nonnegative positive definite kernels, each (under this restriction uniquely determined) nth root IqJn I again being an infinitely divisible positive definite kernel. Let ~(X) denote the closure of all real-valued negative definite kernels on X x X in the space] - 00, oo]X x x.
~2.
Relations Between Positive and Negative Definite Kernels
77
qJ ~
0 on X x X thefollowing
2.7. Proposition. For a positive definite kernel conditions are equivalent: (i) qJ is infinitely divisible; (ii) -log qJ E ~(X); (iii) qJt is positive definite for all t > O. PROOF. "(i) => (ii)" Let
t/J := -log qJ, then
t/J = lim n[1 - exp( -t/J/n)J E ~(X). "(ii) => (iii)" Let (t/J ex) be a net of (finite) negative definite kernels converging pointwise to t/J = -log qJ. Then for any t > 0 we have exp{ - tt/Jex) -+ exp( - tt/J) = qJt, so that qJt is positive definite by Theorem 2.2. "(iii) => (i)" Take t = !, t, i, ... .
D
2.8. Remarks. (I) The above proof shows that ~(X) is in fact the monotone sequential closure of the subset of all negative definite kernels, bounded above. (2) For
Z E
iC, IZ I ~ 1 the 2 x 2 matrix
G;)
is an infinitely divisible
positive definite kernel with nonuniquely determined positive definite "roots ". We conclude this section by indicating a large class of functions which operate on negative definite kernels. For J1 E M +(]O, oo[) we define g: D(J1) -+ C by
1 00
g(z)
where D{J1) is the set of Z
E
=
(l - e- AZ) df1(A.),
C for which A 1---+ 1 - e-;'z is J1-integrable.
2.9. Proposition. Let t/J: X x X -+ C be a negative definite kernel and let J1 E M +(]O, oo[). Ift/J(X x X) ~ D(J1) then g t/J is negative definite. Furthermore, for X o E X the kernel (x, y) 1---+ g[t/J(x, x o) + t/J(y, xo)J - g[t/J(x, y) + t/J{x o , xo)J is positive definite provided (t/J(X x X) + t/J(x o , x o)) u (t/J(X, x o) + t/J{xo, X)) ~ D(J1). If So A{t + A)-l dJ1{A) < 00 and t/JI~ ~ 0 then g t/J is negative definite and g[t/J(x, x o) + t/J(y, x o)] - g[t/J(x, y)J is positive definite for all X o E X. 0
0
= 1 - e-;'z where A E ]0,00[, X and C t , •.. , Cn E C such that L ci = 0
PROOF. It suffices to prove the result for g(z)
i.e. J1 = we get
G;',
D{J1) = C. If Xl'
L ci Ck{1
i, k
... , X n E
- e-;'t/J(Xj,Xk») = -
L ciCke-;'t/J(Xj,Xk) ~ 0 i, k
as an immediate consequence of Theorem 2.2. For any X o E X the kernel t/J{x, x o) + t/J{y, xo) - t/J(x, y) - t/J(x o , x o) is positive definite by Lemma 2.1,
78
3. General Results on Positive and Negative Definite Matrices and Kernels
as is therefore the kernel exp(A[t/J(X, x o)
+ l/J(y, xo) - t/J(x, y) - t/J(x o , x o)]) -
1
by 1.14, and multiplying with the positive definite kernel exp( - At/J(X, x o)) exp( - At/J(y, x o)) gives exp( - A[t/J(x, y)
+ t/J(xo, xo)]) -
exp( - A[t/J(x, x o)
+ t/J(y, xo)])
= g[l/J(x, x o) + t/J(y, x o)] - g[l/J(x, y) + t/J(x o , x o)], which is positive definite by Theorem 1.12. If J
+ t/J(y, x o)] - g[t/J(x, y) + t/J(xo, x o)] is positive definite. Using the fact that the kernel t/J(x, x o) + t/J(y, x o) - t/J(x, y) is positive definite by Lemma 2.1, it is seen as above that g[l/J(x, x o) + l/J(y, x o)] g[t/J(x, xo)
D
g[t/J(x, y)] is also positive definite.
2.10. Corollary. If l/J: X x X -+ C is negative definite and satisfies l/J I ~ then so are t/JCX for 0 < ~ < 1 and 10g(1 + t/J).
~
0
PROOF. The assertions follow by Proposition 2.9 and the formulas
~
JOO
dA
Z~=r(1-IX) 0 (1-e-'
1 X
log(l
+ z) =
'(1 - e-'
e~'< dA
which are valid for Re z ~ O. Each formula can be established by showing that both sides of the equation have equal derivatives. D
2.11. Corollary. Iff: X kernel
-+
C satisfies Re f ~ 0 then for each ~ E [1, 2] the
is negative definite. PROOF. An equivalent formulation is that the kernel - (x + y)CX is negative definite on C + x C + . This is clear when ~ = 1 and ~ = 2. Integrating with respect to z in the formula for ZCX (0 < ~ < 1) in the previous proof we get
- z
CX + 1
J
~(1 + ~) 00 = r(1 _ IX) 0 ( 1 - e
AZ
dA - AZ) A~ + 2'
79
§2. Relations Between Positive and Negative Definite Kernels
and the assertion follows since
I
-II whenever Zl' ... , Zn E C+ and C 1' ... , Cn E C with L c Cj~(1
-
e-;'(Zj+Zk) -
A(Zj
+ ;)) =
cje-;' Zj
j
=
I2 ~ 0,
D
0.
2.12. Exercise. Show by using only the results of this and the preceding section that the following kernels are positive definite. qJ(x, y) = qJ(x, y) = qJ( x, y) = qJ(x, y) = qJ(x, y) = qJ(x, y) = (g) qJ(x, y) = (h) qJ(x, y)=
(a) (b) (c) (d) (e) (f)
cos(x - y) on!R x !R. cos(x 2 - y2) on!R x !R. (1 + Ix - y I)- 1 on!R x !R. exp[ -(1 - e-JfX=Yf)] on!R x !R. (1 + F+Y)-l on!R+ x !R+. t(x- y)2 on 7L x 7L, where t E [ -1, 1]. (x + y)-l on ]0, oo[ x ]0, 00[. (lxl P + IYIP)CX - Ix - ylCX P on !R x !R, where
o<~~1.
°< f3
(i) qJ(x, y) = (x P + yP)CX - (x + y)CX P on!R+ x !R+, where ~, f3 (j) qJ(A, B) = peA (1 B) - P(A)P(B) on .91 x .91, where
~ 2, E
]0, 1].
(n, .91, P)
denotes a probability space. (k) qJ(x, y) = X /\ Y - xyon [0, 1] x [0, 1].
2.13. Exercise. Show that the following kernels are negative definite. (a) (b) (c) (d) (e)
l/f(x, l/f(x, l/f(x, l/f(x, l/f(x,
y) = [sin(x - y)]2 on!R x !R. y) = Ilx - yilP on H x H, H being a Hilbert space, < p y) = l]o,oo[(x + y) on!R+ x !R+. y) = log(x + y) on ]0, oo[ x ]0,00[. y) = 1 - <x, y) on H x H, H being a Hilbert space.
(f) l/f(x, y) (g) l/f(x, y)
°
=
~1 + Jlx - yl on!R x !R.
=
l{o}(xy)
~
2.
on!R x !R.
2.14. Exercise. Let qJ be a positive definite kernel bounded by 1 and let o < ~ < 1. Then (1 - ~qJ) - 1 is an infinitely divisible positive definite kernel. 2.15. Exercise. Let qJ be an infinitely divisible positive definite kernel on := {qJ O}. Then IT is positive definite, too. This does not hold in general without the assumption of infinite divisibility.
+
X x X and denote T
+
2.16. Exercise. For X 0 the set of infinitely divisible positive definite kernels on X x X is closed with respect to pointwise convergence. Hint: Use universal subnets. 2.17. Exercise. Any negative definite kernel with nonnegative real part is the pointwise limit of a sequence of bounded negative definite kernels with nonnegative real part.
80
3. General Results on Positive and Negative Definite Matrices and Kernels
2.18. Exercise. Let the function g: C+
g(z) = where~,
IX
+ pz + yz +
{3, y E ~+ and J1
E
f
r
J~~
-+
C be given by
[1 - exp( -sz - tz)] dJ1(s, t)
M +(~~) satisfies
_s_+_t_ dJ1(s, t) <
J~~ 1 + s +
t
If t/J is a negative definite kernel with Re t/J
~
00.
°
then so is the kernel g 0 t/J.
2.19. Exercise. Let g: JO, oo[ -+ [0, oo[ be a Borel measurable function such that g(A) = A + O(A 2) for A -+ 0. Show that if J1 E M + (JO, 00 [) satisfies
f
..1.2 dJ1()') <
00
and
fX' max(l, g().» dJ1()') <
00,
then
is well defined for C+ x C+.
ZE
C+, and t/J(x
+ y)
is a negative definite kernel on
2.20. Exercise. Let X be nonempty, X o E X, and let the linear transformation T be defined on the real vector space of all hermitian kernels t/J: X x X -+ C by
Denoting by 9 and % the cones of all positive (resp. negative) definite kernels on X x X, the result of Lemma 2.1 is that T- 1(9) = .A( Show that ker(T) = {t/J13f: X -+ C such that t/J(x, y) = f(x) + fey)} and show also that T(%) = {qJ E 9 IqJ(x, x o) = for all x EX}.
°
2.21. Exercise. Let t/J be a negative definite kernel with strictly positive real part. Then 1/t/J is positive definite. For the case Re t/J ~ 0, the result still holds if t/J does not assume the value zero. 2.22. Exercise. Given the negative definite n x n matrix (b jk ) put
where bj = L~= 1 bj,;" j = 1, ... , nand b = Lj,k= 1 bjk. Show that (ajk) is positive definite, and derive from this another proof that (e- tbjk ) is positive definite for all t > 0.
~3.
81
Hilbert Space Representation of Positive and Negative Definite Kernels
2.23. Exercise. Let A be a hermitian n x n matrix, let v E en satisfy 1 and define an n x n matrix P v with ij's element bij - Vj. Show that - A is negative definite if and only if P v AP: is positive definite.
Ii= 1 Vj =
§3. Hilbert Space Representation of Positive and Negative Definite Kernels Around 1940 Schoenberg published three fundamental papers (1938a, b, 1942) all of which were very closely connected with positive and negative definite kernels. The main motivation for deriving these results at that time was to decide which metric (or more general semimetric) spaces (X, d) can be imbedded into a Hilbert space H, i.e. when does there exist a mapping <1>: X ~ H such that 1I(x) - (y)1I = d(x, y)
for all
x, y
E X.
It turned out that this property of (X, d) is equivalent to d 2 being negative definite. We are now going to give an easy derivation of this result using the so-called reproducing kernel Hilbert space (RKHS) associated with a positive definite kernel.
3.1. Let X be a nonempty set and
°
L Jkf(Yk) = L cjak
j, k
(1)
j
evidently does not depend on the chosen representations off, g (which may be not unique) and is denoted . Then = CjCk is linear in its first and antilinear in its by assumption and the form second argument, implying in particular
L
<., .
j.~! ZjZk<./j, 1,,> = (t/ Jj ,jt!Zj./j) ~ 0 <., .
°
> is a positive definite kernel for j'1' ... , fn E H and z l' . . . , Zn E C, i.e. on H o x H o . An immediate consequence of(1) is the reproducing property <.f, = .f(x)
for all f
which implies in particular < I
E
H o and
x
E
X
=
f(x) 1 ~ .
° °
so that = 0 if and only iff is identically zero. Therefore, H is a preHilbert space and its completion H is a Hilbert space in which H is a dense subspace. The transformation H~ex
f
r--+
(x r--+ <.f,
82
3. General Results on Positive and Negative Definite Matrices and Kernels
being linear and injective, the Hilbert space H can even be thought of as a linear subspace of ex, i.e. as a space of functions and not, as mostly, as a space of equivalence classes of functions. This Hilbert function space is usually called the RKHS associated with qJ. If qJ is real-valued then, of course, H can be chosen as a real function space. To sum up there is a Hilbert space H £ ex and a mapping x ~ qJxfrom X to H such that
qJ(x, y) = (qJx' qJy)
for x, y E X.
The Hilbert space representation of general negative definite kernels looks a little bit more complicated.
3.2. Proposition. Let X be a nonempty set and t/!: X x X -+ e be negative definite. Then there is a Hilbert space H £ ex and a mapping x ~ qJxfrom X to H such that
where f: X -+ e is a certain complex function on X. If there is some X o E X such that t/!(x, x o) E IR for all x E X and if t/! vanishes on the diagonal L\ = {(x, x)lx EX}, thenfmay be chosen to be zero. 1ft/! is real-valued, then H may be chosen as a real Hilbert space and equation (2) becomes
where f: X -+ IR. The.function f is nonnegative whenever t/! is. If t/J is real-valued and vanishes on L\ then f = 0 and is a semimetric such that x ~ qJx is an isometry. If,furthermore, {t/! = O} = L\ then is a (hilbertian) metric on X.
fi
PROOF.
fi
We fix some X o E X and define
qJ(x, y) := ![t/!(x, x o)
+ t/!(y, x o)
- t/!(x, y) - t/!(x o , x o)],
which is a positive definite kernel by Lemma 2.1. Let H be the associated RKHS for qJ and again put qJx(Y) = qJ(x, y). Then
IlqJx - qJ y l12 = qJ(x, x) + qJ(y, y) -
2 Re qJ(x, y)
= Re t/!(x, y) - ![t/!(x, x) + t/!(y, y)],
IIqJxl1 2 + IIqJ y ll2 - 2(lfJx' qJy) = IlqJx - qJyl12
- 2i
Im(qJx' qJy)
= t/!(x, y) - ![t/!(x, x) + t/!(y, y)] - i Im[t/!(x, x o) Settingf(x) := !t/!(x, x)
+ t/!(y, x o)]·
+ i 1m t/!(x, x o), we therefore obtain
The other statements can be derived immediately.
o
§3. Hilbert Space Representation of Positive and Negative Definite Kernels
~
The negative definite 2 x 2 matrix (
-l
83
i) shows that not every negative
0
definite kernel t/! vanishing on the diagonal can be represented in the form
for some Hilbert space valued mapping x 1--+ qJx, since this would imply Re t/!(x, y) = IIqJx - qJ yll2 = 0, hence qJx = qJy and then t/!(x, y) = o.
3.3. Corollary. Let qJp(x, y) = exp( -Ix - yiP), x, Y E IR and 0 < p < Then lfJp is positive definite if and only if p ~ 2.
00.
PROOF. By 1.10 the kernel t/!(x, y) = (x - y)2 is negative definite, hence so is t/!a for 0 < (l ~ 1 by 2.10, and from Theorem 2.2 we get that lfJp is positive definite for all p ~ 2. Suppose now there is some p > 2 such that qJp is positive definite. Then for any t > 0, Xl' ... , X n E ~ and C I , ... , Cn E ~ we have n
n
L
j,k=
cjckexp(-tlx j - xkl P) I
=
L
j,k=
cjckexp(-Itl/Pxj - tl/PxkI P) ~ 0, I
so by Theorem 2.2 the kernel IX - yiP is negative definite, and by the proposition above Ix - y1P/2 is a metric on ~. However,
10 - 11 P/ 2 = 1 = 11 - 2I P/ 2 , contradicting the triangle inequality.
10 - 21 P/ 2 = 2P/ 2 > 2
D
3.4. Exercise. Let the kernel qJp be defined on IR+ x ~+ by qJp(x, y) = exp[ -(x + y)P] where 0 < p < 00. Show that qJp is positive definite if and only if p
~
1.
3.5. Exercise. Let H be the RKHS associated with the positive definite kernel qJ: X x X -+ C. Show that any closed subspace of H is the RKHS for some positive definite kernel on X x X.
3.6. Exercise. Let qJ, t/!: X x X
-+ C be two positive definite kernels. Then RKHS(qJ) £ RKHS(t/!) if and only if At/! - qJ is positive definite for some A> 0.
3.7. Exercise. Let qJ: X x X -+ C be a positive definite kernel not identically zero. Show that the following conditions are equivalent: (i) qJ generates an extreme ray in the convex cone 9 of positive definite kernels on X x X (i.e. {AqJ IA ~ o} is an extreme subset of 9). (ii) RKHS( qJ) is of dimension 1. (iii) qJ(x, y) = f(x)f(y) for some functionf: X -+ C not identically zero.
84
3. General Results on Positive and Negative Definite Matrices and Kernels
3.8. Exercise. Let qJ: X x X ~ C be a positive definite kernel such that qJ(x, y) =+= 0 for all (x, y) E X X X. Show that if also l/qJ is positive definite then qJ(x, y) = f(x)f(y) for some functionf: X ~ C\{O}. 3.9. Exercise. Given a positive definite kernel qJ: X x X ~ C and a nonempty subset X' ~ X, consider the restriction qJ' := qJ IX' x X' and the two RKHS's Hand H' associated with qJ (resp. qJ'). Show that H' = {f IX' If E H}. Hint: There is an isometry U: H' ~ H (not necessarily onto) mapping qJ~, to qJx' for each x' E X'. Use U as well as the adjoint operator U*. 3.10. Exercise. If p > 2 then for no A > 0 the kernel exp[ - AIx - yiP] is positive definite on ~ x ~. Nevertheless there are mixtures cp(x, y)
=
LX' exp[ -Alx -
yiP] dj1(A),
Jl E M~(]O, OO[),
which are positive definite.
3.11. Exercise. Let qJ be a positive definite kernel defined on X x X where X is a topological space. Show that qJ is continuous if and only if Re qJ is continuous at each point of the diagonal.
Notes and Remarks Positive (semi-) definite matrices have a long history and we have not traced this history back to its origins. The first instance where a positive definite kernel on a nonfinite set has been considered, seems to be within the theory of integral equations, and the first systematic treatment in this connection was given by Mercer (1909). Mercer defines a continuous and symmetric real-valued function qJ on [a, b] x [a, b] ~ ~2 to be of positive type if and only if
ff
c(x)c(y)cp(x, y) dx dy
~0
(1)
holds for all continuous functions c: [a, b] ~ ~, and he shows that this condition is equivalent to qJ being a positive definite kernel in our terminology; cf. Exercise 1.24. Property (1) had already been singled out a few years earlier by Hilbert (1904) who called the function qJ definite in this case. Mercer also defined a function t/J to be of negative type if and only if - t/J is of positive type. The idea leading to the notion of a negative definite kernel (in our terminology) goes back to Schoenberg (1938b); however, he requires these kernels to vanish on the diagonal-a condition appearing natural in his context: imbedding of metric spaces in a Hilbert space. The product stability of positive definiteness (Theorem 1.12) has been shown by Schur (1911, Satz VII). A remarkable result in this connection was
85
Notes and Remarks
found by Fitzgerald and Horn (1977): if (a jk ) is a positive definite n x n matrix with nonnegative entries then for all real ~ ~ n - 2 the matrix (a lk ) is positive definite, too, and this lower bound for ~ is sharp. The fundamental connection between positive and negative definite kernels expressed in Theorem 2.2 goes back to Schoenberg (1938b) in case the negative definite kernel vanishes on the diagonal. The general case seems to have been proved first in Herz (1962). Infinitely divisible positive definite kernels appear at many places in analysis and-particularly important-in probability theory. Infinitely divisible complex-valued kernels are studied in some detail by Horn (1969a). A study of positive and negative definite kernels with invariance properties under a group action may be found in Parthasarathy and Schmidt (1972). The possibility of representing a positive definite kernel qJ: X x X -+ C as qJ(x, y) =
= IIF(x) - F(y)11 2
for some Hilbert space valued mapping F on X, a result due to Schoenberg (1938b).
CHAPTER 4
Main Results on Positive and Negative Definite Functions on Semigroups
§1. Definitions and Simple Properties In the present book we will deal mainly with positive (and negative) definite functions on abelian semigroups, but nevertheless we will introduce the concepts for arbitrary semigroups with involution. The subject of positive definite functions on locally compact groups splits into two completely different theories for abelian and nonabelian groups. At present there exists a rather satisfactory theory of positive definite functions on abelian semigroups whereas very little is known about the nonabelian case. 1.1. Definition. A semigroup (S, 0) is a nonempty set S equipped with an associative composition and a neutral elementt e. A semigroup with involution or a *-semigroup (S, 0, *) is a semigroup (S, 0) together with a mapping *: S -+ S, called involution, satisfying (i) (s t)* = t* s* for s, t E S; (ii) (s*)* = s for s E S. 0
0
0
One should note that the axioms imply e
= e*, in fact: e* = e* e = 0
(e* e)* = (e*)* = e. 0
For an abelian (= commutative) semigroup the composition and neutral element are always denoted + and 0, and the neutral element is called zero. We now list some examples showing the great generality of the above concept.
t Many authors do not assume the existence of a neutral element in the definition of a semigroup. With the exception of Chapter 8 we always assume that semigroups have a neutral element.
87
§1. Definitions and Simple Properties
1.2. Examples. (a) Any group (G, 0) is a semigroup with involution when we define s* = s- 1. (b) Any abelian semigroup (5, +) is a semigroup with involution when we define s* = s, the identical involution. (c) Let (T, +) be an abelian semigroup. The product semigroup S = TxT is a semigroup with involution if (t l' '2)* := (t 2 , t 1). (d) The closed unit disc D := {z E c z I ~ I} with z 0 w := zw and z*:= Z is a semigroup with involution.
II
Concrete examples of semigroups occur throughout mathematics. We will study some in detail in §4.
1.3. Let (S, 0) denote a semigroup with neutral element e. A subsemigroup of S is any subset T s; S such that e E T, and which contains Sot whenever s, t E T. If S has an involution, then T s; S is called a *-subsemigroup, if T is a subsemigroup which contains t* whenever t E T. An element W E S is called absorbing if sow = W 0 S = w for all s E S; obviously there can exist at most one absorbing element in S and clearly w* = w. In the additive semigroup ([0, 00], +) we have w = 00 and in the closed unit disc (D, .) we have w = 0. Each semigroup 5 can be enlarged to a semigroup T with absorbing element defining T:= S U {OJ} (where OJ is some element not contained in S) and OJ 0 s = s 0 OJ := OJ =: OJ 0 OJ, S E S. Note that if S contains the absorbing element w then S\ {w} is not necessarily a subsemigroup of S. Let (S, 0), (T, 0) be semigroups with neutral elements es and eT. A mapping J: S -. T is called a homomorphism if J(es) = eT and J(x 0 y) = J(x) 0 J(y) for x, yES. Ifboth Sand T have involutions we add the requirementJ(s*) = J(s)* for s E S in order to call J a homomorphism. Sometimes we use the word *-homomorphism in this case. If J: S -. T is a homomorphism which is one-to-one and onto it is called an isomorphism. 1.4. Let (S, 0) be a semigroup. If S is equipped with a topology (9 we call (9) a topological semigroup provided the composition mapping (s, t) 1-+ Sot is continuous from S x S into S. If S in addition has an involution * we
(S,
0,
require s 1-+ s* to be continuous, too. In the rest of this section S = (S, 0, *) denotes a semigroup with involution. 1.5. Definition. A function qJ: S -. C is called positive definite if (s, t) 1-+ qJ(s* 0 t) is a positive definite kernel on S x S, i.e. if n
I
CjCk
qJ(sj
0
Sk) ~ 0
j,k= 1
for all n E N, {SI' ... ' Sn} S; S, {C 1 , . . . , Cn} positive definite if the kernel qJ(s* 0 t) is so.
S;
C, and it is called strictly
88
4. Main Results on Positive and Negative Definite Functions on Semigroups
The set of positive definite functions qJ: S -. C is denoted &J(S) and E &'(S) qJ(e) = I}. The above concept is not changed if the kernel qJ(s* 0 t) is replaced by the kernel qJ(s 0 t*). On an abelian group (G, +) we can consider two different involutions, the identical involution and the involution x* = -x for x E G. Corresponding to these two involutions we have two different notions of positive definiteness for functions on G. We say that qJ: G -. C is positive definite in the group sense if it is positive definite when G carries the involution x* = - x for x E G, and we say that qJ is positive definite in the semigroup sense if G is equipped with the identical involution. Later on, in particular in Chapter 5, both of these notions of positive definiteness will occur simultaneously. Likewise for negative definite functions on G, to be introduced in Definition 1.8 below, we have the two notions of negative definiteness in the group sense and in the semigroup sense.
&' 1(S) = {qJ
1
1.6. By 3.1.5 it follows that any positive definite function qJ is hermitian, i.e. qJ(s*) = qJ(s) for s E S, and by 3.1.8 we have qJ(s*
0
s)
~
0
and
1
qJ(s*
0
t) 12 ~ qJ(S*
0
s)qJ(t*
0
t),
s, t
E
S,
in particular,
qJ(e)
~
0
and
I
qJ(S) 12 ~ qJ(e)qJ(s*
0
s),
S E
S.
The last inequality implies that qJ == 0 if qJ(e) = O. The set &J(S) is a convex cone in the vector space C S of complex-valued functions on S. By 3.1.12 and 3.1.11 it follows that &>(S) is stable under products and closed in the topology of pointwise convergence. The set &'1 (S) is closed and convex and a base for &'(S), i.e. for every qJ E &'(S)\ {O} there exists a unique pair (A, qJo) with A > 0 and qJo E &'1 (S) such that qJ = ,1qJo, namely A = qJ(e) and qJo = A-lqJ.
1.7. Remark. Let H be a complex Hilbert space and B(H) the set of bounded operators on H. Following Sz.-Nagy (1960) a function <1>: S -. B(H) is called of positive type if for all n E N, all {Sl"'" sn} ~ S and all {~1"'" ~n} ~ H n
L j,k=
<<1>(sj
0
Sk)~k' ~j> ~ O.
1
If <1> is of positive type, then <1> is positive definite in the sense that all the scalar functions <1>~(s) = <<1>(s)~, ~>, ~ E H, are positive definite. Examples are known showing that the converse is not true, cf., e.g. Arveson (1969). If H = C the functions of positive type are the same as positive definite functions. More generally if <1>(S) is contained in a commutative C*-algebra of B(H), it can be shown that the two notions coincide.
~ I.
89
Definitions and Simple Properties
A C*-algebra S with unit can be considered as a semigroup with involution, the composition being the multiplication of the C*-algebra. Then linear mappings <1>: S -. B(H) of positive type are widely studied under the name of completely positive mappings, whereas a linear <1> is positive definite if and only if it is positive, i.e. maps positive elements in S into positive elements of B(H). There exists a vast literature about these operator-valued functions, see Arveson (1969), Evans and Lewis (1977) and Mlak (1978), and references therein, but a treatment of this subject falls outside the scope of this book.
1.8. Definition. A function t/J: S -. C is called negative definite if (s, t) H t/J(s* t) is a negative definite kernel on S x S, i.e. if t/J is hermitian and 0
n
L
Cj
0ct/J(sj
0
Sk)
~ 0
j,k= 1
for all n ~ 2, {Sl' ... , sn} ~ Sand {c 1 , .•. , cn} ~ C with Ij= 1 set of negative definite functions t/J: S -. C is denoted JV(S).
Cj
=
O. The
The set JV(S) is a closed convex cone in C S containing the real constants. For t/J E JV(S) we have by 3.1.7 that
2 Re t/J(s*
0
t)
~
t/J(s*
0
s)
+
t/J(t*
0
t),
s, t
E
S,
in particular
2 Re t/J(t)
~
t/J(e)
+ t/J(t*
0
t),
t
E
S.
(1)
As an application of Lemma 3.2.1 we get
1.9. Proposition. Let t/J: S -. C be a hermitian .function sati~fying t/J(e) ~ O. Then t/J is negative definite if and only if the kernel (s, t) H t/J(s) + t/J(t) t/J(s* t) is positive definite. 0
In the further development of the theory we need to consider certain boundedness conditions for positive definite functions.
1.10. Definition. A function a: S -. (i) (X(e) = 1; (ii) (X(s 0 t) ~ (X(s)a(t) for s, t (iii) (X(s*) = (X(s) for s E S.
E
[R + is called an
absolute value if
S;
We will later see many examples of absolute values. At present we note that the constant function s H 1 is an absolute value. If a, 13 are absolute values and k > 0 then (Xf3, max((X, 13) and ak are again absolute values.
90
4. Main Results on Positive and Negative Definite Functions on Semigroups
1.11. Definition. A function J: S -. C is called bounded with respect to an absolute value a (shortly: a-bounded) if there exists a constant C > 0 such that
IJ(s) I ~ Ca(s)
s E S,
for
and J is called exponentially bounded if there exists an absolute value with respect to whichJis bounded. The set of exponentially bounded functions is an algebra.
1.12. Proposition. Let qJ a. Then
E
&J(S) be bounded with respect to an absolute value
IqJ(s) I ~ qJ(e)a(s)
Jor
s E S.
PROOF. Without loss of generality we may assume that I
qJ(S) 12 ~ qJ(S*
0
qJ(e) = 1, so that
s). By iteration we get for n E N I
qJ(s) 12n ~ qJ«s* 0 S)2 n - 1),
and using IqJ(t) I ~ Ca(t) for some constant C > 0 we get 1
qJ(s) 12n ~ Ca«s* 0 s)2 n - 1) ~ Ca(s)2 n ,
hence
IqJ(S) I ~ a(s)
lim C 2 -n
= a(s).
D
1.13. Let qJ E &J(S) and let H 0 be the linear subspace of CS generated by the functions {qJs Is E S}, where qJs(t) = qJ(s* t). Then H 0 is equipped with a scalar product
0
<., .) such that
s, t E S, s and the completion H of H 0 (realized in C ) is the RKHS associated with qJ, 0
t),
cf. 3.3.1. For each that
SE
S there exists a linear transformation n(s): H 0 s, t
E
-.
H 0 such
S,
and n is a representation of S in the space Hom(H 0) of linear transformations of H o , i.e.
n(s
0
t) = n(s)n(t),
n(e) = 1,
n(s*) = n(s)*,
where the last equation means that
for
J,gEH o .
We see that
qJ(s) =
s E S.
91
§l. Definitions and Simple Properties
Conversely, if K o is a pre-Hilbert space, ~ E K o and n: S -. Hom(Ko) is a representation, then it is easily seen that qJ(s) := <~, n(s)~) is positive definite. We shall now characterize the positive definite functions qJ for which the operators n(s) are bounded on H o' If this is the case, then n(s) can be uniquely extended to a bounded operator n(s) on the Hilbert space H, and fl is a representation of S in B(H), the C*-algebra of bounded operators on H.
1.14. Theorem. Let qJ E &>(S), and let n: S -. Hom(H 0) be the representation as above. Then qJ is exponentially bounded if and only if n(s) is a bounded operator on H 0 for each s. Suppose first that n(s) is bounded on H 0 for each s E S. Then a(s) := is an absolute value on Sand IqJ(s)/ = l
Iln(s)11
Lj=1
n
g(s) =
L CjCk qJ(xj
Then g E &>(S), for if {Yl' ... , Ym} m
L p, q = 1
m
dp~g(y;
0
Yq) =
0
s
0
j,k= 1
~
S, {d 1 ,
s E S.
x k), ••• ,
d m}
~
([ we have
n
L L (dpcj)(dqck)qJ((Yp
0
x j )*
0
(Y q 0 x k)) ~
o.
p, q = 1 j, k = 1
Furthermore, g is bounded with respect to a since Ig(s)
I~ (ttl I
r
q>(e)cx(s).
cj cx(x)
By Proposition 1.12 it follows that
"n(s)j'" 2 = g(s* 0 s) ~ g(e)a(s)2
= II f 1 2a(s)2,
which shows that n(s) is a bounded operator on Hoof norm ~ a(s).
D
1.15. Exercise. Let E s denote the shift operator on ([s defined by E s f(t) = f(s t), s, t E S,j' E ([s. Let qJ E &>(S) and let H 0 ~ H ~ ([s be the associated 0
pre-Hilbert space and Hilbert space. Use the closed graph theorem to prove that Es.(H) ~ H if and only if n(s) is a bounded operator on H o. If n(s) is a bounded operator on H 0' then the unique continuous extension n(s) of n(s) to H is given byn(s).f = Es.ffor f E H.
1.16. Exercise. Suppose that S is a group with s* = s- 1. Show that every qJ
E
&>(S) satisfies IqJ(s) I ~ qJ(e), s
E
S.
1.17. Exercise. Let 11(S) denote the set offunctionsf: S -. ([ for which
Ilfli =
L I f(s)1 seS
<
00,
92
4, Main Results on Positive and Negative Definite Functions on Semigroups
and define j'
* g(s) =
L
.f*(s) = f(s*),
.f(a)g(b),
SE
S.
(a, b) E S2
aob=s
Show that [1(S) is a Banach algebra with involution and unit. Let qJ: S -. C be a bounded function and let LqJ: [1(S) -. C be defined by
LqJ(f) = LSES f(s)qJ(s). Show that qJ for allf E [1(S).
E
&P(S) if and only if LqJ(f* * f) ~ 0
1.18. Exercise. Let (S, 0, *) denote a semigroup with involution and let H s; S be a nonempty subset of S. Then if I H E &P(S) it follows that H is a *-subsemigroup of S. In case S is an abelian group and s* = -s the converse also holds, i.e. 1H is positive definite in the group sense if and only if H is a subgroup of S (cf. Exercise 3.1.23). Try to find a counterexample for the converse statement on a semigroup with identical involution. 1.19. Exercise. Let (S, *) be a semigroup with involution and add an absorbing element w to S (cf. 1.3). If qJ E &P(S) then iP: S u {w} -. C defined by iPlS = qJ and iP(w) = 0 is positive definite on S u {w}. For any positive definite extension ((J of qJ we have 0 ~ ((J(w) ~ infsEs qJ(s* ° s). 0,
§2. Exponentially Bounded Positive Definite Functions on Abelian Semigroups Throughout this section, S = (S, +, *) is an abelian semigroup with involution, which mayor may not be the identity.
2.1. Definition. A function p: S -. C is called a semicharacter if (i) p(O) = 1; (ii) pes + t) = p(s)p(t) for s, t (iii) p(s*) = pes) for s E S.
E
S;
Note that if (ii) holds, then (i) is equivalent with p $ o. A semicharacter is a homomorphism of (S, +, *) into the semigroup (C, " -), where the composition and involution are multiplication and conjugation. If the involution * is the identity, a semicharacter is automatically realvalued. If (S, +) is an abelian group and s* = - s, a semicharacter has its values in the circle group lr = {z E C z I = I} and is a group character.
II
2.2. The set of semicharacters on S is denoted S*. We equip S* with the topology inherited from C S , having the topology of pointwise convergence. In particular S* is a Hausdorff space. (Since C S is completely regular and s* is a (closed) subset ofCs we have in fact that S* is a completely regular space.)
~2. Exponentially Bounded Positive Definite Functions on Abelian Semigroups
93
Note that S* is a topological semigroup under pointwise multiplication, the mapping P H P is an involution and the function 1 is the neutral element. We call S* the dual semigroup of S.
2.3. The semigroup structure of S* leads to a convolution in M~ (S*), cf. 2.1.16. Let n: S* x S* -. S* be the continuous mapping given by rc(Pl' P2) = PlP2. For J.1, v E M~(S*) the convolution J.1 * v E M~(S*) is by definition the image measure of the Radon product measure J.1 (8) v under the mapping rc, I.e.
J.1
* v(B)
=
J.1 (8) v(rc- l(B))
Note that by 2.1.24 supp(J.1
* v) =
for
B
E
&l(S*).
supp(J.1)· supp(v).
2.4. Every semicharacter is positive definite. Our main concern is whether every positive definite function can be represented as an integral of semicharacters. We shall return to this question in Chapter 6, where it will be shown that the answer is in general negative. However, exponentially bounded positive definite functions admit such an integral representation as we shall see next. For J.1 E MC+(S*), the set of Radon measures on S* with compact support, the function
cp(S) = is positive definite. In fact, for
1
j,t CjCk cp(Sj
r p(s) dJ1(p),
S E
Js *
{Sl' ... ,
+ st)
=
sn}
fiJI
S
cn}
£;
S, {c l ,
Cj
p(Sj) 1 dJ1(p) ?; O.
... ,
2
£;
C we find
Furthermore, qJ is exponentially bounded. To see this we define an absolute value lJ.K associated with a compact subset K £; S* by
lJ.K(S)
=
II
sup{ Ipes) P E K},
SE
S.
In the special case of K = supp(J.1) we find
IqJ(s) I ~ J.1(K)lJ.K(S)
=
qJ(O)lJ.K(S)
for
s
E
S,
showing that qJ is bounded with respect to lJ.K. This establishes the" only if" part of the following result :
2.5. Theorem. A function qJ: S -. C has an integral representation of the form
cp(S)
=
r p(s) dJ1(p),
Js *
(1)
with J.1 E M C+ (S*) ([and only if it is positive definite and exponentially bounded. If qJ has these properties there is exactly one measure J.1 E M + (S*) such that (1) holds.
94
4. Main Results on Positive and Negative Definite Functions on Semigroups
As a preparation to the proof we fix an absolute value a and consider the set gJll(S) of a-bounded functions qJ E gJ(S), and the subset gJ~(S) of functions qJ E 9 1l (S) satisfying qJ(O) = 1. Clearly gJll(S) is a closed convex subcone of 9(S) and 9~(S) is a closed convex set, which is a base for 9 1l(S). We remark that the set gJe(s) of exponentially bounded positive definite functions is gi ven as
where the union is taken over the set of absolute values, so gJe(s) is a convex cone, stable under products, cf. 1.10. 2.6. Theorem. Let a be an absolute value on S. Then gJ~(S) is a compact convex set whose extreme points are precisely the a-bounded semicharacters. Since the functions in gJ~(S) are pointwise bounded, Tychonoff's theorem implies that gJ~(S) is compact. A semicharacter p is a-bounded if and only if Ipes) I ~ a(s) for s E S. We show below that ex(gJ~(S)) s; S*, and defining y: S -. S by yes) = s* + s we conclude from Corollary 2.5.12 that ex(gJ~(S)) is precisely the set of a-bounded semicharacters. In order to establish ex(gJ~(S)) s; S*, we define for qJ E gJll(S), a E Sand (J E C with I(J I ~ 1 the function ~,(J qJ by PROOF.
~,(J qJ(s)
=
(J
a(a)qJ(s)
(j
+ 2 qJ(S + a) + 2 qJ(S + a*),
n
I
j,k=l
s
E
S.
n
CjCk~, qJ(sj (J
+ Sk) =
a(a)
+
I
j,k=l
CjCk qJ(sj
+ Sk)
Re{a i,t/i('kqJ(Sj + Sk + a)}
= a(a)g(O)
+ Re{(Jg(a)},
where n
g(s):=
I
CjCk qJ(sj
j,k= 1
+ Sk + s),
SE
S.
According to the proof of Theorem 1.14 we have g E gJll(S), so the inequality Ig(a) I ~ g(O)a(a) (cf. 1.12) implies that Ta, qJ E 9(S) and ~,(J qJ is clearly a-bounded. Let qJ E ex(gJ~(S)). We shall show that qJ(s + a) = qJ(s)qJ(a) for s, a E S. This is clear if a(a) = 0 because IqJ(s + a) I ~ a(s + a) ~ a(s)a(a) = 0 and IqJ(a) I ~ a(a) = O. Suppose a(a) > O. Since (J
~,lqJ
+
~,-lqJ
=
~,iqJ
+
~,-iqJ
=
2a(a)qJ,
~2. Exponentially Bounded Positive Definite Functions on Abelian Semigroups
95
all the functions ~. qJ are proportional to qJ for (J E { ± 1, ± i}, and it follows that qJ(s + a) = k(a)qJ(s), S E S, for a certain proportionality factor k(a). Putting S = 0 we get k(a) = qJ(a). D (1
PROOF OF THEOREM 2.5. If qJ is positive definite, exponentially bounded and not identically zero, there exists an absolute value a such that qJ(O) - 1 qJ E £?J'~ (s). By Theorem 2.5.6 there exists a Radon probability measure! on the compact set of a-bounded semicharacters having qJ(O)-lqJ as barycentre. Using the continuous linear functionals f~ f(s) on C S we get
qJ(S) =
f
pes) dJi(p),
S,
SE
s*
where Ji = qJ(O)! belongs to MC+(S*). To prove the uniqueness assertion assume that
qJ(S) =
f
s*
pes) dJi(p) =
f
pes) dv(p),
SE
S,
s*
where Ji E MC+(S*) and v E M +(S*) is such that p ~ pes) is v-integrable for all S E S. The set
of continuous complex-valued functions on S* is an algebra stable under conjugation, containing the constant functions and separating the points of S*. It follows by the Stone-Weierstrass theorem that the set of restrictions to a compact subset K ~ S* of the functions in d is a dense subset of C(K) in the uniform norm. All functions <1> E d are v-integrable and S<1> dJi = S<1> dv. We claim that supp(v) ~ supp(Ji). If not, there exists a compact set L ~ S* disjoint from supp(Ji) such that veL) > O. Choose t; > 0 such that t;2qJ(O) < veL). By the Stone-Weierstrass theorem there exists a real-valued function E.9'1 such that <1> ~ 1 on Land 0 ~ <1> ~ t; on supp(Ji). Then <1>2 E d, <1>2 ~ 0 and
which is a contradiction. Finally, the set of restrictions to supp(Ji) of the functions in .9'1 is dense in C(supp(Ji)), so by the Riesz representation theorem J.1= v. D The uniqueness statement in Theorem 2.5 combined with Theorem 2.6 can be expressed in the terminology of the theory of integral representations. 2.7. Corollary. Let a be an absolute value on S. Then
£?J'~(S)
is a Bauer simplex.
96
4. Main Results on Positive and Negative Definite Functions on Semigroups
As a special case we consider the absolute value b(s) = 1, s E S. The cone f!jJb(S) is equal to the cone of bounded positive definite functions, and we define
S:= S*
n f!jJb(S) = {p
E
S*llp(s)1 ~ 1 for s E S},
which is a compact subsemigroup of S* called the restricted dual semigroup. It is easy to see that Sis the set of bounded semicharacters. For this special case Theorem 2.6 and Corollary 2.7 lead to the following result :
2.8. Theorem. The set f!jJ~(S) of bounded positive definite functions qJ with qJ(O) = 1 is a Bauer simplex and its set of extreme points is S. A bounded positive definite function qJ has an integral representation cp(s) where J.1
E
=
L
p(s) dl1(p),
SE
S,
M +(S) is uniquely determined.
In the special case of S being an abelian group with s* = - s, Theorem 2.8 reduces to Bochner's theorem for discrete abelian groups, cf. Rudin (1962):
2.9. Theorem. Let G be a discrete abelian group. A function qJ: G -. C is positive definite in the group sense if and only if it is the Fourier transform of a (nonnegative) Radon measure on the compact dual group G. 2.10. The set of signed measures of the form J.11 - J.12 + i(J.13 - J.14) with J.1j E MC+(S*),j = 1, ... ,4 will be denoted MC(S*). Extending the convolution of measures in MC+ (S*) to MC(S*) by bilinearity, MC(S*) becomes an algebra. For /1 E MC(S*) we denote by (t the function
(t(s) =
f
p(s) dJ.1(p),
sES
s*
and {t is called the generalized Laplace transform of J.1. The mapping J.1 H (t of MC(S*) into C S , the generalized Laplace transformation, has the following properties, where a, f3 E C, J.1, v E MC(S*):
(i) (aJ.1
+
f3V)A = a{t
v;
(ii) (J.1 * v) = {t . (iii) {t = 0 => J.1 = 0; A
+
f3v;
i.e. /1 H (t is an injective algebra homomorphism. Since MC+(S*) is mapped onto the cone f!jJe(s), MC(S*) is mapped onto the subspace of C S spanned by f!jJe(s). Among the above properties (i) and (ii) are straightforward to establish, and (iii) follows from the proof in Theorem 2.5. Indeed, if J.1 = (/11 - J.12) + i(J.13 7114)' with J.1j E M C+ (S*), and K s; S* is compact such that supp(J.1 j) S; K, j = 1, ... ,4, then J dJ.1 = 0 for all E d, and the
§2. Exponentially Bounded Positive Definite Functions on Abelian Semigroups
97
restrictions I K form a dense subspace of C(K). By Theorem 2.2.4 it follows that J.1 = o. We identify M +(S) with the set of J.1 E MC+(S*) for which supp(J.1) £; S. For J.1, v E MC+(S*) with support in S, J.1 * v also has its support in S, so we can consider the convolution as a composition in M + (S). In the following result M +(S) is equipped with the weak topology, which is equal to the vague topology. The result is analogous to the Levy continuity theorem in probability theory, but simpler.
2.11. Theorem. The transformation J.1 H P, is a homeomorphism of M +(S) onto q;>b( S). Only the continuity of the inverse mapping p, H J.1 needs some explanation. Let (J.1~) be a net in M + (S), let J.1 E M +(S) and assume p,~(s) -.. P,(s) for each s E S. In particular there exists a o and C > 0 such that J.1~(S) = P,~(o) ~ C for a ~ ao , showing that (J.1~) is eventually in the compact set of measures in M +(S) with total mass ~ C. It therefore suffices to show that J.1 is the only accumulation point for (J.1~). Indeed, if (J is an accumulation point for (J.1~), then p, = fj by the continuity of hence J.1 = (J. D PROOF.
A ,
2.12. Exercise. Let S be an abelian semigroup with involution. Show that the Banach algebra [l(S) from Exercise 1.17 is abelian. Let ~ be the Gelfand spectrum of nonzero multiplicative linear functionals (Rudin 1973, p. 265), and let ~h denote the set of hermitian elements, i.e. the set of L E ~ such that L(f*) = L(f) for f E [l(S). Let r denote the set of nonzero bounded functions y : S --+ C satisfying y(s + t) = y(s) y(t) for s, t E S with the topology of pointwise convergence. For y E r let L y : [l(S) --+ C be defined by Ly(f) =
L f(s)y(s). seS
Show that y H L y is a homeomorphism of r onto ~ which maps S onto ~h. It can be proved that [l(S) is semisimple if and only if r separates the points of S, cf. Hewitt and Zuckermann (1956).
2.13. Exercise. Let (S, +, *) and (T, +, *) be abelian semigroups with involution, and let (S x T, +, *) be the product semigroup defined by (s, t) + (u, v) = (s + u, t + v), (s, t)* = (s*, t*). Show that there exists a topological semigroup isomorphism of S* x T* onto (S x T)*, which maps
S x tonto s;:-r.
2.14. Exercise. Let (S, +, *) and (T, +, *) be abelian semigroups with involutions and let h: S --+ T be a homomorphism. The dual map h: C T -.. C S is defined by h(f) = f h. Show that h(&>(T)) £; &'(S), h(&>e(T)) £; (!Je(s), h(T*) ~ S*, h(T) ~ S and that h* := hi T* is a continuous homomorphism of T* into S* which, furthermore, is one-to-one if h is onto. Let qJ E &>e(s) have the representing measure J.1 E MC+(S*). Show that C h qJ = f h for some f E &>e(T) if and only if J.1 = v * for some v E M +(T*). 0
0
98
4. Main Results on Positive and Negative Definite Functions on Semigroups
2.15. Exercise. Let a be an absolute value on S and let cp E ,qI>~(s) satisfy 2 I cp(s) 1 = cp(s + s*) for all s E S. Show that cp E S*. Show also that if cp E ,qI>~(S) satisfies Icp(s) I = a(s) for all s E S, then cp E S*. 2.16. Exercise. Let (S, +, *) be an abelian semigroup with involution. Show that S* is a linearly independent subset of C S and conclude that if S is finite then card S* ~ card S. Give an example where card S* < card S < 00. 2.17. Exercise. Show that ([ -1, 1], .).. . can be described as T u sgn· T u {sgn 2 } where T := {t 1-+ It I~, a E [0, oo]}, sgn:= 1]0,1] - 1[-1,0[, and Itl o := 1, Itl oo := 1{-1,1}(t) for t E [-1,1]. 2.18. Exercise. Let cp E ,qI>~(S). Then {s E S Icp(s) = a(s)} {s E S II cp(s) I = a(s)} are *-subsemigroups of S.
as well as
2.19. Exercise. Let (S, +, *) be an abelian semigroup with involution and let T:= {p E 511 p I == I}. If cp E ,qI>b(S) has the representing measure J.1 then J.1(T) = infses cp(s + s*). 2.20. Exercise. Let (S, +, *) be an abelian semigroup and consider S u {iiJ} as in Exercise 1.19. Let cp E ,qI>b(S) have the representing measure J.1. Then an extension iP of cp to S u {iiJ} is positive definite if and only if 0 ~ i1J( iiJ) ~ J.1( {Is})· 2.21. Exercise. Let S be an abelian semigroup with involution, let n E N /'.... -. and let S" be identified with (S)" in accordance with Exercise 2.13. Show that if cp E ,qI>b(S") has the representing measure J.1 E M + (5"), and if (J is a permutation of n elements, then cp 0 (J has the representing measure J.1u- I. Conclude that cp E ,qI>b(sn) is symmetric, i.e. cp 0 (J = cp for all permutations (J, if and only if J.1 is symmetric. 2.22. Exercise. Let S = [0, 1] and define Sot = (s + t - 1)+ for s, t E S. Show that (S, 0) is an abelian semigroup with neutral element 1 and absorbing element o. Show further that S is 2-divisible (cf. 6.8) and that S* = {I, 1{1}}.
§3. Negative Definite Functions on Abelian Semigroups In most of this section, we consider negative definite functions on an abelian semigroup with involution (S, +, *). From 3.20 onwards, we will specialize to semigroups with the identical involution.
99
§3. Negative Definite Functions on Abelian Semigroups
For tjJ E %(S) we know by Theorem 3.2.2 that e-tl/J E [!lJ(S) for t > O. Since we only have an integral representation for exponentially bounded positive definite functions and, in particular, for bounded positive definite functions, the following simple result is of some interest.
3.1. Proposition. Let tjJ E %(S). Then e-tl/J E [!lJe(s) (resp. E q;b(S)) for all t > 0 if and only if there exists an absolute value a ~ 1 and a constant C E ~ such that Re tjJ(s)
~
C - log a(s)
(resp. Re tjJ(s)
~
C
If this is the case then C
s E S,
for
s E S).
for
= tjJ(O) can be used.
PROOF. If e-l/J is exponentially bounded (resp. bounded) there exists an absolute value a, where we may assume a ~ 1 (resp. a = 1), such that
Ie-l/J(s) I =
e-Rel/J(s)
~
s E S,
e-l/J(O)a(s),
and the stated inequalities follow. Conversely, if the first inequality of the proposition holds, we get for t > 0
/e-tl/J(S) I ~ e-tCa(sr, which shows that e- tljl absolute value.
q;e(s) (resp.
E
s
E
S,
[!lJb(S) when a
E
=
1) since at is an
D
The set of negative definite functions tjJ for which Re tjJ is bounded below will be denoted %/(S).
3.2. Corollary. For tjJ
E
%/(S) we have Re tjJ(s)
3.3. Proposition. Let tjJ
E
~
tjJ(O),
SE
S.
%/(S) satisfy tjJ(O) ~ O. Then
JltjJ(s+t)I~~+~, PROOF. By Proposition
s,tES.
1.9 the matrix
2 Re tjJ(s) - tjJ(s* ( tjJ(s) + tjJ(t) - tjJ(s
+ s) + t*)
tjJ(s) + tjJ(t) - tjJ(s* + t)) 2 Re tjJ(t) - tjJ(t* + t)
is positive definite, hence, using inequality (1) of §1, I
tjJ(s)
+ tjJ(t)
- tjJ(s*
+ t)
2
1
~ (2 Re tjJ(s) - tjJ(s*
+ s)) + t))
x (2 Re tjJ(t) - tjJ(t* ~
41 tjJ(s) " tjJ(t) I.
Replacing s by s* and extracting the square root gives
ItjJ(s
+ t)1 ~
and the inequality follows.
/tjJ(s)1
+
/tjJ(t)1
+ 2~~, D
100
4. Main Results on Positive and Negative Definite Functions on Semigroups
In some later applications we shall need the following boundedness result for negative definite functions on semigroups with absorbing element (cf. 1.3).
3.4. Proposition. Let S contain the absorbing element w. Then the.functions in JV'(S) are automatically bounded. More precisely if IjJ E JV'(S) and 1jJ(0) ~ 0 then 111jJ1100 ~ fil/J(w). PROOF.
For fixed s
(
E
S the 3 x 3 matrix
1jJ(0)
ljJ(s*)
ljJ(s) ljJ(w)
ljJ(s + s*) ljJ(w)
IjJ(W)) ljJ(w) ljJ(w)
= (ajk)j,k= 1, 2, 3
is negative definite. Writing down explicitly the inequality for C 1 = C 2 = 1, C 3 = -2 gives
L cjC;a jk ~ 0
+ 2 Re ljJ(s) + ljJ(s + s*) ~ 41jJ(w), + s*) = Re ljJ(s + s*) ~ 0 by Corollary 1jJ(0)
l/J(s 3.2 we get 21jJ(w). Similarly, the choice of C 1 = 1, (2 = i, C3 = -1 - i gives - 1m ljJ(s) ~ ljJ(w), and the coefficients C 1 = 1, C 2 = - i, c 3 = i - I yield 1m ljJ(s) ~ ljJ(w). Hence 11m ljJ(s) 1~ ljJ(w) and therefore 1IIjJI 100 ~
and
since
o ~ Re l/J(s)
~
D
fil/J(OJ).
We shall next establish an important relationship between functions JV'(S) and convolution semigroups of Radon measures on the compact semigroup 5, which is the restricted dual semigroup of S.
l/J E
3.5. Definition. Let T be a Hausdorff topological semigroup with neutral element e. A convolution semigroup on T is a family (J.1t)t~O from M~(T) such that t ~ J.1t is a continuous homomorphism from the semigroup (IR+, +) into the semigroup (M~ (T), *), i.e. such that: (i) J.10 = Be; (ii) J.1t * J.1r = J.1t+r for t, r E IR+ ; (iii) t ~ J.1t is weakly continuous. In the case of T = 5 condition (iii) can be replaced by a seemingly weaker one as the following lemma shows.
3.6. Lemma. Let t ~ J.1t be a homomorphism from (IR+, +) into M +(5) such that lim t -+ 0 J.1t
=
B1
weakly, then (J.1t)t ~ 0 is a convolution semigroup on
5.
By Theorem 2.11 it suffices to prove thatjs(t):= fils) is a continuous mapping from IR+ into C for each s E S. By assumption we have js(t + r) = js(t)js(r) and lim t -+ o js(t) = 1, so there exists A > 0 such that Ijs(t) I ~ 2 for t E [0, A]. It follows that Us(t) I ~ 2n for t E [0, nA], in particular js is locally PROOF.
~3.
101
Negative Definite Functions on Abelian Semigroups
bounded. This combined with the equations (t, r, t - r > 0)
Ijs(t
+ r)
- js(t) I = Ijs(t) IIjs(r) -
11, 11
Ijs(t - r) - js(t) I = Ijs(t - r) Iljs(r) -
D
implies that js is continuous.
3.7. Theorem. There is a one-to-one correspondence between convolution semigroups (J.1t)t ~ 0 on S and negative definite functions l/J E %l(S) established
via the.formula (It(S) = e - tl/l(s)
for
t
~
0,
S E
s.
(1)
PROOF. Let (J.1t)t~O be a convolution semigroup. For s E S the mapping
js: t t---+ (It(s) is continuous by Theorem 2.11 and satisfiesjs(t + r) = js(t)js(r), hence of the form js(t) = e-tl/l(S) for a uniquely determined complex number t/J(s). Since e-tl/l E g>b(S) for each t > 0 it follows by Theorem 3.2.2 and Proposition 3.1 that t/J E %(S) and that Re t/J ~ t/J(O). Conversely, if t/J has these properties there exists a uniquely determined family (J.1t)t > 0 from M +(S) such that (1) holds. Clearly
(l,(s)(lr(s) = (It+r(s) for s
E
(J.1t)t~O
S, so
=
lim (It(s)
and
t-O
= 1 = £1(S)
is a convolution semigroup by 2.10, 2.11 and 3.6.
D
We will now consider two special types of functions t/J E %l(S), namely, purely imaginary homomorphisms and nonnegative quadratic forms. 3.8. Definition. A function t/J: S --+ C is called a purely imaginary homomorphism if it has the form t/J = ii, where I: S --+ IR is *-additive, i.e. I(s + t) = I(s) + I(t) and I(s*) = -/(s) for s, t E S. A function q: S --+ IR is called a quadratic form if
+ 2q(t)
2q(s)
=
q(s
+ t) + q(s + t*)
for
s, t
E
S.
If the involution on S is the identical, every purely imaginary homomorphism is identically zero, and a quadratic form is a homomorphism of (S, +) into (IR, +). If S is an abelian group, with the involution being s* = -s for s E S, the functional equation for a quadratic form q is
+ 2q(t)
2q(s)
=
q(s
+ t) + q(s
- t)
for
s, t
E
S.
3.9. Theorem. Purely imaginary homomorphisms and nonnegative quadratic forms belong to %l(S). PROOF. Let
t/J: S
~ Sand {c 1 ,
--+
•.. ,
C be a purely imaginary homomorphism, let {SI' ... , sn} LJ= 1 Cj = O. Then
cn } ~ C with
n
L
j.k=l hence t/J
E
%l(S).
n
CjCkt/J(sj
+ Sk) = L
j,k=1
CjCk(t/J(Sj)
+ t/J(Sk)) =
0,
102
4. Main Results on Positive and Negative Definite Functions on Semigroups
Let q: S --+ [R be a quadratic form. Note that q(O) = 0 and q(s*) = q(s). We define B: S x S --+ [R by
B(s, t) = q(s)
+ q(t)
- q(s
+ t*),
s, t
E
S,
and claim that B has the following properties for s, t, rES:
(i) (ii) (iii) (iv)
B(s, t) = B(t, s); B(s*, t) = B(s, t*) = -B(s, t); B(s + r, t) = B(s, t) + B(r, t); tB(s, s) = lim n oo (q(ns)/n 2 ). -+
Here (i) is clear, and
B(s*, t) = B(s, t*) = q(s)
+ q(t)
- q(s
= q(s + t*) - q(s) - q(t)
=
+ t)
-B(s, t).
To see (iii), we first remark that
+ q(t + t*)
q(s)
=
q(s
+ t + t*)
for
s, t
E
S,
and therefore we have
B(s, t)
+ B(r, t) =
+ q(t) - q(s + t*) + q(r) + q(t) - q(r + t*) = q(s) + q(r) + 2q(t) - t[q(s + t* + r + t*) + q(s + t* + t + r*)] = q(s) + q(r) + 2q(t) - tq(s + r*) - tq(t + t*) - tq(s + r + t* + t*) = tq(s + r) + 2q(t) - tq(t + t*) - t{2q(s + r + t*) + 2q(t) - q(s + r + t + t*)} = q(s + r) + q(t) - q(s + r + t*) = B(s + r, t). q(s)
To see (iv) we remark that for n
E
q(ns) = n 2 q(s) -
N n(n - 1)
2
q(s
+ s*).
This formula is clearly true for n = 1,2, and assuming it is correct for n - 1, n we get
q((n
+
l)s) = 2q(ns)
+ 2q(s)
- q((n - l)s
+ (s + s*))
= (2n 2 + 2)q(s) - n(n - l)q(s + s*) - q((n - l)s) - q(s + s*) = (n + 1)2q(s) -
(n
+
2
l)n
q(s
+ s*).
It follows that
. q(ns) lIm - 2 - = q(s) - tq(s n
-+00
n
+ s*) =
tB(s, s).
103
§3. Negative Definite Functions on Abelian Semigroups
If q is a nonnegative quadratic form it follows by (iv) that B(s, s) s E S. For {Sl' ... , sn} ~ S, {c 1 , ... , cn} ~ 7L we now get n
I
j,k= 1
~
0 for all
n
cjck(q(Sj)
+ q(Sk)
- q(sj
I
+ Sk)) =
j,k= 1
cjckB(Sj' Sk)
B(t, t)
=
~
0,
where
I
t =
CjSj
I
+
j,Cj~O
(-cj)sj.
j,Cj
We conclude from Proposition 1.9 and Exercise 3.1.21 that q is negative D definite.
3.10. Remark. If a quadratic form q is bounded below then it is nonnegative and negative definite. In fact, the proof shows that B(s, s) ~ 0, hence q E JVI(S) and, finally, q(s) ~ q(O) = 0 by Corollary 3.2. For.!
E
C S and a E S, we define raf
raf(s)
E
C S by
= !(f(s + a) + f(s + a*)),
SE
S.
3.11. Proposition. Let t/J E JVI(S) and let (Jlt)t>o be the corresponding convolution semigroup on S. For a E S we have r at/J .: t/J E p;>b(S), and if (Ja E M + (S) is the unique Radon measure such that r al/J - l/J = fj a then .
1
(Ja = hm(l - Re p(a)) - Jlt t
t-O
weakly. PROOF. If 1 is the identity operator on CS we have -1 (ra
t
-
1)(1 - e-t"')(s)
= -1 (1 - ra)(e-t"')(s) = -1 t
t
1 s
p(s)(l - Re p(a)) dJlip),
showing that (ra - 1)«1/t)(1 - e- t",)) E p;>b(S). For t -.. 0 this function tends pointwise to rat/J - t/J, which then necessarily belongs to p;>b(S). The assertion about the measure follows immediately from the Levy continuity theorem (2.11). D We now consider an arbitrary hermitian function t/J: S --+ C with the property that r at/J - t/J E p;>b(S) for all a E S, and we denote by (Ja the unique Radon measure on S such that rat/J - t/J = fJa.
3.12. Lemma. Let t/J: S --+ C be a hermitianfunction such that r at/J - t/J E p;>b(S) for each a E S. There exists a unique Radon measure Jl E M + (S\ {I}) such that (1 - R e p( a)) Jl
= (J a I(S\ { 1})
for
a E S.
(2)
104
4. Main Results on Positive and Negative Definite Functions on Semigroups
If tjJ E JVI(S) and (J.1t)t> 0 is the corresponding convolution semigroup then J.1 = lim t.... o(l/t)J.1tl(S\{l}) vaguely. PROOF.
For a E S we let
(!Ja:= {p
E
SIRe p(a) < I},
+
which is an open subset of S. Since (!Ja = {p E Slp(a) I} the family «(!Ja)aeS is an open covering of the locally compact space S\ {I}. On each (!Ja we consider the Radon measure
Since -(ra
-
I)(r b
-
I)tjJ(s) = -(r b
=
L
=
L
I)(r a
-
-
I)tjJ(s)
p(s)(l - Re p(a)) dUb(P)
p(s)(l - Re p(b)) dua(p),
we get by the unicity assertion of Theorem 2.8 that (1 - Re p(a))ab = (1 - Re p(b))aa'
a, b E S,
so the compatibility condition of Theorem 2.1.18 is satisfied for the measures Consequently, there exists a uniquely determined Radon measure J.1 on S\{I} such that J.11 (!Ja = La for a E S, and it is clear that J.1 is the only Radon measure satisfying (2). To finish the proof it is enough by Theorem 2.4.9 to prove that for each
(La)aeSo
aES . 1 1Im-J.1tl(!Ja=L a t .... O
g(p)
:=
vaguely on (!Jao
t
{f(P )(1
- Re p(a)) -
1,
0, is a bounded continuous function on
S, so by 3.11, D
3.13. Definition. Let tjJ be as in Lemma 3.12. The measure J.1 is called the Levy measure for tjJ or for the corresponding convolution semigroup (J.1t)t~O in case tjJ E JV'(S). -
105
§3. Negative Definite Functions on Abelian Semigroups
The Levy measure might have infinite total mass, but (2) implies that
(l - Re p(a)) djl(p) <
fA
for all
00
a
E
S.
(3)
JS\{l}
Let £ denote the set of J.1 E M + (S\ {I}) for which (3) holds. With the same assumptions as in Lemma 3.12 we have:
3.14. Lemma. Thefollowing conditions are equivalent: (i) J.1 = 0; (ii) a t/J - t/J is constant for each a E S; (iii) t/J = t/J(O) + q + il, where q is a nonnegative quadratic form, and I is
r
*-additive. PROOF. The equivalence "(i) <=> (ii)" is obvious. Suppose that (ii) holds. Defining q = Re(t/J - t/J(O)), I = Im(t/J - t/J(O)), we have
raq - q = (Ja({l})
~
0,
ral - 1=0
and
q(a*)
I(a*) = -/(a)
q(a),
=
for all a E S, and it follows easily that q is a nonnegative quadratic form and I is *-additive. In fact
raq(s) - q(s) = raq(O) - q(O) = q(a),
s
E
S,
which is the functional equation for a quadratic form. Furthermore,
ral(s) - I(s) = t[/(s
+ a) +
I(s
+ a*)]
- I(s) = 0,
SE
S,
so by interchanging the roles of a and s we have
+ a) + I(s + a*) = 2/(s), I(a + s) + I(a + s*) = 2/(a), = -/(a + s*) we find I(a + s) = I(a) + I(s). I(s
and using I(s
+ a*)
Conversely, if (iii) holds, it is easy to see that rat/J - t/J
= q(a).
D
As another use of the Levy measure we can completely characterize the set %b(S) of bounded negative definite functions.
3.15. Proposition. Let t/J J.1 (S\{l}) ~ Iit/Jiloo and "'(s)
E
%b(S). Then the Levy measure J.1 for t/J satisfies
= "'(0) +
fA
(l - p(s)) djl(p),
s
E
S.
JS\{l}
In particular t/J has the form c -
qJ
where c E IR and
qJ E
[!lJb(S).
106
4. Main Results on Positive and Negative Definite Functions on Semigroups
PROOF. Without loss of generality we may assume 1/1(0) = O. The corresponding convolution semigroup (/It)t ~ 0 consists of probability measures, and we have for t > 0 -
1 - e-tl/J(s) Using Re I/I(s)
~
I. (l
=
(l - p(s» dJI-t(p).
JS\{l}
0 (cf. 3.2) and
o ~ I.
I.
=
- p(s» dJI-t(p)
Js
11 - e-zi
~
(1 - Re p(s» dJI-t(p)
JS\{l}
Izl for
Re
z~
0, we find
~ tlt/J(s)1 ~ t 1It/J1l
00'
For any molecular measure a on S we define
1i(p)
i
=
p(s) da.(s),
pES.
If a is a molecular probability measure on S we have
o~ =
f.s ( I.
(l - Re p(s» dJI-t(P») da(s)
JS\{l}
I. (l JS\{1}
- Re &(p» dJI-t(p)
~ tllt/Jlloo'
Let C £; S\ {1} be a compact set. For any Po E C there exists a E S such that Re po(a) < 1, and there _xists an open neighbourhood U of Po in S such that Re pea) < 1 for all p E U. By compactness of C there exist finitely many points a l ' ... , an E S such that 1
n
- L Re p(ai)
< 1
for all
p E C.
ni=l
If we define the molecular probability a by
then
1
n
&(p)=- LRep(ai)E[-l,l],
pES
ni=l
and &(p) < 1 for p E C. For every natural number p the convolution power a*P is again a molecular probability measure and (a*P)A(p) = (&(p))P, pES. Applying Fatou's lemma gives
JI-t(C)
~ I.
lim inf(l - (1i(p»2 P + 1) dJI-t(p)
JS\{l}
~ lim inf p--.oo
P-OO
J
(l - (1i(p»2 p + 1) dJI-t(p)
~ tllt/JII
00'
107
§3. Negative Definite Functions on Abelian Semigroups
and C being arbitrary, we get Jlt( 5\ {1}) ~ t I t/J I 00 .
Since (l/t)Jlt I(5\ {I}) ~ Jl vaguely for t ~ 0 we get by Proposition 2.4.4 that Jl(5\ {I}) ~ II t/J 1100 and that ljJ(s)
= lim ~ (l - e-tt/J(S») t
t ..... O
f
= lim
=
(1 - p(s)) ~ dJlt(p) t
t ..... O
S\{1}
fA
(1 - p(s)) dJl(p),
S\{1}
D
because P r-+ 1 - p(s) belongs to CO(5\ {I}).
3.16. Corollary. Let S contain an absorbing element w. Then the transformation &'~(S) ~ {t/J
E
%'(S)It/J(O) = 0, t/J(w) ~ I},
qJr-+l-qJ
is onto, i.e. is an affine homeomorphism. PROOF. Let t/J E %'(S) fulfil t/J(O) = 0 and t/J(w) ~ 1. By Proposition 3.4 t/J is bounded and therefore t/J has the representation
t/J(s)
=
fA
(1 - p(s)) dJl(p),
s
E
S,
S\{1}
where t/J(w) M~(S), then
= Jl(5\ {I}) ~ 1. Put a:= 1 - Jl(5\ {I}) and Jl':= Jl + ae 1 E 1 - t/J(s) = a
+ fA
p(s) dJl(p)
=
fA p(s) dJl'(p) S
S\{1}
showing that 1 - t/J
E
&'~ (S).
D
Our goal is to extend the integral representation in Proposition 3.15 to functions t/J E %'(S), thus looking for a representation analogous to the classical Levy-Khinchin formula in probability theory, cf., e.g. Loeve (1963). If Jl E M +(5\ {I}) is such that
fll -
p(s) I dJl(p) <
00
for
s E S,
(4)
SES
(5)
then t/JJl(s):=
fA S\{1}
(1 - p(s)) dJl(p),
108
4. Main Results on Positive and Negative Definite Functions on Semigroups
belongs to %'(S), as is easily seen. However (4) need not hold for all Levy measures, and therefore it is necessary to introduce a compensating term in the integral in (5). This is made precise in the following.
3.17. Definition. A function L: S x
8~
[R
is called a Levy function for S if
(i) L(·, p) is a *-additive function from S into [R for each p E 8. (ii) L(s, .) is Borel measurable on 8 for each s E S. (iii) 11 - p(s) + iL(s, p)1 dJ.l(p) < 00 for all s E S and all measures J.l (i.e. such that (3) holds).
J
E
S!
We do not know if every semigroup with involution has a Levy function, and it is probably a difficult question. If S is an abelian group and s* = - s there exists a Levy function. This may be seen from Parthasarathy et al. (1963) and depends on the structure theory for abelian groups. See also Forst (1976). If S is an abelian semigroup with s* = s the function L == 0 is a Levy function. This also holds (cf., 3.4 and 3.15) if S contains an absorbing element. We shall later give other examples of Levy functions.
3.18. Proposition. Suppose S has a Levy function L. F'or any measure J.l E S! the function ljJ,ls):=
I.
(1 - p(s)
+ iL(s, p»
djl.(p),
sES
JS\{l}
belongs to %'(S) and the Levy measure for t/J Jl is J.l. PROOF. The function t/J Jl is easily seen to be negative definite and Re t/J ~ O. Furthermore,
(ra
-
I)ljJ,is) =
I.
p(s)(l - Re p(a» djl.(p),
a, s E S,
JS\{l}
which shows that (J~ E M +(8), determined such that fJ~ = (ra - I)t/J Jl , is concentrated on 8\ {I} and equal to (1 - Re p(a»J.l. By Lemma 3.12 it D follows that J.l is the Levy measure for t/J Jl' We can now state and prove the '" Levy- Khinchin " integral representation for %'(S).
3.19. Theorem. Suppose S has Levy function L. The following conditions are equivalent for a function t/J: S ~ C: (i) t/J E %'(S). (ii) t/J is hermitian and r a t/J - t/J E &>b(S) for each a E S. (iii) There exists a triple (I, q, J.l) where I: S ~ [R is *-additive, q is a nonnegative quadratic form and J.l E £ such that
ljJ(s) = ljJ(O)
+ il(s) + q(s) + I. JS\{l}
for all s E S.
(1 - p(s)
+ iL(s, p»
djl.(p)
109
§3. Negative Definite Functions on Abelian Semigroups
The triple (I, q, J.l) is uniquely determined by t/J measure for t/J and _ l' Re t/J(ns) 1m 2
q(s ) -
n
" _ 00
E
%'(S), J.l being the Levy
+ I'1m t/J(n(s + s*» " -+
2n
00
,
SE
S.
PROOF, "(ii) => (iii)" Let J.l be the Levy measure for t/J and let t/J Jl be defined as in Proposition 3.18. The function
r := t/J - t/J(O) - t/J Jl is hermitian and rar - r = fi a - (O"al(S\{I}»A = O"a({1}) for a E S, hence positive definite and constant. By Lemma 3.14 it follows that t/J has the representation stated in (iii). Finally, if (iii) holds, it is clear by 3.9, 3.14 and 3.18 that t/J E %'(S), and that the Levy measure for t/J is J.l. This shows that the triple (I, q, J.l) is uniquely determined by t/J, and we find
= t/J(?) + q(~s) +
Re t/J2(ns)
n t/J(n(s
n
+ s*» =
n
t/J(O) n
n
~ (1
f
_ Re(p(s)"» dJ1.(p),
JS\{l} n
+ q(n(s + s*» + f
JS
n
\{l}
! (l
_ Ip(sW") dJ1.(p).
n
By the proof of Theorem 3.9 we have
,
hm "-+00
q(ns) -2-
n
= q(s) - tq(s + s*),
and by the functional equation for quadratic forms
q(n(s
+ s*» = nq(s + s*),
which gives the formula for q(s) provided the integrals tend to zero. That this is true follows from the dominated convergence theorem because of the inequalities
1 n2 (1 - Re(p(s)"»
~
n2
4
(1 - Re p(s»,
1 -(1 -lp(s)1 2 ") ~ 1 - Re p(s
n
Putting p(s)
= re i8 , r E [0, 1], 0 E ]
1
n2 (1 - r" cos(nO» Clearly 1 - r"
~
~
n2
4
-
+ s*).
n, n], these inequalities become n EN.
(1 - r cos 0),
n(1 - r) for r E [0, 1], n E N, and using the inequalities
ISi: x I ~ 1,
X E ~,
sin x 2 --2x
- n
for
n
Ixl -~ -2'
110
4. Main Results on Positive and Negative Definite Functions on Semigroups
we find for f)
E ]
-n, n]
.
= 2 SIn
1 - cos(nf)
2
nf)
2 =
hence
1
2 (1 -
n
r" cos(nf))
1
~ 2
n
(1 - r")
1
~ ;; (1 -
r)
~
+2
n
(1 - cos nO)
n2
+ 4 r"(1 -
cos f) ~
n2
4
(1 - r cos f).
D
Remark. In the above representation q and J1 are independent of the Levy function L, whereas I depends on the choice of L. In the rest of this section, we assume that S is an abelian semigroup with the identical involution. Then negative definite functions are real-valued and %'(S) is the set of negative definite functions t/J which are bounded below or equivalently satisfy t/J(s) ~ t/J(O). For f E [Rs and a E S, we define ~af E [Rs by ~af(s)
= f(s + a) - f(s),
s E S,
hence ~af = (ra - I)fwith the previous notation. The function L == 0 is a Levy function for S, and from the previous results it is easy to get the following: 3.20. Theorem. Let S have the identical involution. The following conditions are equivalent for a function t/J: S -+ [R: (i) t/J E %'(S). (ii) ~a t/J E pjJb(S) for each a E S. (iii) There exist an additive function q: S J1 E M + (8\ {I}) such that
t/J(s)
= t/J(O) + q(s) +
-+
r
JS
[0, oo[ and a Radon measure
(l - p(s)) dJ1.(p).
\{l}
The decomposition in (iii) is uniquely determined, J1 being the Levy measure for t/J and q (s)
.
= 11m "-+00
t/J(ns) --, n
SE
S.
If q == 0 then t/J is bounded if and only if J1(S\ {I}) <
00.
111
§3. Negative Definite Functions on Abelian Semigroups
3.21. Corollary. Every t/J
E ~ (S)
satisfies the inequalities
It/J(s) - t/J(t) I ~ tjJ(s + t)
~
tjJ(s)
+
tjJ(t)
for
s, t
E
S.
PROOF. The set of functions t/J: S ~ [0, oo[ satisfying the above inequalities is a closed convex cone C in [Rs containing the nonnegative constants, the additive functions q: S ~ [0, 00 [ and the functions 1 - p, pES. By 3.20 it follows that ~(S) £; C. D
The following result is inspired by a remark of van Harn and Steutel (1980). 3.22. Proposition. Let t/J: S ~ [R and define t/Ja(s):= t/J(a) + t/J(s) - t/J(a + s) .for a E S. ~f t/J E %'(S) then t/Ja E %b(S) for a E S. Conversely, if t/Ja E %b(S) for all a E S then t/J E %(S). PROOF.
If t/J E %'(S) has the representation of (iii) in 3.20 we get I/Iis) = 1/1(0) +
I. JS
(1 - p(s»(l - p(a» djJ.(p),
\{l}
and since (1 - p(a»J.l is a finite measure, t/Ja E %b(S). Conversely, if t/Ja E %b(S) with Levy measure J.la' we have by 3.15 that I/Ia(s) = 1/1(0)
+ I.
JS
(1 - p(s»djJ.a(P)'
\{l}
Now ~bt/Ja(S)
=
~at/Jb(S)
= t/J(a + s) + t/J(b + s) - t/J(s) - t/J(a + b + s)
and Abl/la(s) =
I. JS
p(s)(1 - p(b» djJ.a(P),
\{l}
Aal/lb(s)
= I
JS
p(s)(1 - p(a» djJ.b(P).
\{l}
By uniqueness of representing measures we get a, bE S.
(1 - p(b»J.la = (1 - p(a»J.lb'
As in the proof of Lemma 3.12, there exists a unique Radon measure J.l on
S\ {1} such that (1 - p(a»J.l = J.la
for
a
E
S,
and since J.la is finite we have J.l E £. The function h(s) := I/I(s) - 1/1(0) -
f
(1 - p(s» djJ.(p),
sES
112
4. Main Results on Positive and Negative Definite Functions on Semigroups
is real-valued, and we get
h(a)
+ h(s)
- h(a
+ s) =
ljJis) - ljJ(O) -
= ljJis) - ljJ(O) -
I Io -
(l - p(s»(l - p(a» dJl(p)
p(s» dJlip) = 0,
which shows that h is additive, hence h E %(S). The formula
ljJ(s) = ljJ(O)
+ h(s) +
Io -
p(s» dJl(p),
SE
S,
D
shows that t/J is negative definite.
3.23. Exercise. Let S be an abelian semigroup with involution and I: S ~ IR a function such that t/J(s) = il(s) is negative definite. Show that I is *-additive. 3.24. Exercise. Let S, T be abelian semigroups with involution having Levy functions. Construct a Levy function for S x T. 3.25. Exercise. Let S be an abelian semigroup with involution and Levy function. Show that if t/J E %'(S) then Re t/J(a) - (r a - I)t/J E %b(S) for all a E S. Let t/J: S ~ C be a hermitian function such that Re t/J(a) - (r a - I)t/J E %b(S) for all a E S. Show that t/J is sum of a quadratic form and a function in %'(S). Show finally that t/J E %'(S) if Re t/J in addition is bounded below. 3.26. Exercise. Let S have the identical involution and let
E
q>~ (S) be
(a)
3.27. Exercise. Let S have an absorbing element w. Ift/J E %'(S) and t/J(O) = 0, then II t/J II ~ 2t/J( w), i.e. the constant in Proposition 3.4 can be amended to 2 in case t/J(O) = O.
J5
00
+, *) be an abelian semigroup with involution and S be a generator set (see 5.7). Show that if J1 E M + (8\ {I}) satisfies
3.28. Exercise. Let (S, let G
£;
f.
JS
(1 - Re p(a» dJl(p) <
\{l}
then J1 E £ (see 3.13).
00
for all
aE G
113
§4. Examples of Positive and Negative Definite Functions
3.29. Exercise. Let S be an abelian semigroup with the identical involution and let G £; S be a generator set. Show the following generalization of Theorem 3.20: If t/J: S ~ [R satisfies ~a t/J E &>b(S) for all a E G then t/J E %1(S).
§4. Examples of Positive and Negative Definite Functions 4.1. S = ([R+, +). The closed half-line [R+ is an abelian semigroup. As involution we use the identical involution x = x*. Since [R+ is 2-divisible (i.e. every x E [R+ can be written x = 2y for y E [R+), we see that positive definite functions on [R + are nonnegative; if they are bounded they are even completely monotone, cf. Corollary 6.8. For a E [0,00] we define Pa: [R+ ~ [R by Pa(s) = e- as with the convention that 0· 00 = 0, a . 00 = 00 for a > 0, i.e. Poo = I{o}' Then Pa is a bounded semicharacter, continuous when a E [0, oo[ and discontinuous when a = 00. Conversely, if P is a bounded semicharacter, then ~ p(s) ~ 1, which implies that P is decreasing. Let a E [0, 00] be determined such that p(l) = e- a. Then it is easy to see that P = Pa' so a ~ Pa is a bijection of [0, 00] onto S, and it is continuous when [0, 00] is considered as the one-point compactification of [0, 00[. It follows by a well-known theorem from topology that a ~ Pa is a homeomorphism. It is not possible to give a similar simple representation of S*. However, if h: [R ~ [R is any solution of the functional equation h(x + y) = h(x) + h(y), x, Y E [R, then exp(h) I[R+ is a semicharacter, and all P E S*\ {Pool are given in this way. By the axiom of choice there exist nonmeasurable solutions h of the functional equation. An application of Theorem 2.8 leads to the following result:
°
4.2. Proposition. A .function
and only
qJ(s) =
where J-l
if
if it has the .form
E M~([R+)
LX> e- as df1(a) + bpoo(s),
and b
~
s
~ 0,
0. The pair (J-l, b) is uniquely determined by
An additive function q: [R+ ~ [0, oo[ is necessarily of the form q(s) with c = q(l) ~ 0. By 3.20 we then have: 4.3. Proposition. A function t/J: [R + ~ [R belongs to %1([R +)
has the form
L
if and only if it
oo
ljJ(s) = ljJ(O)
where b, c
~
°
+ cs + bl lO • OO[(s) +
(l - e- as ) df1(a),
= cs
s
and J-l E M +(]O, oo[) are uniquely determined by t/J.
~ 0,
114
4. Main Results on Positive and Negative Definite Functions on Semigroups
Remark. The measures J.l E M + (]O, oo[) which occur in the above formula can be characterized by the single integrability condition
f.o
oo
The functions of the form f(s)
=
y
+ cs +
_x_ dJ.l(x) < 1+ x
f)
00.
(l - e- as ) dJl(a),
s
~ 0,
where c, y ~ 0 and J.l E M +(]O, oo[) such that J~[x/(1 + x)] dJ.l(x) < 00, are called Bernstein functions by some authors, cf. Berg and Forst (1975). By Proposition 4.3 we see that f: IR + -+ IR is a Bernstein function if and only if it is continuous and belongs to ~ (IR +). The expression for a Bernstein function is well defined in the right half-plane {z Eel Re z ~ O}. It is continuous there and holomorphic in the interior. Bernstein functions f operate on negative definite kernels t/J: X x X -+ C with t/J I~ ~ 0 in the sense that f 0 t/J is again a kernel of this type, cf. Proposition 3.2.9.
4.4. The function S 1--+
LX> e -sa dJl(a)
is called the Laplace transform of J.l and is denoted 2 J.l. This function is actually well defined in the right half-plane Re z ~ 0 by the formula 'pJl(z)
= {OO e- za dJl(a),
and it is easily seen to be continuous, and holomorphic in the open half-plane Re z > O. Furthermore, (.pJlyn)(z)
= {OO (-a)"e- za dJl(a)
for
Re z > 0,
n
~
0,
so in particular for
x >
o.
We have the following corollary of Proposition 4.2.
4.5. Corollary. A function qJ: IR + -+ IR is continuous, positive definite and bounded &(' and only if it is the Laplace transform ofa measure in M~(IR+). In Theorem 6.13 we will give a completely different characterization of the functions in Corollary 4.5.
4.6. S = (IR~, +). For s = (Sl' .•• , Sk) E IR~ and a = (a 1 , .•• , ak ) E [0, oo]k we put <s, a) = L~=l sja j and define Pa: IR\ -+ IR by Pa(s) = e-(s,a). Then
115
§4. Examples of Positive and Negative Definite Functions
the mapping a ~ Pa is a homeomorphism and an isomorphism of ([0, ooJ k, +) onto (8, .). The semicharacter Pa is continuous on [R~ when a E [R~ and discontinuous when a E [0, 00 Jk\[R~. Clearly we have results analogous to 4.2 and 4.3. For J.l E M~([R~) the function S E [R~
is the (k-dimensional) Laplace transform of J.l. This formula makes sense for s E e~ and defines a continuous function which is holomorphic in {z E eiRe z > O}k. A k-dimensional version of Corollary 4.5 is
4.7. Proposition. A function
qJ(s)
=
r
e- (s, a) dp.(a),
SE
1R"r .
)[0, oo]k
If 1r j : [0, OOJk ~ [0, ooJ denotes the projection onto the jth coordinate, we have
L oo
+
e- Sj ' dai t ),
where (Jj is the restriction of the image measure J.l1t j to [R + , and it follows that J.l({a E [0, OOJk/ aj = oo}) = for j = 1, ... , k, hence J.l([0, OOJk\[R~) = 0, and
°
4.8. S = (Nt, +). Here No = {O, 1,2, ... } and k ~ 1. The involution is the identical mapping. For x = (Xl' ... , x k ) E [Rk the function Px: Nt ~ [R given by n = (n l' . . . , nk ) E Nt, is a semicharacter, and it is easy to see that the mapping X ~ Px is a topological semigroup isomorphism of ([Rk, .) onto (Nt)*, where the composition in [Rk is x· Y = (X1Yl' ... ,XkYk)' Under this isomorphism [-1, l]k corresponds to Nt. A measure J.l E M + ([Rk) such that x n E !fJl(J.l) for all n E Nt is said to have moments of all orders. The function qJ(n)
=
J
x n dp.(x),
116
4. Main Results on Positive and Negative Definite Functions on Semigroups
is called the moment function of J.l, and the number
moment of J.1. The classical moment problem consists in characterizing the set of moment functions. We return to this in Chapter 6. Using that n ~ anI + ... +nk is an absolute value on N~, for each a > 0, we get the following partial solution of the moment problem from Theorem 2.6. 4.9. Proposition. A function
I
-+ ~
is positive definite and verifies
Ca nl +···+ nk
for
nE
N~,
where C, a > 0, if and only if
J.l
=
r
Jr- a ,
x" dJl(x)
nEN~.
for
a]k
In particular
ljJ(n)
-+ ~
is negative definite and bounded
if
below if and only
= ljJ(O) + +
r
(1 - x") dJl(x),
n
Jr-l,l]k\{l}
with b = (b l ,
... ,
E
N~
bk) E ~~ and J.l EM +([ -1, 1]k\ {I}), where 1 = (1, ... , 1).
Remark. The measures J.l E M +([ -1, l]k\ {I}) which can occur in the above formula are characterized by
r
(k - (Xl
+ ... + x k » dJl(x) <
00.
Jr-l,l]k\{l}
In fact, this condition is equivalent with the k conditions
r
Jr-
(1 -
Xj)
dJl(x) <
j
00,
= 1, ... , k,
l, 1 ]k\{ I}
which are clearly necessary. They are also sufficient (cf. 3.28), as is seen from the inequalities
o~
k
1 - X~I
•••
X~k ~
I
nj (1 -
Xj),
j= 1
4.11. S = (N5, +, *). In this case we equip the additive semigroup (N5, +) with the involution (n, m)* = (m, n). For Z E C, the function pz: N5 -+ C given by for
(n, m)
E
N5
117
§4. Examples of Positive and Negative Definite Functions
is a semicharacter, and it is easy to see that the mapping z r-+ pz is a topological semigroup isomorphism of (C, .) onto S*. Under this isomorphism the unit disc D = {z E C II z I ~ I} corresponds to S. For a measure j.1 E M +(C) such that znzm E !£l1(j.1) for all (n, m) E ~5 the function ({)(n, m)
=
L
z"zm dj1(z),
(n, m) E ~5
is called the complex moment function of j.1, and the number cp(n, m) is called the (n, m)th complex moment of j.1. The complex moment problem consists in characterizing the set of complex moment functions. We return to this in Chapter 6, cf. 6.3.5. Using that (n, m)r-+a n + m is an absolute value on for each a > 0, we get the following partial solution of the complex moment problem from Theorem 2.6.
N5
4.12. Proposition. A complex-valued function cp is positive definite on (~~, +, *) and verifies
Iq>(n, m)1 ~ Ca n + m
for
(n, m) E ~5,
where C, a > 0, if and only if q> is a complex moment function of a measure E M +({z E Clizi ~ a}), i.e. is of the form
Il
({)(n, m)
=
r
J1z1 ~a
z"zm dj1(z),
(n, m)
In particular q> is positive definite and bounded moment function of a measure j.1 E M + (D).
E
~5.
if and only if cp is a complex
In preparation for the Levy-Khinchin representation in this case we need the following two lemmas. 4.13. Lemma. The *-homomorphisms I: (~5,
l(n, m) = a(n - m), and the quadratic forms q: (~5, +, *) q(n, m) = a(n
+ m) +
0 ~land only ~la, fJ ~
-+
+, *)
-+
a
IR,
E
IR are given by
IR are given by
fJ(n - m)2,
a, fJ
E
IR.
o.
Here q
~
PROOF.
The assertion about the *-homomorphisms is easily established. Let
q be a quadratic form and put a = q(1, 0), b = q(1, 1). Since q(s q(s) + q(t) if t = t* we have q(n, m)
=
q(n - m,O)
+ mb
if
n ~ m,
+ t) =
118
4. Main Results on Positive and Negative Definite Functions on Semigroups
and from the proof of Theorem 3.9 we know that 2
q(n, 0) = n a For n
~
n(n - 1) 2 b,
n
~
o.
m we therefore have q(n, m)
=
(n - m)2
(a - ~) + (n + m) ~'
and since q(n, m) = q(m, n) this holds for all (n, m) E N5· Conversely, it is clear that any function of the form q(n, m) = a(n + m) + f3(n - m)2, a, f3 E lR, is a quadratic form, which is nonnegative if a, f3 ~ 0; and if q ~ 0 then q(l, 1) = 2a ~ 0 and lim n ---. oo (ljn 2 )q(n, 0) = f3 ~ o. D
4.14. Lemma. Thefunction L:
~5 x
D -+ lR given by
L«n, m), z) = y(n - m),
Z
= x + iy
+, *).
E
M + (D\ {I}) satisfies (3) of
is a Levy function for (~5, 3.13 if and only if
r
A measure J.1
(1 - x) dJ1(z) <
00.
JD\{l}
PROOF. The condition given on J.1 is clearly necessary. Suppose next that J.1 satisfies the given condition. For (n, m) E N5 such that n + m > 0 and Z = x + iy E D we have
1 - Re(z"zm)
= 1-
L (;)(;)x"+m- p- q( -1)q( -1)(P+Q)/2 y P+q,
where the sum is over (p, q) such that p ~ n, q ~ m and p + q is even. For p = q = 0 we have the term x n + m • In all other terms we have a factor yp+q, where p + q is even ~ 2, so yp+q ~ (1 - x 2 )(p+q)/2. Therefore
o~
1 - Re(z"zm)
~
1 - x"+m
+L
(;)(;)(1 -
x 2)(p+q)/2,
which shows that there exists a constant K > 0 (depending on n, m) such that 1 - Re(znzm) ~ K(l - x) so (3) of 3.13 holds. (This result also follows from Exercise 3.28.) The function L clearly satisfies (i) and (ii) of 3.17, and (iii) follows if we establish
11 -
znzm
+
i(n - m)yl
~
K(1 - x)
for
z
= x + iy E D
and a suitable constant K depending on (n, m). This is easily seen for n + m and for n + m ~ 2 we find 1-
znzm
+
i(n - m)y = 1 - x n + m
+
+ y2p(X, y),
iy(n - m)(1 - x n + m -
1
)
~
1
119
§4. Examples of Positive and Negative Definite Functions
where P is a polynomial, hence 11 -
znzm
+ i(n
~
- m)yl
(1 - x n + m )
+ In -
+ (1
max IP(x, y)1
- x
2
)
ml(1 - x n + m -
1
)
D
D
and the inequality follows. We can now give the Levy-Khinchin representation. 4.15. Proposition. A function has a representation
t/!(n, m) = t/!(O, 0)
t/!: N6 -+ C
+ ia(n
+ f
- m)
(1 - z"zm
belongs to JVI(N6)
+ b(n + m) + c(n
+ iy(n
-
m»
if and only if it
- m)2
dj1(z),
JD\{l}
where a E [R, b, c
~
0 and J1
E
M + (D\ {I}) satisfies
f
(1 - x) dJ1(z) <
00.
JD\{l}
4.16. Definition. A commutative semigroup (S, +) is called idempotent if + s = s, for every s E S.
S
We shall always equip an idempotent semigroup with the identical involution. Let (S, +) be an idempotent semigroup. We define an ordering ~ on S by s ~ t if s + t = t. It is easy to see that ~ is reflexive, transitive and antisymmetric, and that 0 ~ s, s + t = sup(s, t) for s, t E S. If S has a greatest element then it is absorbing and vice versa. A semicharacter p on S is 0-I-valued so S = S*. Furthermore, the set I = p -l( {I}) is a subsemigroup which is hereditary on the left, i.e. for s, t E S with s ~ t and tEl we have s E I. Conversely, it is easy to see that if I ~ S is a subsemigroup and hereditary on the left then 11 E S*. Therefore S* is isomorphic with the set !/ of subsemigroups which are hereditary on the left, considered as a semigroup under intersection. The neutral element is Sand {O} is an absorbing element. The topology from S* transported to !/ is the coarsest topology under which the mappings (Xs)seS from !/ to {O, I} are continuous, where
{I
if s E I, ( ) = 0 if s $ I,
Xs I
for
IE!/.
A function qJ E £!J>(S) is nonnegative, decreasing and bounded, so, in particular, &>(S) = &>b(S). In fact qJ(s) = qJ(s + s) ~ 0, and if S ~ t then qJ(t)2 = qJ(s + t)2 ~ qJ(s + s)qJ(t + t) = qJ(s)qJ(t) so qJ(t) ~ cp(s). In
120
4. Main Results on Positive and Negative Definite Functions on Semigroups
particular, qJ(t) ~ qJ(O). Similarly, ljJ ljJ(O) ~ ljJ(s) so %(S) = %1(S).
E
%(S) is increasing and satisfies
By specialization of the Theorems 2.8 and 3.20 we get:
4.17. Proposition. Let (S, +) be an idempotent semigroup. For qJ E &>(S) there is a unique j.1 E M + (Y') such that
qJ( s) andfor ljJ
E
=
j.1( {I
E
Y' IS E I}),
%(S) there is a unique j.1
ljJ(s)
=
ljJ(O)
SE
EM +(Y'\{S})
+ j.1({I E Y'\ {S} Is $ I}),
S,
such that SE
S.
A function qJ: S ---+ [0, oo[ which is decreasing and bounded need not be positive definite, cf. Exercise 4.25. However, if S has the further property of the order being total, i.e. for any s, t E Seither S ~ t or t ~ s, then we have
4.18. Proposition. Let (S, +) be an idempotent semigroup for which the ordering is total. Then qJ: S ---+ [R is positive (resp. negative) definite if and only if qJ is nonnegative and decreasing (resp. increasing). PROOF. Let qJ: S ---+ [0, 00 [ be decreasing and let s l' , Sn E S, C l' ... , Cn E ~. By 3.1.2 it is no restriction to assume s 1 ~ S2 ~ ~ Sn. From probability theory we know the existence of n independent normally distributed random variables X l' . . . , X n with mean zero and variances qJ(SI)' qJ(S2) - qJ(SI)' . .. , qJ(sn) - cp(sn - 1). (If one of the variances is zero the corresponding normal distribution is degenerate.) Put ~ = L~= 1 Xj' k = 1, ... , n and denote the expectation by IE; then ifj ~ k,
IE( lj ~)
= IE( YJ) = qJ(S 1) + qJ(S2) - qJ(S 1) + ... + qJ(Sj) - qJ(Sj_ 1) = qJ(Sj) = qJ(Sj + Sk)
implying
If qJ: S ---+ [R is increasing then e - tep is nonnegative and decreasing for each t > 0, and it follows by Theorem 3.2.2 that qJ is negative definite. D
4.19. As an example of an idempotent semigroup whose ordering is total we consider S = [0, 1], the composition being maximum. The neutral element is zero and 1 is an absorbing element. The ordering of 4.16 is the usual ordering. The set Y' of Proposition 4.17 has the following elements: {O}, [0, a], [0, a[ where a E ]0, 1], and we will now describe the topology on Y' induced by the isomorphism between S* and !/.
121
§4. Examples of Positive and Negative Definite Functions
a'" [0, a]
b '" [0, b [ Figure 1
A base of neighbourhoods of {O}, [0, a] and [0, bE, where a, bE ]0, 1], is given, respectively, by
°
(i) {{o}} u {CO, s] Is E ]0, e[} u {CO, s[ Is E ]0, 8[} for < 8 < 1; (ii) {CO, S]ISE [a, a + 8[n[0, I]} u {CO, s[lsE]a, a + 8[n[0, I]} for < 8 < 1; (iii) {CO, s] Is E ]b - 8, b[n[O, I]} u {CO, s[1 s E ]b - 8, b]n]O, I]} for < 8 < 1.
° °
We remark that f/ is totally ordered under inclusion and that the corresponding order topology on f/ is the topology just described. It is useful to think of f/ as two copies of [0, 1] glued together in the point as Figure 1 illustrates. On the upper segment a E ]0, 1] represents [0, a] and on the lower segment b E ]0, 1] represents [0, be. We know that f/ is a compact Hausdorff space and see that [0, 1] is an isolated point. Furthermore f/ is one of those strange nonmetrizable spaces which are first countable (i.e. every point has a countable neighbourhood base). In fact, if f/ was metrizable, then a classical theorem from topology (cf. Bauer (1978, p. 217)) implies that the Banach space C(f/) of continuous functions on f/ is separable, which is not true: For a E [0, 1] let ba E C(f/) be the function I ~ l I (a). Then Ilb a - bbll = 1 for a =+= b, so the open balls B(b a , t) in C(f/), a E [0, 1] form a noncountable family of disjoint open sets contradicting the separability. As an application of 4.17 and 4.18 we get
°
4.20. Proposition. To every decreasing function qJ: [0, 1] exists a unique Radon measure J1 on f/ such that qJ( s)
= J1( {I
E
f/ Is
E
I}),
S E
-+
[0, oo[ there
[0, 1].
4.21. Exercise. Let (N, .) be the multiplicative semigroup of natural numbers. Show that the dual semigroup and the restricted dual semigroup are isomorphic to IR lPl and [-1, l]lPl where lP = {2, 3, 5, ...} denotes the set of primes.
122
4. Main Results on Positive and Negative Definite Functions on Semigroups
4.22. Exercise. Let qJ E g>b(N o) have the representing measure J.1 on [ - 1, IJ. Show that L~ I qJ(n) 1 < 00 if and only if(1 - Itl)-l E 2 1(J.1). 4.23. Exercise. Let A = (au) be a real positive definite k x k matrix. Show that there exists a function qJ E g>b(Nt) such that for
i,j=I, ... ,k,
where e 1 = (1,0, ... ,0), ... , ek = (0, ... ,0,1). Hint: Assume laul ~ 1. There exists a real k x k matrix B = (b ij ) such that A = B*B. Let Xl' ... ' X k E IRk be the row-vectors of B, and define a measure /l E M +([ - 1, IJk) by putting mass 1 at each of the points X l' . . . , X k .
4.24. Exercise. Let qJ: [0, IJ -+ [0, 00 [ be decreasing and let J.1 be the representing measure on f/ according to Proposition 4.20. Show that qJ(a) - qJ(a
+ 0) =
°
~ a < 1,
/l( {CO, aJ}),
O
qJ(a - 0) - qJ(a) = J.1( {CO, a[}),
Show further that qJ(S) = qJ(l)
+
L J.1({[0, aJ})I[O,a](s) + I aeA
J.1({[0, b[})I[o,b[(S)
beB
+ J.1c( {I E f/ IS E I}), where A is the at most countable set of a E [0, 1[ such that J.1( {CO, aJ}) > 0, B is the at most countable set of b E JO, IJ such that p( {CO, b[}) > and J.1c is the continuous part of p. The function s ........ /lc( {I E f/ IS E I}) is continuous and decreasing.
°
4.25. Exercise. Let X be a nonempty set. The set IP(X) of subsets of X is an idempotent semigroup under intersection. Describe the semicharacters of IP(X) explicitly in the case where X has two elements, and show that qJ = 1 - 1{0} is decreasing but not positive definite. 4.26. Exercise. On IR + we define X 0 Y = X + Y + xy. Show that (IR +, 0) is an abelian semigroup and that x ........ log(1 + x) is an isomorphism of (IR+ , 0) onto (IR+, +). Show that the bounded semicharacters on (IR+ ,0) are given by x ........
(_1 )t + 1
x
for
°
~ t ~
00.
4.27. Exercise. An infinite matrix (a jk ) of the form ajk = qJ(j + k) with qJ: No -+ IR is called a Hankel matrix. It is positive (resp. negative) definite if and only if qJ is positive (resp. negative) definite on the semigroup (No, +), and it is infinitely divisible if and only if qJt is positive definite for all t > 0. Show that the so-called Hilbert matrix (1/(j + k + 1))j,k~O is infinitely divisible. -
123
§5. r-Positive Functions
4.28. Exercise. Let t/J: IR + -+ IR + be given. Show that if qJ 0 t/J E g>b(1R +) for all qJ E g>b(1R +).
4.30. Exercise. For M equivalent:
£;
t/J E LN(IR +) if and only
M~ (IR~) the following three conditions are
(i) M is relatively compact with respect to the weak topology. (ii) {fe III Il E M} is uniformly equicontinuous on IR~ . (iii) {fellill E M} is equicontinuous in O.
Hint: Use Prohorov's compactness criterion mentioned in the Notes and Remarks to Chapter 2. 4.31. Exercise. (Continuity Theorem for Laplace Transforms). Let a sequence Ill' Ill'··· E M~(IR~) be given such that qJ(t) = lim n_ oo (feJ1n)(t) exists for all t E IR~ and assume that qJ is continuous at O. Then qJ = feJ1 for some Il E M ~ (lR k+) and Iln -+ Il weakly. 4.32. Exercise. If M ~ (IR~) is equipped with the weak topology and g>~ (IR~) n C(IR~) with the topology of uniform convergence on compact subsets of IR~, the Laplace transformation fe: M ~ (IR~) -+ g>~ (IR~) n C(IR~) is a homeomorphism.
§5. r-Positive Functions In this section, we shall present an approach due to Maserick (1977) to the integral representation of positive definite functions. Let A be a real or complex commutative algebra with identity e and involution *. If A is a real algebra we always assume a* = a for all a E A. The basic idea is to find an integral representation of certain "positive" linear functionals on A, the representing measure being concentrated on a set of multiplicative linear functionals. The connection to semigroups is obtained if A is chosen to be the algebra of shift operators generated by the shifts E s' s E S, where (Esf)(t) = f(s + t) forf E C S . By the relation L(Es) = f(s) functionsfon S are in one-to-one correspondence with linear functionals Lon A. The positivity concept for linear functionals depends on the notion of an admissible subset r of A. 5.1. Definition. A nonempty subset r
(i) a* = a for all a E r; (ii) e - a E alg span+(r) for all a E r; (iii) alg span(r) = A.
£;
A is called admissible if
124
4, Main Results on Positive and Negative Definite Functions on Semigroups
Here alg span(r) (resp. alg span +(r)) is the set of linear combinations ~ 0) and each Xi is a finite product of elements from r. Note that alg span +(r) is the smallest convex cone in A, stable under multiplication and containing r. Furthermore e E alg span+(r) because of (ii).
I?= 1 AiXi, where each Ai is a scalar (resp.
5.2. Lemma. Let r be an admissible subset of A. Then: (i) e - Xl ... Xn E alg span +(r)for Xl' ... ' Xn E r; (ii) for X E alg span +(r) there exists 8(X) > 0 such that e for 8 E [0,8(X)]. PROOF.
8X E
alg span +(r)
The assertion (i) follows from the algebraic identity
"n
n
n
n
e-
=
Xj
(1'( XjJ
~
e-
(1)
X j )1-(1'J,
(1$1 j=l
j=l
where the summation is taken over all a = (a l ' ... , an) E {O, 1}n\ {(I, ... , I)} (note that XO := e), and this identity follows in turn by the formula n
e=
n(x
j
+ (e
-
Xj)).
j= 1
If X = LJ= 1 Ai Yj' where A j > 0 and Yj is a product of elements from r, we put 8(X) = (IJ= 1 Aj ) - 1. For 8 E [0, 8(X)], we find
e-
8X
=
(1 -
(8 8 X
))e + e .± Aie- Yj)
E
alg span+(r)
J= 1
D
because of (i). A linear functional L: A
L(a)
~ 0
-+
C is called r-positive, where r is admissible, if
for all
a E alg span +(r).
This holds if and only if
L(a 1
•••
an)
~
0
{a 1 ,
for all finite sets
••• ,
an}
~
r.
The set Ai of r-positive linear functionals on A is a convex cone in the algebraic dual space A*, closed in the topology a(A*, A). Note that L(x*) = L(x) for X E A when L E Ai. By Lemma 5.2 it follows that L(e) > 0 for L E Ai\ {O}, so
B:= {L is a base for
E
AiIL(e) = I}
Ai.
5.3. Lemma. The base B is compact. PROOF.
For
X E
A there exists a constant K x > 0 such that
for
L
E
Ai.
~5.
125
r-Positive Functions
In fact, if x E alg span +(r) we can use K x = 8(X)-l with 8(X) from 5.2(ii), and everyxEAcanbewrittenx = Xl - X 2 + i(x 3 - x 4 )withx j Ealgspan+(r), j = 1, ... ,4. If (L(X) is a universal net on B the above inequality shows that L(x) := lim(X L(X(x) exists for every x E A, and this implies the compactness
D
ofB.
Let ~ denote the set of r-positive, multiplicative linear functionals on A, which are not identically zero. Clearly ~ is a compact subset of B.
5.4. Theorem. A linear functional L on A is r-positive ~{and only ({there exists a (necessarily unique) Radon measure J.1 E M +(~) such that L(x} =
i
l5(x} dJ1(l5}
for
The base Bfar Ai is a Bauer simplex with ex(B) =
x
A.
E
(2)
~.
A functional of the form (2) is clearly r-positive. Given L E Ai, the existence of J1 such that (2) holds follows from Theorem 2.5.6, once we have shown that ex(B) £; ~. Suppose L E ex(B) and define for a E A the translate La of L by La(x) = L(ax). For a E alg span+(r) we have La E Ai. Since L = La + L e - a and La' L e - a E Ai for a E r, La is proportional to L. The set of elements a E A for which La is proportional to L is a subalgebra of A containing r, hence La = AaL for all a E A. Evaluating at x = e shows Aa = L(a), hence L E ~. Defining y: A ---+ A by y(x) = xx* an application of Corollary 2.5.12 yields ex(B) = ~. The set of functions £5 ~ £5(x), x E A, is a point-separating *-subalgebra of C(~, C) containing the constant functions, hence uniformly dense in C(Ll, C) by the Stone-Weierstrass theorem. This implies that the representing measure is uniquely determined. D PROOF.
From the integral representation in Theorem 5.4 we immediately get:
5.5. Corollary. Any functional L for all x E A.
E
Ai is positive in the sense that L(xx*)
~
0
It is interesting to note that the corollary can be proved directly without use of the integral representation. This was done by Maserick and Szafraniec (1984) as follows: For a function .1': [0, 1] ---+ C and x E A we define
Bn(x;f) = kt/(~)(;)xk(e - x)"-k, and for f'(t) = 1, t, t 2 we find (cf. Davis (1963, p. 109))
Bn(x; 1) = e,
Bn(x; t) = x,
1 Bn(x; t 2 ) = - X n
n- 1
+ - - x2• n
126
4. Main Results on Positive and Negative Definite Functions on Semigroups
Similarly, for a functionf: [0,1]2
-+
C and Xl' X2 E A we define
Bn(x x ;f) = j.~/(~' ~)(;)(:)x{(e - x1)n- jx~(e - x )n-k, 1,
and for f(s, t)
2
=
2
(s - t)2, we find
Bn(x l' x 2 ; (s - t)2) = Bn(x l' X 2 ; S2)
+ Bn(x l' X 2 ; -
= Bn(x l ; S2) - 2B n(x l ; s)B n(x 2 ; t)
= (Xl -
If Xl' X2 E A such that Xl' LE Ai
X 2)
X2'
2
+ -1 (Xl + X2 n
-
+ Bn(x l' X 2 ; t 2) + Bn(x 2 ; t 2)
2st)
2
Xl -
2
X 2 )·
e - Xl' e - X2 E alg span+(r), we have for
hence for n -+ 00: L«x l - X 2)2) ~ O. For Xl' X2 E alg span +(r) there exists 8 > 0 such that 8X l , 8X 2 , e - 8X l , e - 8X 2 E alg span +(r), cf. 5.2, and therefore L(8 2 (X l
-
X2)2) ~ 0,
so L«x l - X2)2) ~ O. An element X E A can be written X = Xl - x 2 + i(X3-X4) with xjEalgspan+(r), j=1, ... ,4, so xx*=(X l -X 2)2+ (x 3 - X 4 )2 and finally L(xx*) ~ o. Let S be a commutative semigroup with involution, and let
A
=
{.f
cjEsjln EN, cj
)=1
E
C, S} Sj
E
be the algebra of shift operators on C S . We recall that Esf(t) = f(s + t) for s, t E S, f E ([s. Clearly, A is a commutative algebra with unit I = Eo, and defining
we get an involution on A. Since (Es)ses is a basis for A, functionsf: S -+ C and linear functionals L: A -+ C are in one-to-one correspondence via the formula L(E s ) = f(s). Iffand L correspond to each other and n
L cjEsj E A
T =
j= 1
then n
L(T) =
L cjf(Sj) = j= 1
Tf(O).
127
§5. r-Positive Functions
It follows that n
L cjCkf(sj + st),
L(TT*) =
j,k= 1
so L is positive if and only iff E (!P(S). If r ~ A is admissible, we call f r-positive if the corresponding linear functional L is r-positive, i.e. if and only if
r.f(O)
~
for all
0
T
E
alg span + (r),
or if and only if
T1 ... T"f(O)
~
0
for all
T1 ,
•• • ,
T" E r.
Note that a semicharacter p is r-positive if and only if Tp(O) T E r, since T1 ... T"p(O) = T1P(O) ... T"p(O).
~
0 for all
5.6. Theorem. Let r be an admissible subset ofthe algebra A ofshift operators.
Every r-positive function f: S -+ C is positive definite, exponentially bounded and has an integral representation f(s)
=
f.
p(s) dJ1(p),
S·
where J1 E M +(S*) is concentrated on the compact set of r-positive semicharacters. PROOF. By Theorem 5.4 the linear functional L corresponding to the rpositive functionfhas a representation
L(T)
=
i
(j(T) dj1«(j),
TEA,
where fi E M +(~). For b E ~ the function s H b(E s ) is a r-positive semicharacter, and the mapping j: ~ -+ s* given by j(b)(s) = b(E s ) is a homeomorphism of ~ onto the compact set j(~) of r-positive semicharacters. The image measure J1 := fij of fi under j is a Radon measure on S* with compact support contained in j(~), and replacing T by Es we get
f(s) =
f.
p(s) dJ1(p),
s
E
S.
S·
This shows thatfis exponentially bounded and positive definite.
D
5.7. A subset G ~ S is called a generator set for S, if every element in S\ {O} is a finite sum of elements from G u {a* Ia E G}. Let ~: S -+ ~+ be an absolute value such that ~(a) > 0 for all a E S. For (J E C, a E S, we define
l(
a) n... a = "2 I + 21X(a) Ea + 21X(a*) Ea· · (J
128
4. Main Results on Positive and Negative Definite Functions on Semigroups
The family
=
rG
{Qa,alaE
{±1, ±i},aEG}
is easily seen to be admissible if G is a generator set. The following converse of Theorem 5.6 holds:
Iff is positive definite and exponentially bounded there exists an admissible r such that f is r-positive. In fact, there exists an absolute value tJ, such that f is tJ,-bounded, and we may assume tJ,(a) > 0 for all a E S. Then
f(s) =
f
pes) djJ.(p),
s E S,
where J.1 E M + (S*) is supported by the compact set of tJ,-bounded semicharacterse With r = rG as above, where G is a generator set, we get
na,af(s) =
f
p(s)t[l
+ lX(a)-l
Re(O"p(a))] djJ.(p),
and since Ipea) I ~ tJ,(a) for p E supp(J.1) it follows that Qa, a f E (!lJCX(S). Hence (!lJCX(S) for a j E {± 1, ±i}, aj E G, in particular,
Qa1,a1 ... Qan,anf E
Qa1,a1 ... Qan,anf(O) ~
showing thatfis rG-positive. Using the absolute value tJ,
==
1 we get" (i)
=>
0 (ii)" of the following result:
5.8. Proposition. F or a function f: S -+- C the following conditions are equivalent: (i) f E (!lJb(S); (ii) f is r-positive, where r
= {Q a , a I a E {± 1, ± i}, a E S} and Qa, a
=
1(I + 2a E + 2(j) E
2"
a
a", .
PROOF. To see" (ii) => (i)" it suffices by Theorem 5.6 to verify that a r-positive semicharacter is bounded. But p E S* is r-positive if and only if 1 + Re(ap(s)) ~ 0 for s E S, a E {± 1, ± i}, i.e. if and only if Re pes), 1m pes) E [ -1, 1] for all s E S, which is equivalent with pES. D
5.9. Remark. In the above proposition it is not possible to replace r by rG
=
{Qa,ala E
{± 1, ±i}, a E G},
where G is a generator set for S. The implication" (i) => (ii)" is, of course, still true, but" (ii) => (i)" might fail to hold. In fact, if S = (N6, +, *) is the semigroup studied in 4.11, then G = {(I, O)} is a generator set and the
129
§6. Completely Monotone and Alternating Functions
rG-positive functions are given by f(n, m)
=
r
znzm dJl(z),
(n, m)
E
)[-1,1]2
N~,
which is a larger class than q;b(S). In the case where the involution is the identical this remark does not apply:
5.10. Proposition. Assume S has the identical involution and let G be a generator set for S. Then the following conditions are equivalent: (i) f E q;b(S); (ii) (1 ± Eat) ... (1
± Ean)f(O)
~ 0
for all aI' ... , an E G, n E N.
It is clear that" (i) => (ii)" holds, and (ii) is equivalent with f being r-positive, where
PROOF.
r = {tel
± Ea)/a E G}
is admissible. Finally every r-positive semicharacter p is bounded, because Ip(a)1 ~ 1 for a E G implies Ip(s) I ~ 1 for all s E S. Note that Q± 1, a = ± Ea) and Q± i, a = tl, and the families
tel
{t(l±Ea)laEG}
and
{Qa,alaE{±l,±i},aEG}
D
lead to the same r-positive functions.
5.11. Exercise. Let S = (N~, +) and let A(k) be the algebra of real polynomials in k variables Xl' •.• , X k . Show that the set r =
{Xl' ... ,
xk , 1 -
Xl -
... -
xk}
is admissible. There is a one-to-one correspondence between functions
f: N~ ~ ~ and linear functionals L: A(k) ~ ~ established via L(x n ) = fen), n E N~. Show that f is r-positive (in the sense that the corresponding linear functional L is r-positive) if and only if there exists J.1 f(n)
=
L
n
x dJl(x),
E
M +(K) such that
n E N~,
§6. Completely Monotone and Alternating Functions In this section, S is an abelian semigroup with the identical involution. We have already introduced the shift operator Ea: ~s -+ ~s defined by Eaf(s) = f(s + a) and ~a = Ea - 1, where 1 denoted the identity operator. Since {E a Ia E S} is a commuting family of operators, the algebra generated by this
130
4. Main Results on Positive and Negative Definite Functions on Semigroups
family is commutative, too; in particular, we have for
a, b E S.
Sometimes it is convenient to use also the operators
Va:=I-Ea=-L\a'
aES,
which are, of course, again commuting; note that Va Vb
n [1 -
= L\aL\b' and that
n
Val···
VanP(S) = p(S)
(1)
p(a i )]
i= t
for p E S*. In his fundamental work on capacities Choquet (1954) studied functions on S which he called monotone (resp. alternating) 0.[ infinite order. Here we shall call these functions completely monotone (resp. completely alternating).
6.1. Definition. A function qJ: S -+- ~ is called completely monotone if it is nonnegative and if for all finite sets {at, ... , an} ~ Sand S E S Val ... Van qJ(S) ~ O. A function and S E S
t/J: S
-+- ~
is called completely alternating if for all {at, ... , an}
~
S
The set of completely monotone (resp. alternating) functions is denoted .A(S) (resp. d(S)). It is clear that .A(S) and sl(S) are closed convex cones in ~s, and the nonnegative (resp. real) constant functions are contained in .A(S) (resp. scl(S)). If qJ E .A(S) then EaqJ E .A(S) and, similarly, if t/J E scl(S) then Eat/J E d(S) for all a E S, since Val ... Van(Eaf) = E aV a1 ... Vanfforf E ~s and since E a is a positive operator. Notice that qJ E Jt(S) and t/J E d(S) satisfy 0 ~ qJ ~ qJ(O) and t/J ~ t/J(O).
6.2. Remark. The following terminology is frequently used for s, at, ... , an E S: Vtf(s; at) = f(s) - f(s and inductively for n
~
+ at),
2
Vn.[(s; at, ... , an) = V n-1.[(s; at, ... , an-t) - Vn-l.[(s
+ an; at,···, an-t).
The following result connects the two concepts of completely monotone and alternating functions, and it is easily established.
131
§6. Completely Monotone and Alternating Functions
6.3. Lemma. A function t/!: S -+- ~ belongs to deS) a E S the function ~a t/! belongs to vII(S). PROOF.
1ft/!
E
deS) then L\at/! = -Vat/!
if and only if for every
~ 0 and
Val··· Van(L\at/!) = ~a Val··· Vant/! = - Va Val··· Vant/! ~ O. On the other hand, if ~a t/!
E
vII(S) for all a E S, then
Val··· Vant/! = -Val··· Van_I(L\ant/!) ~ O.
D
The relation of completely monotone functions to r-positive functions is described in the following:
6.4. Theorem. Let G ~ S be a generator set. The family r = {E a , 1 - Ea Ia E G} is admissible, and for a function qJ: S -+- ~ the following conditions are equivalent: (i) qJ is completely monotone. (ii) qJ is r-positive. (iii) There exists a measure J.1 E M +(S +) such that cp{s)
=
f.
pes) dJl{p),
Js
s
E
S.
+
(I - Ea) ... (1 - Ean)E SI ... Esrn qJ(O) = Val ... Van qJ(S 1
+ ... + Sm)
so (i) => (ii). The implication "(ii) => (iii)" follows from Theorem 5.6 since P E s* is r-positive ifand only if 0 ~ pea) ~ 1 for a E G, but this is equivalent with pES + because every s E S\ {O} is a finite sum of elements from G. Finally, if (iii) holds, then qJ(s) ~ 0 and by (1) Val'"
so qJ
qJ
E
vII(S).
Vancp{s)
=
f.
Js
p{s).n (1 - pea)) dJl{p) +
J= 1
~ 0, D
A function qJ E vII(S) is bounded by qJ(O). The set vIIl(S) of functions JI(S) such that qJ(O) = 1 is therefore a compact convex base for the
E
cone vII(S).
6.5. Theorem. The cone vII(S) is an extreme subset of g>b(S) and Jll(S) is a Bauer simplex with ex(vIIl(S)) = S+ . For qJl' qJ2 E vII(S) also qJl . qJ2 E vII(S). A function qJ E g>b(S) is completely monotone if and only if the representing measure J.1 is concentrated on S+ . The previous theorem shows that A(S) ~ g>b(S). If qJ = qJl + JI(S) and qJi E g>b(S) has representing measure J.1i E M +(S), i = 1, 2, then J.1 = J.1l + J.12' where J.1 E M +(S +) is the representing measure for qJ. It PROOF.
qJ2
E
132
4. Main Results on Positive and Negative Definite Functions on Semigroups
follows that J.11' J.12 are concentrated on S+ so qJl' qJ2 E ..4t(S), and we have shown that vII(S) is an extreme subset of g>b(S). By transitivity of extremality an extreme point of ..4t 1(S) is also an extreme point of g>~ (S), hence ex(vII I(S)) ~ S +, and in fact there is equality since S+ ~vlll(S)nex(g>~(S)). For J.1, vEM+(S+) we have supp(J.1*v)~S+, cf. 2.3, and it follows that ..4t(S) is stable under multiplication. By unicity of the representing measure for qJ E g>b(S) it follows that vII 1(S) is a simplex, and that the representing measure for qJ is concentrated on S + if qJ E g>b(S) is completely monotone. D In analogy with Theorem 6.4 it suffices to check the defining conditions for a completely alternating function for elements in a generator set.
6.6. Proposition. Let G be a generator set for S. A function t/J: S -+pletely alternating if and only if Vat ...
Vant/J(s)~O
for
SES
and
a 1, ... ,an EG,
~
is com-
n~1.
PROOF. Suppose the conditions of the proposition hold. For s E S\ {O} there exist aI' ... , an E G with s = a 1 + ... + an, and the identity (1) in the proof of Lemma 5.2 gives in this case
I - Es =
n
n
L
E~f(I
(1"*1 j==1
- Eaj )I-(1j.
Applying the above procedure to elements s l '
(I - ESt) ... (I - ESk)t/J = VSt hence
... , Sk •••
E S\ {O} we find
VSkt/J ~ 0,
D
t/J E deS).
6.7. Theorem. The cone .s:I(S) is an extreme subset of JVI(S). A function t/J E JV'(S) is completely alternating if and only if the Levy measure J.1 is concentrated on S+ \ {I}. PROOF. If t/J E deS) we have by 6.3 and 6.5 that L\a t/J E g>b(S) for a E S, hence t/J E JVI(S) by Theorem 3.20. If t/J E deS) and t/J = t/Jl + t/J2' where t/Jl' t/J2 E JV'(S), then L\a t/J = L\a t/J 1 + L\a t/J2 E ..4t(S) and L\a t/J l' L\a t/J2 E g>b(S) for a E S. As vII(S) is extreme in g>b(S) we get L\a t/J l' L\at/J2 E vII(S), and by Lemma 6.3 we conclude that t/Jl' t/J2 E deS), and have thereby shown that
deS) is extreme in JV'(S). Let t/J E JV'(S) have the Levy-Khinchin representation t/J(s}
= t/J(O} + q(s} +
f.
JS
(1 - p(s)} dJl.(p}. \{I}
It is easy to see that t/J(O), q E .s:I(S) and that 1 - P E deS) when pES +\ {I}, so if J.1 is concentrated on S +\ {I} then t/J E deS). Let us next suppose that t/J E deS). Since deS) is extreme in JV'(S) we conclude that
t/Jis}:=
i
Ta
(1 - p(s)} dJl.(p}
~6.
133
Completely Monotone and Alternating Functions
belongs to .s:#(S), too, where Ta = {p ~at/Ja E Jt(S), hence
o ~ dat/Jia) =
i
E
SIpea) <
O}. By Lemma 6.3 we have
p(a)(1 - p(a)) djJ.(p).
Ta
But the integrand is strictly negative on Ta and therefore J.1(Ta) = O. Using that 1;, is open for every a E Sand UaeS 1;, = S\S + we get ,u(S\S +) = O. D
6.8. Corollary. Suppose that S is 2-divisible, i.e. every s E S is of the form 2t for some t E S. Then vII(S) = g>b(S) and deS) = JVl(S). PROOF. If Sis 2-divisible every semicharacter is nonnegative, so the assertions follow from 6.5 and 6.7. D
6.9. Remarks. (1) It can happen that vII(S) = g>b(S) without S being 2divisible. Let S = {O, a, b} be the commutative semigroup with neutral element 0 and a + a = a + b = b + b = a. Then S is not 2-divisible, but S* = S has two elements Po == 1, Pl(O) = 1, Plea) = Pl(b) = 0 so vII(S) = q>b(S) = g>(S). (2) Suppose G is an abelian group considered as a semigroup with the identical involution. Then G is the group of homomorphisms of G into the multiplicative group { - 1, I}. In this case vII(G) = g>b(G) if and only if G is 2-divisible. In fact, if G is not 2-divisible then G/G 2 has at least two elements, where G2 = {2glg E G}, and every element s E G/G 2 satisfies 2s = o. If s E G/G 2 , S t 0, then b(G) t vII(G). 0
By Corollary 6.8 and Theorem 3.2.2 it follows that on a 2-divisible semigroup S a function t/J: S ~ ~ is completely alternating if and only if exp( - tt/J) is completely monotone for all t > O. The next result shows that we can waive the assumption of 2-divisibility. 6.10. Proposition. Let t/J: S A(S)jor all t > O.
~ ~.
t/J~E
Then
deS)
if and
only
if exp( -tt/J) E
/
PROOF. If exp( - tt/J) E Jt(S) for all t > 0 then 1 - exp( - tt/J) E deS) so t/J = lim t -+ o t- 1(1 - exp( - tt/J)) E deS). For the converse it suffices to prove that exp( - t/J) E vII(S) for t/J E deS) with the representation
t/J(s) = t/J(O)
+ q(s) + f. JS+\{l}
(1 - p(s))djJ.(p).
134
4. Main Results on Positive and Negative Definite Functions on Semigroups
Since vII(S) is closed under multiplication and exp( - q) prove that SH
ex p [-
r.
E
S+ ,
it suffices to
(1 - p(s)) dj1(P)]
JS+\{l}
belongs to vII(S). Approximating J.1 by measures with compact support (2.3.9) and approximating these by molecular measures (2.3.5) the problem is reduced to the implication
°
S+,
C
~
This is in fact true because p
E
Jt(S) and
p E
exp(cp)
=>
vII(S).
1
00
exp(cp)
E
L ---, cnpn.
=
n=O
D
n.
For the case S = N~ we have that S+ is homeomorphic with [0, 1]k via the mapping x H Px' px(n) = x n = X~l ... X~k, where x = (Xl' ... ' X k ) E [0, 1]k, n = (n l , , nk) E N~, cf. 4.8. The set G = {e l , .•. , ek}' where ej = (0, ... , 0, 1, 0, , 0) with 1 on the jth place, is a generator set for N~. We put E j = Eej,j = 1, ... , k, and
Ip I =
(;) = (;:) ... (;:}
P1
+ ... + Pk
n, p E N~.
for
Specializing 6.4, 6.6, 6.7 and 4.10 we get the following two results, the first of which is due to Hildebrandt and Schoenberg (1933); in the one-dimensional case it is the solution of the so-called Hausdorff moment problem. 6.11. Proposition. For a function qJ: N~ equivalent:
(i) qJ is completely monotone. (ii) (1 - E 1)n 1 ••• (1 - Ek)nkqJ(m l ,
... ,
-+-
E
the following conditions are
mk)
o~~~n (_l)IPI(;)
~
for all n, m E
N~.
M + ([0, 1]k) such that
=
r
x n dj1(x),
J[O, llk
6.12. Proposition. For a function equivalent:
(i) l/J is completely alternating. (ii) (I - El)n 1 • • • (1 - Ek)nkl/J(m l ,
l/J:
••• ,
N~
-+-
nE
N~.
~ the following conditions are
mk)
o~~~n (-1)IPI(;)l/J(m + p) ;£ 0
for all m E
N~,
nE
N~\{O}.
135
§6. Completely Monotone and Alternating Functions
(iii) There exist a E ~,b tjJ(n}
E ~~
and J.1
= a +
E
M +([0, 1]k\{I}) such that
r
(l - x n ) dJl(x}.
J[O,l]k\{I}
In the case of the 2-divisible semigroup S = ~ + the continuous functions in J((~+) = &,b(~+) can be characterized in yet another way, cf. 4.5:
6.13. Theorem. For a continuous function cp: are equivalent: (i) (ii) (iii) (iv)
~+ -+ ~
the following conditions
cp E uIt(~ +). cp E &,b(~ +). There exists J.1 E M~(~+) such that cp = 2J.1. cp is Coo on ]0, oo[ and ( -1)ncp(n>(s) ~ 0 for n ~ 0, s >
o.
PROOF. We have already established the equivalence of (i), (ii) and (iii), and "(iii) => (iv)" follows from 4.4. Suppose (iv) holds. For a ~ 0 the function Vacp is again continuous on [0, oo[ and satisfies (iv). In fact, for n ~ 0, s > 0 the mean value theorem gives
(-1)n(V acpyn>(s) = (_1)n[cp(n>(s) - cp(n>(s + a)] = ( _1)n+ lacp(n+ l)(~) ~ 0 for ~ between s and a + s. By iteration we find that f := Vat ... Van cp satisfies (iv) whenever at, ... , an ~ 0, in particular,f(s) ~ 0 for s > 0, and by continuity f(s) ~ 0 for s ~ 0, hence cp E J((~+). D The equivalence of (iii) and (iv) is a famous result of Bernstein, cf. Widder (1941), and can be given a slightly more general formulation:
6.14. Corollary. For a function cp: ]0, oo[ equivalent:
-+ ~
the following conditions are
(i) There exists J.1 E M +(~+) such that cp = 2J.1. (ii) cp is Coo and ( _1)ncp(n> ~ 0 for n ~ o.
If(i) and (ii) are satisfied lims -+ o cp(s) =
J.1(~+) ~ 00.
PROOF. It is easy to see that (i) => (ii). Suppose that (ii) holds and let h > o. The function EhCP is continuous on ~+ and satisfies (iv) of 6.13, hence of the form 2(J.1h) for a uniquely determined J.1h E M~(~+). For h, k > 0 we then have
L oo
cp(s
+ h + k} =
L oo
e-(s+k)x
dJlh(X}
=
e-
SX
so by uniqueness of the representing measure
e
-kx
J.1h = e
-hx
J.1k = J.1h + k.
dJlh+k(X},
s
~
0,
136
4. Main Results on Positive and Negative Definite Functions on Semigroups
Defining the measure J.1 on [0, oo[ by J.1 h > 0, we find
= ehxJ.1h' which is independent of
cp(s} = {"e- sx dJ.L(x}, hence lim s -+ o qJ(s)
=
s > 0,
J.1(~+).
D
The functions of the corollary are, in the literature, called completely monotone functions. In Chapter 8 we will see that they are the completely monotone functions on the semigroup without zero element (]O, 00[, +).
6.15. The abstract theory of capacities introduced by Choquet (1954) fits into the theory of completely monotone and alternating functions. Let X be a locally compact space and let % = %(X) denote the set of compact subsets of X. Then (%, u) is an idempotent semigroup with 0 as zero element and the induced ordering is inclusion. The set f/ of subsemigroups I ~ ff which are hereditary on the left, i.e. families I of compact sets verifying K, LEI => K u LEI
is a semigroup under intersection, and I %*, cf. 4.16, 4.17. Defining
K=
K
and
{IEf/IKEI}
~
~
L,
LEI => K
E
I,
I] is an isomorphism of f/ onto
for
K
E
%
and equipping f/ with the coarsest topology in which the sets K and f/\K are open (and hence clopen) for all K E ~ the mapping I ~ I] is a homeomorphism. A function qJ: % -+- ~ will be called continuous on the right if for every K E % and e > there is an open neighbourhood G of K such that IqJ(K) qJ(L) I ~ e for all L E r(K, G) = {L E %IK ~ L ~ G}. Note that the functions continuous pn the right can be considered as the functions on % continuous in the coa~sest topology for which the sets j/'(K, G) are open. This topology is not Hausdorff and 0 is an isolated point. We see below that the completely monotone and alternating functions on % which are continuous on the right admit an integral representation over the set of semicharacters on % which are continuous on the right. We first show that the set of semicharacters on ff which are continuous on the right can be identified with the set ~ = ~(X) of closed subsets of X. Let!/,. denote the set of I E f/ for which I] is continuous on the right.
°
6.16. Lemma. For F E ~ let IF = {K E fflK n F = 0}. Then FI-+ IF is a bijection of ~ onto!/,. and the inverse mapping is I 1-+ X\ UKe] K. PROOF. It is obvious that IF E f/ and easy to see that I] F is continuous on the right because a compact set K not intersecting F has an open neighbourhood
~6.
137
Completely Monotone and Alternating Functions
which does not intersect F. Conversely, if IE!/,. we have
UK = UK, Kef
Kef
for if K E I the continuity on the right implies that I contains a compact neighbourhood of K. Therefore F ~= X\UKef K is closed and I ~ IF. For L E IF we ~ave L ~ UKef K so there exist finitely many K 1 , ••• , K n E I with L ~ K 1 U ... u Kn ~ K 1 U ... u Kn , and I being stable under finite unions and hereditary on the left we see that LEI, hence I = IF. Finally, if F l' F 2 are different closed sets and x E F 1 \F 2 then {x} E I F2 \1 Ft. D We will now introduce a topology on ~ Let rg open subsets of X and define for a subset B ~ X ~B ~B
= {FE~IF n B = 0}, = {F E ~IF n B =i= 0} =
=
rg(X) denote the set of
~\~B.
We equip ~ with the coarsest topology in which the sets ~K, K E $' and ${;, G E rg are open. Since ~Kt n ~K2 = g;K t uK 2 we see that sets ~K n ~ Gt n ... n ~ G n with K E ~ G 1 , ••• , Gn E rg form a base for the topology on ff. It is easy to see that ~ is a Hausdorff space, and using Alexander's subbase theorem it can be proved that ~ is compact, cf. Matheron (1975). The compactness is, however, also a consequence of the following result.
6.17. Proposition. The mapping e: !/ --+
e(l)
=
~
defined by
U K,
X\
I
E
!/
Kef
is continuous and maps!/,. bijeetively onto ff. PROOF. By Lemma 6.16 we already know the last statement. For the continuity of e it suffices to prove that e-l(~K) and e- 1(${;) are open in !/ for K E .Y{ and G E rg. We first remark the following biimplication for I E !/ and K E $':
e(I) n K =
0
~ 3L
E
I: K ~
L.
(2)
Here "<=" is clear, and if e(I) n K = 0 there exist by compactness K 1 , ••• , K n E I such that K ~ K1 U ... u Kn and then K ~ L with L = K 1 U ... u K n E I. By (2) it follows immediately that e-l(~K)
UL
=
(3)
Le:K
KSL
which shows that e-l(~K) is open in !/. Similarly, we have the following biimplication for I e(I) n G =i=
0
~
3K
E
$'\1: K
~
E
!/ and G E rg:
G,
hence
e- 1(${;) =
U (!/\K),
(4)
Ke:K
KSG
showing that e - 1(${;) is open in !/.
D
138
4. Main Results on Positive and Negative Definite Functions on Semigroups
6.18. Theorem. For every qJ E A(:It, u) which is continuous on the right there exists a unique J1 E M + (ff) such that
for
K
E
:It,
andfor every J1 E M +(ff) this.formula defines afunction qJ is continuous on the right. PROOF. A decreasing function
E
A(Yi, u) which
qJ: .% -+ IR is continuous on the right if and
only if
qJ(K) = sup{qJ(L)IL E Yi, K ~ L}
(5)
for all K E $: Suppose first that qJ: .% -+ IR is completely monotone and continuous on the right, in particular, decreasing. By Proposition 4.17, there exists v E M +(Y) such that
qJ(K) = v({I
E
YIK
for K E $: Using that the family L, L by (3) and Theorem 2.1.5 that
E
E
I}) = vCR)
:It, K ~
sup{v(L)ILEYi,K ~ L}
L is upwards filtering we get
v(c- 1(ffK»,
=
hence qJ(K) = J1(ffK) where J1 = VC is the image measure of v under the continuous mapping c: Y -+ ~ Conversely, if J1 E M +(ff: then qJ(K) = J1(ffK) is a positive decreasing function satisfying (5) because the family ffL, L E Yi, K ~ L is upwards filtering with union ffK. To see that qJ is completely monotone we use the following formula which is easily established by induction:
V K1 V K2 ··· VKnqJ(K) = J1(ffK n ~l n ... n ~n) for K, K 1, . .. ,K n E $: By Theorem 2.1.5 we have for K
E
Yi, G 1, . .. , Gn E
(§
J1(ffK n ~l n ... n ff G n )
= sUp{J1(ffK n = sUp{J1(~K n
~l n
n ~n)IKi
~l n
n ~n)IKi
E E
Yi, K i ~ Gi , i = 1, Yi, K i ~ Gi , i = 1,
, n} , n}.
If J11' J12 E M +(ff) agree on the sets ffK, K E Yi, the above formulas show that they agree on all finite intersections of the sets ~K, K E .% and ~, G E rg, hence on the algebra .91 of sets they generate. Since any open set in ff is union of an upwards filtering family of open sets from .91, another application of 2.1.5 shows that J11 and J12 agree on the open sets in ff, hence
J11=J12.
D
In order to derive the corresponding integral representation for the set of completely alternating functions which are continuous on the right, we first notice that c(I) = 0 for lEY if and only if I = .x. Therefore, c induces a continuous mapping of the locally compact space Y\ {.%} onto the locally
139
§6. Completely Monotone and Alternating Functions
compact space ~\ {0} and this mapping, still denoted c, is proper. It follows by Proposition 2.1.15 that every v E M + (!/\ {X}) has a Radon image measure J1 = VC and every J1 E M + (ff\ {0}) is of the form VC for some v E M +(!/\{X}), cf. 2.2.13. A completely alternating function ljJ: X -+ [R which is continuous on the right and satisfies ljJ(0) = 0 is called a (completely alternating) capacity on x. By exactly the same method as in the preceding theorem it is possible to get the following result which contains Choquet's integral representation of the completely alternating capacities on X (Choquet 1954, Chap. VII). The details are left to the reader.
6.19. Theorem. For every ljJ E sd(Yi, u) which is continuous on the right there is a unique J1 E M + (ff\ {0}) such that ljJ(K) = ljJ(0)
+ J1(ffx),
K
E
Yi,
andfor every J1 E M +(ff\{0}) this formula defines a function ljJ which is continuous on the right.
E
d(Y(, u)
= Va.! . Eag + f . Vag for f, g E this for another proof that A(S) is stable under multiplication.
6.20. Exercise. Show that Va(fg)
[Rs.
Use
E f!l>~(S) and assume CP2(S) > 0 for all s. Show that if CPl CP2 E A(S) then CPl E vII(S). Show that if cP E f!l>~(S) is such that cpk E Jt(S) for some kEN then cP E A(S).
6.21. Exercise. Let CPl' CP2
6.22. Exercise. Assume cP E f!l>b(S), ljJ E ..;VI(S). Show that cP is completely monotone if and only if n 1---+ cp(na) is completely monotone on No for each a E S, and that ljJ is completely alternating if and only if n 1---+ ljJ(na) is completely alternating on No for each a E S. 6.23. Exercise. Given cP E f!l>b(N o) define Tcp: No -+ [R by Tcp(O) = 0, Tcp(n) := Ii: 6cp(j) for n ~ 1. Then T is an affine isomorphism from f!l>b(N o) onto {t/J
E ~(No)It/J(O)
= O}, mapping ..b(N o) onto {t/J
6.24. Exercise. Show that cp(n)
=
_1__ log n n+ 1 n
+2= +1
II
Xn
(1 + 1 -
E
.sd(No)lljJ(O) = O}.
x) dx,
n
log x
0
is completely monotone on (No, +), and that
ljJ(n) (ljJ(O)
:=
= 1
1
1
+ 2 + ... + ~ -
log(n
+
1),
n
~ 0
0) is completely alternating with the Levy measure J1
=
(-1--1-t + og_1_)1]0 1[(t) dt, t ' -1
~0
140
4. Main Results on Positive and Negative Definite Functions on Semigroups
which is finite. It foHows that y := lim n _
1
oo
ljJ(n) exists and that
1
y=
o
(-1_1- t
+ -1_1_) og t
dt <
00.
The number y is called Euler's constant.
6.25. Exercise. Let
qJ E
f?}Jb(S). Show that the function qJ(2s) belongs to
u1t(S).
6.26. Exercise. Try to find a decreasing nonnegative positive definite function on No which is not completely monotone. 6.27. Exercise. Show that Theorem 6.13 can be extended to k dimensions: For a continuous function alent:
(i) (ii) (iii) (iv)
qJ: IR~ -+
IR the following conditions are equiv-
qJ E u1t(IR~); qJ E f?}Jb(IR~); qJ
= 2/1 for some Ji E
qJ
is
COO
on ]0,
OO[k
M~(IR~);
and (-I)I~ID~qJ(s) ~ 0 for
r:J. E
N~, s E ]0,
oo[k.
Extend Corollary 6.14 to k dimensions.
6.28. Exercise. Let qJ: ]0, oo[ -+ IR be the Laplace transform of Ji E M +(IR+). Then for each x E ]0, oo[ the sequence n ~ qJ(n)(x) belongs to &P(N o). 6.29. Exercise. A function qJ: S -+ IR is called absolutely monotone if qJ ~ 0 and L\al ... L\a n qJ ~ 0 for all a l' ... , an E S; and ljJ: S -+ IR is by definition absolutely decreasing if L\al ... L\a n ljJ ~ 0 for all a l' ... , an E S. (a) Show that both classes of functions are closed convex cones in IRs and that the absolutely monotone functions are, furthermore, stable under multiplication. (b) A function ljJ is absolutely decreasing if and only if Va ljJ is absolutely monotone for each a E S. (c) Show that ljJ is absolutely decreasing if and only if exp( - tljJ) is absolutely monotone for each t > o. (d) A semicharacter P E S* is absolutely monotone if and only if p ~ 1. (e) If r:J.: S -+ IR+ is additive and kENo then r:J.k is absolutely monotone.
6.30. Exercise. Let ljJ: S -+ IR be a lower bounded negative definite function and let (Jit)t ~ 0 be the corresponding convolution semigroup on S, cf. Theorem 3.7. Show that ljJ is completely alternating if and only if supp(Jit) £; S+ for all t ~ o. This result should be compared with Theorem 6.7. For a generalization of these results see Berg (1984).
Notes and Remarks
141
Notes and Remarks The algebraic theory of semigroups is treated in Clifford and Preston (1961) and Redei (1965). Concerning topological semigroups see Paalman-de Miranda (1964) and Hofmann and Mostert (1966). The concept of a semigroup with involution is general enough to include abelian semigroups and groups. It can be found in the appendix by Sz.-Nagy (1960) to the famous functional analysis monograph by Riesz and Sz.-Nagy. The concept is however so simple, that it might have appeared elsewhere long ago. Positive definite functions on groups form a very important subject within harmonic analysis, and it is discussed in every monograph on the subject; see Stewart (1976) for an excellent survey of this topic. Operator valued functions of positive type occur in Sz.-Nagy (1960), and have been frequently studied since, see, e.g. Mlak (1978). It should be noticed that the negative definite functions t/J in the group sense form a slightly more general class than those usually considered, cf., e.g. Berg and Forst (1975), where in addition it is required that t/J(O) ~ O. The notion of exponentially bounded functions on a semigrGup was introduced in Berg and Maserick (1984), but somewhat similar conditions are implicit in Szafraniec (1977). Gelfand's theory of commutative Banach algebras was applied in connection with commutative semigroups by Hewitt and Zuckermann (1956). Theorem 2.8 was proved by Lindahl and Maserick (1971) and rediscovered by Berg et al. (1976) for semigroups with the identical involution. A special case was treated by Ressel (1974). The generalization of 2.8 given in 2.5-2.7 is due to Berg and Maserick (1984). Theorem 2.5 can also be proved by operator theory since the operators n(s), s E S from Theorem 1.14 generate a commutative C*-algebra for which the spectral theorem can be applied, cf. Rudin (1973, Theorem 12.22). In this connection see also Schempp (1977). The continuous negative definite functions on a locally compact abelian group G are in one-to-one correspondence with the convolution semigroups on the dual group G, cf. Berg and Forst (1975), which in turn are in one-toone correspondence with the G-valued stochastic processes with stationary and independent increments. The integral representation of the cone of negative definite functions is known in the literature as the Levy-Khinchin formula, which was established for G = [R in the late 1930's by Levy and Khinchin. The theory of convolution semigroups has been developed for arbitrary locally compact groups in the monograph by Heyer (1977). Our most general Levy-Khinchin formula is 3.19. The special case in 3.20, where the involution is identical, is due to Berg et ale (1976), and the special case of 3.19 (where the negative definite function is real-valued) is due to Maserick (1978). Bernstein functions as defined after 4.3 occur at many places of analysis but often without the label Bernstein function. In analogy with Theorem 6.13 a continuous function./': [R + -+ [R + is a Bernstein function if and only if E COO(JO, ro[) and (-I)"j<"+ 1) ~ 0 for n = 0,1, .... Bochner (1955) proved
.r
142
4. Main Results on Positive and Negative Definite Functions on Semigroups
that Bernstein functions.r are characterized by the property that cp 0 f is completely monotone for all (continuous) completely monotone functions cp on [R + and used the name completely monotone mappings instead of Bernstein functions. Bochner's result is generalized in Exercise 4.28. Let us mention some places where Bernstein functions appear: In probability theory because a probability measure J1 on [R+ is infinitely divisible if and only if -log!£J1 is a Bernstein function. In potential theory because a positive measure Kon [R + is a potential kernel if and only if I/!£K is a Bernstein function, cf. Berg and Forst (1975, 1979). Similarly, a bounded continuous function p: [R + -+ [R + is proportional to a standard p-function in the theory of regenerative phenomena (cf. Kingman (1972)) if and only if I/!£p is a Bernstein function. Micchelli and Willoughby (1979) have characterized the Bernstein functions as those functions which operate on the class of Stieltjes matrices. This has, in turn, interesting applications in the theory of orthogonal polynomials, cf. Micchelli (1978). An important class of Bernstein functions is the reciprocals of nonzero Stieltjes functions cp. A function cp: JO, 00 [ --+ [0, 00 [ is a Stieltjes function if it has the form
() + 1 -+- ,
cp s = a
°
00
dJ.l(x)
o s
x
s > 0,
°
where a ~ and J1 E M +([R+). The functions cp(s) = S-fl, < rx ~ 1 and 1/10g(1 + s) are Stieltjes functions, cf. 3.2.10. A survey of the proporties of Stieltjes functions can be found, e.g. in Berg (1980). The results of Exercises 4.30 and 4.31 have some" infinite dimensional" analogues, see Hoffmann-J¢>rgensen and Ressel (1977). For further applications of the theory of r-positive functions we refer to Maserick (1977), see also Tonev (1979) and Berg and Maserick (1984). The theory of completely monotone functions on commutative semigroups goes back to Choquet (1954), who found the integral representation in 6.4 by proving ex(vH 1(S)) £; S+ and then using the Krein-Milman theorem. An elementary direct proof for the converse inclusion S+ £; ex(vH 1(S)) can be found in Fine and Maserick (1970) and it also follows from Corollary 2.5.12. Choquet also studied completely alternating functionscapacities are of this type-and proved an integral representation for bounded completely alternating functions. The general representation of functions ljJ E .91(S) as a special case of the Levy- Khinchin formula is due to Berg et ale (1976). Choquet (1954, §49) obtained Theorem 6.19 from the Krein-Milman theorem by considering a suitable vague topology on the cone of increasing functions which are continuous on the right. See also Talagrand (1976). Another approach to the theorem can be found in Matheron (1975), who considers a completely alternating capacity as "distribution function" for a random closed set. The present approach seems to be new. The result of Exercise 6.27 can be generalized by replacing [R ~ by a closed convex cone in [Rk, cf. Hirsch (1972) and Bochner (1955).
Notes and Remarks
143
Absolutely monotone functions on semigroups have hitherto not found particular attention. Widder (1941, p. 146) shows that any absolutely monotone function on IR + is analytic with nonnegative coefficients. The name "absolutely decreasing" has been introduced by us. Functions!: S -+ IR, which can be represented as ({Jt - ({J2 with ({Ji E f?}Jb(S), can be characterized intrinsically and are called BV-functions by Maserick (1975, 1981 and references therein). The beautiful duality theory for locally compact abelian groups does not seem to have been extended to topological semigroups, and it is probably difficult, if at all possible. The compact semigroup S = [0, 1] with maximum as semigroup operation has only one continuous semicharacter, namely, the constant semicharacter. The right dual object to look at might be the set of semicharacters which are continuous at O. In the group case, these are precisely the continuous group characters. Berg et al. (1976, §8) studied a semitopological semigroup having a continuous positive definite function, such that all semicharacters in the support of the representing measure are discontinuous at the neutral element. For a study of representations of semitopological semigroups see Dunkl and Ramirez (1975). See also the historical surveys by Williamson (1967) and Hofmann (1976) which both contain a rich bibliography.
CHAPTER 5
Schoenberg-Type Results for Positive and Negative Definite Functions
§1. Schoenberg Triples We have already mentioned the three fundamental papers of Schoenberg (1938a, b, 1942), all of which are very closely related to positive and negative definite kernels. The main purpose in (1938b) was to show the close connection between (real-valued) negative definite kernels and Hilbert metrics, see Chapter 3, §3. In his first mentioned paper (1938a), entitled" Metric spaces and completely monotone functions", Schoenberg raises the question about the connection between the class of Fourier transforms of (finite, nonnegative) measures in euclidean spaces and the class of Laplace transforms of (finite, nonnegative) measures on the half-line IR + . He states: In spite of the entirely different analytical character of these two classes, a certain kinship was to be expected f0r the following two reasons: 1. In both classes the defining kernel is the exponential function. 2. The less formal reason of the similarity of the closure properties of both classes, for both classes are convex, i.e. alf~ + a2j~ (a l ~ 0, a 2 ~ 0) belongs to the class if 11 and f~ belong to it, multiplicative, i.e. also ./1 belongs to the class, and finally closed with respect to ordinary convergence to a continuous limit function.
·.h
The answer Schoenberg gave to the above question was the remarkable result that to each continuous function cp: IR --+ C, with the property that cp 0 II· lin is a positive definite function on the group IR nfor all n (II· lin denoting the euclidean norm), there exists a finite nonnegative measure on [R + with Laplace transform cp(0); see (1938a, Theorem 2). The proof of this theorem (including that given in the more recent book of Donoghue (1969,
145
§ 1. Schoenberg Triples
pp. 201-206)), is, however, rather complicated and technical in nature. But there exists a further common feature of Fourier and Laplace transforms as shown in the preceding chapter: Laplace transforms, too, are characterized essentially by positive definiteness. Using this fact, we will prove a quite abstract form of Schoenberg's theorem, where we replace both the real line and the half-line [R+ by arbitrary abelian *-semigroups with neutral element. The key to most of the results in this chapter is given by the following, rather elementary, approximation lemma.
1.1. Lemma. Let (ajk) be a hermitian p x p matrix, let X be a nonempty set and let qJ: X x X --+ C be a given kernel. Suppose that for every n E t\J there exists a finite subset {xj~lj = 1, ... , p; r:J. = 1, ... , n} £; X such that qJ
n) ( n kp Xj~' x
=
{a jk
if) =t= k,
if'· - k bU t r:J.
a jj
1 } -
=t=
f3 •
If' qJ is positive definite and bounded, then the matrix (ajk) is positive definite; and if' qJ is negative definite such that Re( qJ) is bounded below, then (a jk) is negative definite. PROOF.
put
dj~
Let :=
C 1 , .•. ,
cj/n, 1
~
c p E C be given. For n E N choose {xj~} as indicated and j ~ p, 1 ~ r:J. ~ n. If qJ is positive definite, then
n
p
o~ I
I
j, k = 1
~,p
dj~dkpqJ(xj~, x~p) = 1
n2
p
=
-
n
PIP
I CjCkajk + --2I j,k=1 n
ICjl2ajj
+ 2"
j=1
j::fk
n
n
I ~=1 I ICjI2qJ(xj~, xj~)
j=1
P
=
I CjCka jk + R n , j,k= 1
where 1
R n :=
-
-
PIP
I
n j=1
ICjl2ajj
+ 2" n
I I
j=1 ~=1
ICjI2qJ(xj~, xj~).
The kernel qJ being bounded, R n tends to zero for n --+ ment follows immediately from Theorem 3.2.2.
00.
The second state-
D
Suppose now we are given a triple (T, S, v), where T and S are *-semigroups with neutral element, both abelian, and where v: T --+ S is a mapping satisfying: (S1) v(O) = 0; (S2) v(t*) = (v(t))* for all t E T; (S3) veT) generates S. Besides the finite powers Tn, n E N, we will also consider the infinite direct sum
146
5, Schoenberg-Type Results for Positive and Negative Definite Functions
which is again a *-semigroup with the canonical operations, and we associate with v the mappings vn: Tn -+ S defined by
nE N U {oo}. Condition (S3) then means the same as voo(T(OO» = S. For a function qJ: S -+ C it is equivalent to state that qJ 0 V oo is positive definite on T(oo) and that qJ 0 Vn is positive definite on Tn for all n E N. Although the three properties (S1)-(S3) seem to be rather weak, they nevertheless lead to the following surprisingly strong conclusion.
1.2. Theorem. Let (T, S, v) be a triple as described above and let qJ: S -+ C be a boundedjunction. Then if'qJ V oo is positive definite on T(oo), itjollows that qJ is positive definite on S. 0
PROOF. We fix a finite subset {S1' ... , sp} ~
S and have to show that the p x p
matrix
+ Sk»
(a jk ) = (qJ(sj
is positive definite. By condition (S3) there exist n j E N, tjm j = 1, ... , p such that
E
T, m = 1, ... , nj ,
nj
Sj
=
L v(t jm ),
j
= 1, ... , p.
m=1
For a fixed n E N we define x}et = (x}et(I), x}et(2), ...) E T(oo) as the columns in the matrix on the next page, where empty places should be read as zeros. Formally, the coefficients x}et(a) are defined by n
x' (a):=
Jet
{tjt 0
if a
= n(n 1 + ... + nj_ 1) + (a - l)n j +
L,
1~
L
~ nj;
if a is not of this form;
where j = 1, ... , p and a = 1, ... , n. On X := T(oo) we consider the kernel (x, y) := qJ(voo(x* + y», being positive definite by assumption. We have
qJ(sj qJ(sj
+ Sk) + Sj)
CPCtl v(t
jm
if j =t= k; if j
= k but a
=t= f3;
+ t jm ») if j = k and ex = {3.
Consequently, being bounded, the matrix (ajk) is positive definite by Lemma 1.1. D
1.3. Example. Consider the triple (T, S, v) = (1R, 1R+, x 2 ) with the involutions t* = - t on IR and s* = s on IR + . If qJ: IR + -+ C is a function such that qJ(II·lIn) is a positive definite function on IR n for every n E N (where 1I·lIn denotes the euclidean norm) then qJ is, of course, real-valued and bounded and the above theorem can be applied to qJ 0 showing by 4.4.2 that
J,
147
§l. Schoenberg Triples
... ... ...
x~n
t 12
... ... ...
0 0 0
0
t 1nl
...
0
0 0
0 0
... ...
t11 t 12
0
...
X~1
X~2
t 11
0 0
(12
X~1
X~2
.. . x~n
. ..
X;1
X;2
. ..
x;n
0 0 1
t 1 nl 0 0
0 t 11
2
11
0
( 1nl (21 (22
n+ 1 ( 2 n2 (21 ( 22
n+2 t 2n2
( 21 ( 22
2n t 2 n2
t p1 t p2 (p - l)n + 1
t pn p ( p1
t p2
(p - l)n ( pn p
( p1
t ~2 ( pn p
pn
+2
148
5. Schoenberg-Type Results for Positive and Negative Definite Functions
qJ(Jt) is the Laplace transform of some Radon measure on [0, measure is supported by [0, 00 [ if and only if qJ is continuous.
qJ:
00].
This
The above example also allows a statement in the other direction. Let ~+ -+ ~ have the form 2
cp(t) = {'" e-
Al
dJ.l(J-)
for some nonnegative finite measure J1 on ~ +. Then qJ( II· lin) is posItIve definite in the group sense on ~n because the square of the euclidean norm is negative definite, cf. 3.1.10 and 3.2.2. It will not surprise now, that the following condition (S4) p v E r!J>(T) for all pES 0
for a triple (T, S, v) as described above will play some role for a converse of Theorem 1.2.
1.4. Definition. A triple (T, S, v), where T and S are abelian *-semigroups, and v: T -+ S is a mapping fulfilling the conditions (S 1 )-(S4)' will be called a Schoenberg triple. 1.5. Theorem. Let (T, S, v) be a Schoenberg triple. If'thefilnction qJ: S -+ C is bounded and positive definite, then qJ Vn is positive definite on Tn for every nE N. 0
In view of Theorem 4.2.8 it is enough to show that p all pES. This, in fact, is true because PROOF.
p vit 0
I' ... ,
tn ) =
0
Vn E
r!J>(T n ) for
pet V(t ») = }]/(V(t)) j
l
n
=
TI p
0
vonit l '
... ,
t n)
j= 1
(where n j denotes the canonical projection) is a product of positive definite D functions. The restricted dual of the semigroup (~+, +) is given by the exponentials A ~ 00, as we have seen earlier. This shows that a triple (T, S, v) where S = ~+, is a Schoenberg triple if and only if v is a nonnegative negative definite function on T for which (Sl) and (S3) hold. In particular, if T is a connected topological *-semigroup and v: T -+ [0, oo[ is a continuous negative definite function satisfying v(O) = 0 and v $ 0, then (T, ~+ , v) is a Schoenberg triple. sHe-AS, 0 ~
1.6. Example. The space IP of all real sequences whose pth power is summable is a Banach space only for p ~ 1, i.e. the function IIxllp:= IxjIP]l/P is a norm only for p ~ 1. Nevertheless IP as well as II· lip is well defined also for
[I
~ 1.
149
Schoenberg Triples
o< p <
1, and /P is a linear space. We consider the /P-spaces for 0 < p ~ 2. Let <1>: /P -+ ~ be a given function depending only on the "norm" II· lip, i.e. <1> = qJ II· /I P' where qJ: ~ + -+ ~ is an arbitrary function. We claim that <1> is positive definite in the group sense on /P if and only if qJ(t 1 / P) is positive definite and bounded on ~+ , i.e. if and only if 0
for some J1 E M + ([0, 00 ]). In fact, from 3.2.10 it follows that x H Ix P is negative definite in the group sense on ~, so (~, ~+ , Ix I P) is a Schoenberg triple, and I
X H
IIxll~
= lim n-HX)
L Ixjl P
j= 1
is negative definite in the group sense on /p. If <1> is positive definite then so is <1> I ~(OO), and Theorem 1.2 implies that qJ(t 1 / P ) is positive definite and bounded on IR + . Conversely, if this is the case, then the integral representation above and the negative definiteness of 11·11 ~ on /P imply that <1> is positive definite. Finally, let us remark that <1> is continuous if and only if qJ is continuous, if and only if J1( { 00 }) = O. The above result (assuming continuity of qJ) has been obtained by Bretagnolle et a/. (1966).
N6, N6
1.7. Example. Consider the triple (Z, v) where n* = - n for nEZ, (n, m)* = (m, n) for (n, m) E and v: 7L -+ is given by v(n) = (n,O) for n ~ 0, v(n) = (0, - n) for n ~ O. We claim that (Z, v) is a Schoenberg
N6
N6,
triple. In fact, we know from 4.4.11 that (N 6)"" is isomorphic (and homeomorphic) with the unit disc D in the complex plane. Let z = re i8 ED be given; then, writing v(n) = (lI(n), w(n)), (1)
and since n H In I is negative definite in the group sense on Z, the function r lnl = e(log r) Inl is positive definite in the group sense (for r = 0 this is the function l{o}), and so is therefore the function (1). Given qJ: N6 -+ C we see that the functions
are positive definite in the group sense on TL P for all pEN if and only if
for some J1 E M +(D).
150
5. Schoenberg-Type Results for Positive and Negative Definite Functions
A related example is given by the triple (Z, No, 1·/) which is also easily seen to be a Schoenberg triple. That is, given an arbitrary sequence qJ: No -+ [R, the functions k
= (k 1 , ••• ,kp)HqJ(lk l l + ... + Ikpl)
are positive definite in the group sense on TL P for all P E
=
f
~
if and only if
n
1
t d/l(t)
for some J.1 E M +([ -1, 1]). The reader will not be surprised that statements similar to 1.2 and 1.5 are also possible for negative definite functions.
1.8. Corollary. Let (T, S, v) be a Schoenberg triple and let t/J: S -+ C be a function whose real part is bounded below. Then the following conditions are equivalent: (i) t/J is negative definite on S. (ii) t/J vn is negative definite on Tn for all n E (iii) t/J V oo is negative definite on T(oo). 0
~.
0
PROOF.
D
Immediate consequence of 1.2, 1.5 and 3.2.2.
1.9. Example. Let t/J:
~o -+ [R
be a sequence of real numbers. Then the
functions are negative definite in the group sense on TL P for all P E
t/J(m) = a
+ bm +
J
(1 - t
rn
)
~
if and only if
dJ.1(t)
[-I,l[
for certain a E [R, b E [R + and a Radon measure J.1 on [ - 1, 1[. This follows from Proposition 4.4.10. Likewise for a doubly indexed sequence t/J: ~6
-+
C the functions
(with v as in Example 1.7) are negative definite in the group sense on TL P for all p ~ 1 if and only if t/J E ..;Vl(N6). In this case t/J has the Levy-Khinchin decomposition given in Proposition 4.4.15.
1.10. Exercise. Show that the following triples (T, S, v) are Schoenberg triples (where on the groups Z, Z2' [R, [Rk the involution t* = - t is used, whereas the other semigroups carry the identical involution apart from (i) where S is given the" product-involution" (s l' S2)* := ( - S l' S2)).
§2. Norm Dependent Positive Definite Functions on Banach Spaces
(a) (b) (c) (d) (e) (f) (g) (h) (i) (j)
151
(Z, ~o, 12~+1)' 1 - cos x). (Z2' No, v) where v(o) = 0, v(l) = 1. (~k, ~+, IIxll~) where < a ~ 2 and 11·11 is the euclidean norm. (~\ ~~, (IXll~t, ... , IXkl~k)) where {a I , · · · , ak} £; JO, 2J. (Z, ~o, n 2 ). (~+, JO, IJ, (1 + t)-I). ([ - 1, IJ, ~ + , 1 - t). (lR, S, (t, t 2 )) where S = (~ x JO, oo[) U {CO, O)}. ((~+, max), (~+, +), id). (~, ~+,
°
1.11. Exercise. Let h: T -+ S be a *-homomorphism between the two semigroups with involution T and S. Then (T, S, h) is a Schoenberg triple if and only if h is onto. 1.12. Exercise. If (T1 , S 1, VI) and (T2 , S2' v2 ) both are Schoenberg triples, then so is (T1 X T2 , SI X S2' VI x v2 ). 1.13. Exercise. Let (T, S, v) and (S, U, w) be two Schoenberg triples. Then W 0 v) is again a Schoenberg triple if w(v(T)) generates U. The special case T = Z, S = ~6, v as in Example 1.7, U = ~o and wen, m) = n + m relates the two parts of 1.7. (T, U,
1.14. Exercise. Let qJ be the Laplace transform of some probability measure J.1 on ~+. Then qJ(I7= 1 xf) is the Fourier transform (or characteristic function) of some probability measure vn on ~n. Show that J.1 is infinitely divisible if and only if all the Vn are infinitely divisible.
§2. Norm Dependent Positive Definite Functions on Banach Spaces The classical result of Schoenberg mentioned at the beginning of §1 can also be reformulated in this way: If H is an infinite dimensional Hilbert space and <1>: H -+ ~ is a positive definite continuous function on the group H depending only on the norm, i.e. (x) = qJ(llxll) for some continuous qJ: ~+ -+ ~, then qJ(Jt) is the Laplace transform of some finite measure J.1 on ~+. Of course, the question arises as to how essential the assumption about H actually is, i.e. can something similar be said if we only assume H to be a Banach space? On finite dimensional spaces, a general result like this cannot be expected, since, for example, on the real line the positive definite function cos(t) only depends on It I. More generally, the characteristic function of the uniform distribution on {xE~nlllxll = I} depends only on the (euclidean) norm Ilxll, but not monotonically. Using a deep approximation theorem of Dvoretzky we can, however, show the following rather satisfactory result :
152
5. Schoenberg-Type Results for Positive and Negative Definite Functions
2.1. Theorem. Let B denote an infinite dimensional Banach space and let ({J: IR+ ~ C be a continuousfunction such that ({J(lIxll) is positive definite on the group B. Then ({J(..jt) is the Laplace transform of'some finite measure on IR+. PROOF. Of course, qJ is real-valued and bounded so that in view of Corollary 4.4.5 we are left with the problem of showing that ({J(..jt) is positive definite on the semigroup IR+. Our approximation lemma (1.1) cannot directly be applied in this case, but a certain modification will suffice. Let {t I , ••• , t k } £; IR+ and {c I , ... , ck } £; IR be given and fixed, and let t > o. We put t* := maxI $j:5k and choose b > 0 such that
J20
s, t E [0, t*],
Is - t I ;£ bt*
I({J(s) - ({J(t) I ;£
implies
t.
Now by a famous theorem of Dvoretzky (1961) there are linear injections TN: IR N ~ B, N = 1,2, ... , such that 1 ;£
IITNII·IITNIII
;£ 1
+ b,
i.e. assuming without restriction II TNII = 1, we have
I TN(x) I N
;£
IIx I
;£ (1
+ b)11 TN(x) I
(1)
N
for all x E IR , where on IR we use the euclidean norm. Given n E N we define vectors Xim E IR kn and real numbers aim for 1 ;£ i ;£ k and 1 ;£ m ;£ n by and where e I , e2' ... are the standard basis vectors in IR kn . Then ~ for (i, m) :f (i', m') and from (1) 0;£ I/x im - xi'm,1I - I/~n(Xim whence
I ~, L
l,l m,m
,aimai'm' ({J(II
xi'm,)11
;£
Ilx im - xi'm,1I
bllx im - Xi'm' I
=
;£ bt*,
~n(Xim) - ~n(xi'm')1/)
- l,l4, m,mL ,aimai'm' ({J(llx im -
Xi'm' II) I ;£
t(.Ll,m Iaim 1)2
=
e(Jllcdr,
where on the left-hand side the first sum is nonnegative by assumption, whereas the second sum is equal to
i.~lCiCjCP(Jti + t) + ~ [CP(O)JIC~ - itCtCP(~)J N ow letting first n tend to infinity and then t to zero we get k
0;£
thus proving the theorem.
L
i,j= I
CiCj({J(~),
D
153
*2. Norm Dependent Positive Definite Functions on Banach Spaces
Being of a general nature, the above result may not in some special cases give the sharpest possible results. Consider, e.g. the Banach space B = [1. If cp(llxlI) is positive definite on [1, then not only is cp(Jt) positive definite, but so also is cp itself, as we have seen in Example 1.6. Another question arises naturally: For which continuous bounded functions cp: [R+ -+ [R is it true that cp(llxll) is positive definite in the group sense on B? As we have seen, a necessary condition is that cp(Jt) be a Laplace transform, and on a Hilbert space this condition is sufficient, too, because 11.11 2 is negative definite. Moreover, if this condition is (necessary and) sufficient for 2 some Banach space, then by choosing cp(t) = e- At , A > 0, we see that II x 11 2 is negative definite on B, and then B is already a Hilbert space, as shown in the next lemma, which is taken from Wells and Williams (1975, Theorem
2.5). 2.2. Lemma. Let B be a Banach space such that the square of the norm is negative definite on B (in the group sense). Then B is already a Hilbert space. PROOF. Let x, y E B and a E [R and consider the four-point set {Xl' ... , x 4 } with Xl = X, X 2 = y, X 3 = - y, X 4 = 0 as well as the coefficients C l = 1 - 2a, C 2 = C 3 = a, C4 = -1. Then, by assumption, 4
o ~! L
cjckllx j - x kl1
2 = (1 - 2a)allx - Yl12
j,k= I
or
Assuming a <
!, dividing out 1 -
2a and then letting a tend to
t, we get
and the same procedure for a > ! gives the reverse inequality. Hence, the norm obeys the parallelogram law and is therefore induced by a scalar product. D For any Banach space B we introduce the convex cone
&>r(B) = {q>:
[R + -+ [R I q>
continuous, q>( 11·11) E &>(B)},
where positive definiteness refers to the group sense. The determination in general of &>r(B) seems to be open. If dim B = 00 then it is a subset of {2 J1( t 2 ) IJ1 E M~ ([R + ) }, and the sets are precisely equal when B is a Hilbert space. The following result of Einhorn (1969) shows that &>r(B) may be degenerate.
154
5. Schoenberg-Type Results for Positive and Negative Definite Functions
2.3. Theorem. Let qJ: [R+ ~ [R have the property that qJ(llxll) is positive definite on the Banach space C[O, 1]. For some rJ. E [R + we then have qJ 1]0, 00 [ == rJ. and qJ(O) ~ rJ.. In particular, .9r(C[O, 1]) consists only oj'the nonnegative constants. PROOF. For n ~ 2 let {(i,j) 11 ~ i
If 0 < s
~ t ~
<j
~
n} be the disjoint union of M and N.
2s we define (i,j) d(i, j) :=
{::
E
M,
(i,j)EN,
and d(j, i) := d(i,j) for i < j as well as dei, i):= O. The finite set {I, ... , n} together with d is a metric space and can hence be embedded in C[O, 1] by a famous theorem of Banach and Mazur (cf. Banach 1932, p. 169). This means that we can find {x b . . . ,xn } £; C[O, 1] such that (i,j)EM, (i,j) EN.
By assumption we have
o~
n
L cicjqJ(IIX i - Xjll) = [qJ(O) - qJ(t)] . L cf
+ qJ(t)(L Ci)2
i,j= I
+ 2[qJ(s)
- qJ(t)] . L CiCj
(2)
M
for all (C l , . . . , Cn) E [Rn. If M = 0 and L Ci = 0 this shows qJ(O) ~ qJ(t). Choosing n = 2m to be even, C 1 = ... = Cm = 1, cm + I = ... = cn = -1 and M := {(i,j)1 i < j, cic j = -I}, inequality (2) reduces to [qJ(O) - qJ(t)] . 2m ~ 2[qJ(s) - (p(t)] . m2 ,
and since this holds for all mEN, we see qJ(S) ~ qJ(t); analogously, the complementary choice of M gives qJ(s) ~ qJ(t). Finally, if all Ci = 1, (2) implies 0 ~ [qJ(O) - qJ(t)] . n + qJ(t) . n2 for all n, hence qJ(t) ~ O. D It seems to be open again, if .9r(B) can be degenerate also for some finite dimensional Banach space B. A closely related question was raised by Schoenberg (1938b): When is it true that exp( -llxlI1') is positive definite on a Banach space for some y > 0; if this is the case, which value of y then is the maximal possible? By homogeneity of the norm exp( - I x I 1') is positive definite if and only if I x I l' is negative definite, so if we put y*(B):= sup{y ~
0lllxll Y is negative definite on B},
Schoenberg's problem is equivalent to the determination of y*(B). By 3.3.3 we always have y*(B) ~ 2 and Lemma 2.2 shows that y*(B) = 2 is characteristic for Hilbert spaces. Schoenberg was particularly interested in
155
§3. Functions Operating on Positive Definite Matrices
the maximum norm on
~n
and stated the following explicit formula for
X,yE ~:
exp[ -(Ixl v Iyl)]
= n22
foo foo_ooexp[i(xu + yv)] -00
du dv x [1
+ (u + v)2][1 + (u
_ V)2]'
showing the negative definiteness of the maximum norm on ~2. In fact, as Herz (1963b) has shown, each norm on a two-dimensional real vector space is negative definite, but the maximum norm on ~3 is not negative definite. Hence, y*(~3, "·1100) < 1; already Schoenberg found that y*(~4, II
·1100) <
9 1 0.
2.4. Exercise. Let B be a Banach space and let qJ E f!J'Jr(B) be nonconstant and
have the form qJ(t) = J
2.5. Exercise. If B then
= LP(J.1) for some measure space (Q, d, J.1) and 1 ~ p ~ 2,
IlxII P is a negative definite function in the group sense on B.
2.6. Exercise. A famous theorem of Mazur and Ulam (1932) says that an isometric mapping between normed spaces which maps 0 to 0, is automatically linear. Use this result and Proposition 3.3.2 to give another proof of Lemma 2.2.
§3. Functions Operating on Positive Definite Matrices In his paper "Positive definite functions on spheres" Schoenberg (1942) proved the following result: Every continuous function qJ: [ -1, 1] -+ ~ having the property that qJ( y is a positive definite kernel on the unit sphere of some infinite dimensional real Hilbert space has a power series representation with nonnegative coefficients. He obtained this result by first proving finite dimensional analogues involving ultraspherical polynomials and then going to the limit. As a by-product he obtained this representation for all continuous functions operating on positive definite matrices. We are now going to derive a slightly more general result (without continuity assumptions) in a completely different way. Our first main result will be that the set of all functions on [ -1, 1], operating on positive definite matrices and normalized in a suitable way, forms a Bauer simplex
<x, »
156
5. Schoenberg-Type Results for Positive and Negative Definite Functions
and that the monomials t ~ tn, together with their two limit points, are precisely the extreme points of this simplex. The approximation lemma (1.1) will then give us Schoenberg's theorem for the real Hilbert sphere. A byprod uct of our considerations will be a new characterization of the so-called generating jiinctions-well known and widely used in probability theorywhich may be compared with Bochner's theorem for characteristic functions and with the characterization of the classical Laplace transforms given in Corollary 4.4.5. The functions t/J on [ - 1, 1] such that t/J( Y is negative definite on the Hilbert sphere turn out to have a very close connection with infinitely divisible probability measures on No. In order to prove the main results we need several lemmas which we prove first.
<x, »
3.1. Lemma. Let t < b < 1, a E ~+ and put qJ(x):= [b + (1 - b)e-X]a, x E ~ +. Then qJ is positive definite on the semigroup ~ + if and only if'a E No. PROOF. The Laplace transform of bco + (1 - b)Cl is given by the function b + (1 - b)e- X , therefore qJ is positive definite for all a = 0, 1, 2, ... , cf. 4.4.5 and 3.1.12. Suppose now that a is not an integer. We have
cp(x) = [1
+ (1
- t5)(e- X
-
1)]~ =
Jo (:)0 -
t5y(e- X
-
ly,
so that qJ is the Laplace transform of the signed measure J1:=
f (a)( 1 -
n=O
b)n( c1
-
co)*n
n
(note that this series converges in total variation). In view of 4.2.10 we only have to show that j1 is not a positive measure. Now
and therefore
wherej'(t):= (1
+ t)a.
We choose m:= [a]
+ 2; then
j'(m)(b - 1) = a(a - 1)· .... (a - [a] - l)b a-[a]-2 < 0 implying j1({m}) < O.
D
157
§3. Functions Operating on Positive Definite Matrices
3.2. Lemma. If n E N is odd then Icos(t) In is not positive definite in the group sense on IR. Likewise, if n E N is even then Icos(t) In sgn(cos(t» is not positive definite. PROOF. Assume that Icos(t) In is positive definite; then by Bochner's theorem there exists a symmetric probability measure J1 on IR such that
Icos(t) In = Loo,,/IA djl(A) = f:oo COS(tA) djl(A),
tE
~.
From Icos(n)/n = 1 we see that J1(2Z) = 1. Now (cos(t»2n is the characteristic function of K:= v*2n where v := !(Gl + G_ 1)' The measure K is concentrated on [-2n, 2n] n Z and J1 * J1 = K, hence J1([ -n, n] n Z) = 1. If n is odd then J1 is even concentrated on [-(n - 1), n - 1] n Z, contradicting the fact that J1 * J1( {2n}) = K( {2n}) > O. The second statement may be proved similarly. D We now introduce C 1 to be the set of all functions qJ: [ -1, 1] --+ IR which operate on positive definite matrices in the sense that whenever (a jk) is a positive definite matrix with entries ajk E [ -1, 1], then (qJ(a jk » is again positive definite. By Schur's theorem (3.1.12) C 1 is a multiplicative closed convex cone in IR[-I, ll. It is immediate that each qJ E C 1 is nonnegative on
[0, 1]. For t
E [
-1, 1] the matrix
G ~)
qJ( 1) ( qJ(t)
if qJE C 1, and so IqJ(t) I ~ qJ(l) for all t K 1 :=
is positive definite, hence so is
qJ( t) )
qJ(l) E
[-1,1]. Therefore
I
{qJ Eel qJ( 1) =
I}
is a compact convex base for the cone C l '
3.3. Lemma. Let qJ E C 1 and t E [-1,1]. Put qJl(S):= qJ(s) + qJ(st), qJ2(S):= qJ(s) - qJ(st) }or s E [ -1, 1]. Then qJl and qJ2 also belong to C l' PROOF. Let (ajk) be a fixed positive definite n x n matrix, where ajk E [ -1, 1] for allj, k = 1, ... , n, and let (c 1 , • •• ,cn ) E IRn also be fixed. Define <1>: [ -1, 1] --+ [R by n
(s):=
L
j, k= 1
cjc k qJ(sajk)'
We claim that E C 1. In fact, for any positive definite m x m matrix (b pq ), Ibpql ~ 1 for all p, q = 1, ... , m, and for all (d 1, ... , dm ) E IRm we get m
L
p,q=1
m
dpdq(b pq)
=
L
n
L
p,q=1 j,k=1
dpdqcjCkqJ(bpqajk)'
158
5. Schoenberg-Type Results for Positive and Negative Definite Functions
which is nonnegative because the tensor product (ajk) ® (b pq ) is positive definite, too, see 3.1.13. In particular, n
I(t) I ~
L
<1>(1) =
for all
CjC k qJ(a jk)
t E [ -1, 1]
j, k= 1
D
and this shows that qJ l' qJ2 both belong to C l'
3.4. Theorem. The set K 1 is a Bauer simplex whose extreme points consist oj' the monomials t ~ tn, n E No, and the two discontinuousjilnctions 1{1} - 1{-1} and 1{- 1 , 1} • Let qJ E ex(K 1) and t A(t) ~ 0 such that
PROOF.
A
=
qJ(s)
hence A = 1
-1, 1]. By Lemma 3.3 there exists a constant
E [
+ qJ(st) =
for all
AqJ(S)
s E [ -1, 1],
+ qJ(t) and therefore qJ(st)
=
qJ(s)qJ(t)
for all
s,
tE [
-1, 1],
i.e. qJ is multiplicative. It is well known that there exists lJ. E [0, 00] such that for all s E ]0, 1], qJ(s) = sa where 100 := 1, Soo := 0 for s E ]0, 1[. If lJ. E [R+ \N o then by Lemma 3.1 there is a positive definite matrix (a jk ) with ajk E]O, 1] such that (aJk) is not positive definite. We conclude that lJ. E No u {oo}. Of course, qJ(O) E {O, I} and qJ(O) = 1 only if qJ == 1, hence for lJ. = 00 we get qJ = 1{-1, I} or qJ = 1{1} - 1{-1}' Let lJ. = n E No be finite. If qJ( - 1 ) = 1 then qJ(t) = I tin; by Lemma 3.2 n has to be even, therefore qJ(t) = t n for all t E [ -1, 1], except for the special case that n = 0 and qJ = 1[-1, 1]\{O}' If qJ( - 1) = -1 then qJ(t) = Itin. sgn(t) and for n > 0 we get from Lemma 3.2 that n must be odd, i.e. qJ(t) = t n for all t E [ -1, 1] holds again. Two cases are left: qJ(t)
=
sgn(t)
and
qJ(t)
= sgn 2 (t) =
1[-1,
1]\{O}(t).
We shall show that these two functions do not belong to K l' Let t 1 = 0, t 2 = n12, t 3 = n and put a jk :=! + ! cos(t j - tk),j, k = 1,2,3. The matrix
is positive definite, but
has not this property, because (1, -1, l)A(l, -1, 1)' = -1 < O. This finishes the first half of the proof.
159
§3, Functions Operating on Positive Definite Matrices
Let E:= {t H t n I n E No} u {1{1} - 1{-1}, 1{l, -I}}. By Theorem 3.1.12 the monomials belong to K 1 and so do their two limit points, hence E ~ K 1. Let S denote the multiplicative semigroup [ -1, 1], then K 1 ~ g>~(S) and E ~ S. In Theorem 4.2.8 we have seen S = ex(g>~(S)), so that ex(K 1) ~ E ~ KIn ex(g>~ (S))
and finally ex(K 1) = E. It is very easy to see that every function in K 1 has a unique integral (= series) representation over E. Therefore K 1 is a Bauer simplex. D
3.5. Corollary. Every qJ E C 1 admits a unique series representation oj' the form 00
qJ(t) =
L antn + a_ 1[1{1}(t)
- 1{_l}(t)]
+ a-2 1{-1, l}(t),
n=O
where all an ~ 0 and L an < 00. The jilnction qJ is continuous if' and only if' a_ 1 = a_ 2 = 0, and qJ is then even an analytic junction. Now let H denote an infinite dimensional real Hilbert space with scalar product and denote by X := {x E H IllxlI = I} the unit sphere of H. Restricted to X x X, the scalar product is a mapping onto the interval [-1,1], and plainly for every qJEC l the composed map qJ«x,y») is a positive definite kernel on X x X. The next theorem shows that the converse of this result also holds.
<', .)
3.6. Theorem. Let qJ: [ -1, 1] -+ [R be a function such that qJ(<x, y») is a positive definite kernel on X x X, then qJ belongs to the cone C l ' PROOF. Let (a jk) be a positive definite p x p matrix with entries ajk E [ -1, 1]. We have to show that the matrix (qJ(ajk)) is positive definite, too, and shall again apply Lemma 1.1. First we choose vectors Xl' ... , x p E IR P such that ajk = <Xj' Xk) for all j, k = 1, ... , p. Without loss of generality we assume H = l2(N) and define x Jex = (x}ex(I), x}ex(2), ...) E H, 1 ~ j ~ p, 1 ~ It ~ n as the columns in the matrix on the next page, where empty places should be read as zeros. Formally, this means
x}ex«j):=
(Note that
IIxjll2 = ajj
{
Xj«j) OJI -
if 1
I xj l1
2
~
(j
~
p,
if (j = p + (j - l)n otherwise.
+ It,
~ 1.) Then
<Xj' Xk) = ajk <x}ex, x~p) = I"Xj112 = ajj {
if j =i= k, ifj = k but It =i= {3, if j = k and It = {3,
160
5. Schoenberg-Type Results for Positive and Negative Definite Functions
-..
I:
I:I:l.
~
N
I:I:l.
~
... 5
---- S
... 5
---- S
... 5
I:l.
~
I:l.
~
-..
I:l.
~
I:l.
~
-..
I:I:l.
~
I:
I:N
~
I:l.
~
I:l.
~
-
I:l.
~
-.. I:l.
-
-.. --- S ~
~
-..
-..
~
~
- ----.. S
... 5
>;
~
~
0
0
0
~
... 5
~
0
-..
... 5
-.. ~
~
~ 0
~
-..
--- S
N
...
-..
-..
>;
0
-..
-..
---~ S~ ... 5~
I:
0
I:l.
~
I:N
1;
. ..
---~ S~ ... 5~
-..
~
0
-..
---~ S~ ... 5~
N
0
~
I:N
~
~
-..
--- S
...
~
o~o
...
0
~o
. ..
0
0
~
~
0
...
0
. ..
0
161
§3. Functions Operating on Positive Definite Matrices
»)
so that {x)a} ~ X and qJ(<x)a' x kP = qJ(a jk ) for all U, 0:) =f= (k, f3). Moreover qJ is obviously bounded by qJ(1) and therefore Lemma 1.1 implies the positive definiteness of (qJ(a jk)). D Let us insert here a short discussion of the so-called generating functionsan adequate and widely used transform method in probability theory. They are defined for probability measures on No (resp. on Nt in the multivariate case). If J1 E M~(No) then its generating function fi,: [ -1, 1] -+ IR is given by 00
fi,(t):=
L J1( {n} )tn. n=O
It is immediate that fi, = vimplies J1 = v and that p*v = fi, . v, so that this transform already shares some well-known properties of Fourier and Laplace transformation. We can add the following Bochner-type characterization theorem: 3.7. Proposition. Afunction qJ: [ -1, 1] -+ IR is the generatingfunction ofsome probability measure J1 on No if and only if (i) qJ( 1) = 1; (ii) qJ is continuous; (iii) for each positive definite matrix (a jk) with entries in [ -1, 1] the matrix (qJ(a jk)) is also positive definite. PROOF. As a consequence of Corollary 3.5 the three conditions (i)-(iii) describe exactly the continuous functions in K l' i.e. the set of functions representable as a power series with nonnegative coefficients summing up to 1. D
3.8. Remark. In view of Theorem 3.6 condition (iii) above may be replaced by (iii)' qJ(<x, y») is a positive definite kernel on X x X, X being the unit sphere of some infinite dimensional Hilbert space.
3.9. Remark. If we replace condition (iii) in Proposition 3.7 instead of (iii)' by (iii)" for all finite subsets is positive definite
{Sl' ... ,
sn}
~
[ -1, 1], n E N, the matrix (qJ(SjSk))
and remove condition (i), then we obtain exactly the family of continuous positive definite functions on the multiplicative semigroup [ -1, 1]. Such functions qJ have the unique integral representation cp(t)
= [ )[0, oo[
It la dfl(a) + [ J]O, oo[
Itill · sgn(t) dv(fJ)
162
5. Schoenberg-Type Results for Positive and Negative Definite Functions
with finite Radon measures Il, v (see Exercise 4.2.17). Sharpening (iii)" to (iii) therefore corresponds to the requirement that Il is concentrated on 2N o and v is concentrated on 2N o + 1. As an application of our Bochner-type characterization, we give a short proof of the continuity theorem for generating functions:
3.10. Proposition. Let Ill' 1l2' ... be a sequence of probability measures on No and suppose the corresponding generating junctions /L 1, /L2' ... to be pointwise convergent to a limit function qJ. Then, if qJ is continuous at t = 1, qJ is also the generating function of some probability measure Il on No and, moreover, Iln ~ Il weakly, i.e. Iln{{m})
~
Il{{m})
jor all
mE No.
PROOF. Looking at Proposition 3.7 we see that the properties (i) and (iii) are preserved in the limit whereas the integral representation given in 3.5 shows that qJ is continuous as soon as it is continuous at t = 1. Hence qJ = p, for some Il E M~{No)' Identifying a natural number with the monomial of the corresponding degree we may view No as a subset of [ -1, 1]..... By Theorem 4.2.11 the measures Iln converge to Il as elements of M~{[ -1, 1].. . ) and they are all concentrated on the Borel subset No of [ -1, 1].. . , therefore by Exercise 2.3.6 lim Iln = Il weakly in the space M~{No), and this convergence is equivalent with Iln{B) -+ Il{B) for all B ~ No, or with
Iln{ {m })
~
Il{ {m} )
for all
mEN
because of the portmanteau theorem (2.3.1), noting that all subsets of No are open and closed. D Again let (ajk) be a positive definite matrix with entries ajk E [ -1, 1], and let qJ E K l' Then the new matrix (I - qJ{ajk)) is negative definite, Cj = 0 implying
L
L cj ck(1
- qJ{ajk)) = -
L cjckcp{ajk) ~ O.
This shows that 1 - qJ belongs to the closed convex cone C 2 of all functions t/J: [ -1, 1] -+ ~ which satisfy the two conditions:
(i) t/J(I) = 0; (ii) t/J{a jk) is negative definite whenever (a jk) is positive definite and ajk E [ -1, 1] for allj, k. If
,I, 0/ E
C2 th . Iar t h e 2 x 2 matrIx . (t/J{I) en 'In partIcu t/J{t)
t/J{t)).IS negatIve . t/J{I)
definite, implying t/J ~ 0 by 3.1.7. It is immediate that C 2 is a (proper) subcone of .H+{[ -1, 1]), and since the semigroup [-1, 1] contains an absorbing element, t/J is bounded by t/J{O), see Proposition 4.3.4, so that K 2 := {t/J E C 21 t/J{O) ~ I} is a compact convex base for the cone C 2'
J5
163
§3. Functions Operating on Positive Definite Matrices
3.11. Theorem. The transformation T: K 1 is an affine homeomorphism.
~
K 2 defined by T(qJ):= 1 -
qJ
PROOF. From Corollary 4.3.16 we know the two Bauer simplices 9~([ -1,1]) and {t/J E %'([ -1, 1]) It/J(1) = 0, t/J(O) ~ 1} to be isomorphic under the mapping qJ ~ 1 - qJ. Therefore, if t/J E K 2 then qJ:= 1 - t/J E 9~([ -1, 1]) and it remains to be shown that qJ even belongs to the smaller set K l' i.e. denoting by v E M~([ -1, 1]") the measure representing qJ, we have to make sure that v is concentrated on ex(K 1)' the closure of the monomials in [ -1, 1]". Since t/J E K 2 it follows e-tt/l E K 1 for all t > 0, hence e-tt/l = {It where Ilt E M~ (ex(K 1)) by Theorem 3.4. The negative definite function t/J is bounded and has the integral representation
t/J(s)
=
r
A
(1 - p(s)) df1(p),
)[-1, 1] \{1}
where the finite Radon measure 11 is the vague limit 11
1
"
= lim - (Ilt I[ - 1, 1] \ {1}) O
t
on the locally compact space [ -1, 1]"\{1}, see Proposition 4.3.15. Since all the measures Ilt I[ -1, 1]"\{1} are concentrated on the closed subset ex(K 1)\{1} of [ -1, 1]"\{1}, the limit measure 11 is concentrated there, too, by Exercise 2.4.13. Now 11
+ [1 -
Il( ex( K 1)\ { 1})] . e1 = v
is the representing measure for qJ, and it is concentrated on ex(K 1) as asserted.
D 3.12. Corollary. Every t/J E C 2 admits a unique series representation of·theform 00
L an(1
t/J(s) =
- sn)
+ a_ 1[2 . 1{_l}(s) + 1]-1, 1[(S)] + a_ 2 1]-1, 1[(S),
n=1
where all an ~ 0 and a_I
=
a-2
=
o.
L an <
00.
The function t/J is continuous
if and only if
3.13. Corollary. A real-valued function t/J on [ -1, 1] belongs to the cone C 2 if and only if· t/J(<x, y is a negative definite kernel on the infinite dimensional Hilbert sphere and t/J( 1) = O.
»
In the preceding chapters we saw already the close connection between negative definite and infinitely divisible positive definite kernels. We shall now give a further example of this kind. Let us agree that calling a probability measure 11 on No infinitely divisible will always tacitly assume 11 to be infinitely divisible inside M~(No), i.e. 11 = (Iln)*n for suitable Iln E M~(No) and all n E N. The following result may be considered as a Levy-Khinchin representation theorem for infinitely divisible generating functions.
164
5. Schoenberg-Type Results for Positive and Negative Definite Functions
3.14. Proposition. A probability measure /1 on No is infinitely divisible if· and only if'P,(t) > Ojor all t E [ -1, 1] and t/J := -log p,julfils the three conditions: (i) t/J(I) = 0; (ii) t/J is continuous; (iii) for each positive definite matrix (a jk) with entries in [ -1, 1] the matrix (t/J(a jk)) is negative definite.
If the conditions are fulfilled there exist uniquely determined numbers bb b 2 , ••• ~ 0, L bn < 00, such that 00
t/J(s) = L bn(1 - sn) n=l
jor aII
sE
[ -
1, 1].
PROOF. If P, > 0 and t/J := -log p, fulfils (i)-(iii) then t/J is a continuous function in C 2 , hence by 3.2.2 exp( -tt/J) is a continuous function in K 1 , i.e. a generating function for all t > O. In particular
p, = [exp( - t/J In)]n for all n E N, showing that /1 is infinitely divisible. Now let us assume /1 = (/1n)*n for suitable probability measures /11' /12' ... on No; equivalently, p, = (p'n)n for the corresponding generating functions. If P,(t o) = 0 for some to E] -1, 1[, then all derivatives of fi in to would vanish and then p, == 0 in contradiction to P,(I) = 1 and to {l's continuity on [-1, 1]. Hence fi(t) > 0 at least for all t E ] -1, 1] and consequently all the functions {In are strictly positive on ] -1, 1]. Now t/J:= -log p, is well defined, continuous on [ -1, 1] and finite on ] -1, 1], but still we have to exclude that t/J( -1) = 00. Let (a jk ) be a positive definite matrix, ajk E] -1, 1] for allj, k. Then (t/J(ajk)) is negative definite, because for all n E N
(e-o/n)!/I(a jk ») = (p,(a jk)l/n) = (Pn(ajk)) is positive definite by Proposition 3.7. In particular, t/J I] -1, 1] is a nonnegative negative definite function on the semigroup ] -1, 1], therefore bounded by Proposition 4.3.4, and since t/J is continuous on [-1, 1] we get t/J( -1) < 00, i.e. P,( -1) > 0, so that t/J is indeed a (real-valued) continuous function in the cone C 2 • The asserted representation of t/J follows immediately from Corollary 3.12. D
3.15. Corollary. Let /1 E M~(No) be infinitely divisible. Then /1 is the distribution of a random variable X = L:= 1 nXn' where the Xl' X 2' ... are independent, and where each X n is Poisson distributed with parameter bn, and L:=l bn < 00. PROOF.
L bn <
By 3.14 P, = exp(-t/J) with t/J(s) = L:=l bn(1 - sn), bn ~ 0 and The Poisson distribution ttb = Lk=O e-b(bklk !)Ck with parameter
00.
~3.
165
Functions Operating on Positive Definite Matrices
b~
°
has the generating function " ( ) _ ~ _b (bs )k _ - b(l - s) TCb S - i...J e k, - e , k=O
•
and if Y is a random variable with distribution TCb' then n Y has the distribution Lk=O e-b(bkjk !)£nk with generating function exp[ -b(1 - sn)]. Choosing now a sequence of independent random variables Xl' X 2, ... , where X n has the distribution TC b", the (well-defined) infinite series X:= L nX n has the generating function
i\
exp[ - bn(l - sn)] = ex p [ -
Jl
bn(l - sn)] = exp( - t/f(s»
D
implying that X has the prescribed distribution Il.
3.16. Exercise. Let Iln be the binomial distribution B(n, Pn), i.e. J1n
=
kt (~)p:(l
- Pnt-kck'
where n E Nand Pn E [0, 1]. Show by using 3.10 that if lim n _ then Iln tends weakly to the Poisson distribution TCb.
oo
npn =
bE
IR+,
3.17. Exercise. Let Ill' 1l2' ... be a sequence of infinitely divisible probability measures on No tending weakly to Il E M~(No). Show that Il is infinitely divisible inside M~(No). 3.18. Exercise. If Il E M~(No) is infinitely divisible then 1l(2N o) > 1l(2N o
+ 1).
3.19. Exercise. Let K be the set of all real-valued functions qJ on the unit interval [0, 1] operating on positive definite matrices with entries from [0, 1] and normalized by qJ(l) = 1. Show that K is a Bauer simplex and determine the extreme points of K. 3.20. Exercise. For a probability measure Il E M~(N6) the (bivariate) generating function It: [ -1, 1]2 ~ IR is defined by 00
(i(s, t):=
L
Il( {(n, m)})sntm.
n,m=O
Show that a given function qJ: [ -1, 1]2 ~ [R is a bivariate generating function if and only if (i) qJ( 1, 1) = 1; (ii) qJ is continuous; (iii) for each pair of positive definite matrices (a jk ), (b jk ) of equal size and with entries in [ -1, 1] the new matrix (qJ(a jk , bjk )) is also positive definite.
166
5. Schoenberg-Type Results for Positive and Negative Definite Functions
3.21. Exercise. If J1 E M~(Ni) is infinitely divisible (inside Ni !), then J1 is the distribution of some random vector X where 00
X = (X 1,···,Xk) =
L
(n1Xnl, ...,nk,···,nkXnl, ...,nk)' nl, ... ,nk = 1 k the {X nl, ...,nk I(n l' , nk) E N } being independent and Poisson distributed with parameters bn1 , ,nk' and L bn1 ,... ,nk < 00.
3.22. Exercise. Let qJ: [ -1, 1] function
~ [R
be given such that for all n E N the
is positive definite in the group sense. Then qJ belongs to the cone C 1 (and therefore has the representation stated in Corollary 3.5). Conversely, if qJ E C b then qJ([]i= 1 cos t j) is positive definite on [Rn for all n.
3.23. Exercise. Show that the extreme points of the Bauer simplex K 1 are characterized by the three conditions: (i) qJ E C 1; (ii) qJ(l) > 0; (iii) qJ2(t) = qJ(t 2) for all t E [ -1, 1].
§4. Schoenberg's Theorem for the Complex Hilbert Sphere In the preceding section we have seen that a continuous function qJ: [ -1, 1] such that qJ( y») is positive definite on the unit sphere of some infinite dimensional real Hilbert space, has a power series representation qJ(s) = ans n with an ~ 0 and an < 00. We are now going to extend this result to the case of a complex Hilbert space H (always infinite dimensional), and of course we have to replace the interval [ -1, 1] by the closed unit disc D:;:::: {z E Clizi ~ I} in the complex plane. Let lr := {z E C/izi = I} denote the torus group. When we consider lr as a discrete group, we shall write lr d instead of lr. In analogy with the real case we introduce C to be the set of all functions qJ: D ~ C operating on positive definite matric'"3 in the sense that (qJ(ajk)) is positive definite whenever (a jk ) is a positive definite matrix with entries in D. By the Schur Product Theorem (3.1.12) C is a multiplicative closed convex cone in CD. It is immediately seen that each qJ E C is nonnegative on [0, 1].
<x,
~ [R,
L
L
167
§4. Schoenberg's Theorem for the Complex Hilbert Sphere
G;)
For zED the matrix
is positive definite, hence so is
qJ(l) ( qJ(z) if qJ E C, and therefore qJ(z) showing, in particular, that
qJ(Z») qJ(l)
= qJ(z) as well as IqJ(z) I ~ qJ(l) for all zED,
K := {qJ E C IqJ( I) = I} is a compact convex base for C. For the proof of Theorem 4.5 below, in which the extreme points of K are determined, we need several lemmas which we prove first. 4.1. Lemma. The set of numbers c E C for which the matrix
MC:=(~eel ~ ~) is positive definite is the" pear-like" subset Q
n of the unit disc (f"igure 2)
= {cEDI[3 - 2 Re(c)JlcI 2
~ I},
having the following properties: (i) {cECllcl ~!} ~ Q. (ii) Q is symmetric about the x-axis and its boundary curve in the upper half-
plane is y
with y'(l) =
=
y(x)
= (1-
f2X+1
x)V~
-J3.
(iii) For t E [!, IJ and
for
eE [ -1t, 1tJ we have c = te lei
iO
x
E [
E Q if and only if
~ e(t):= arccos(~t-l - !t- 3).
y
y(x)
1 x
-1
Figure 2
-!, IJ
168
5. Schoenberg-Type Results for Positive and Negative Definite Functions
PROOF.
We have detG
~) =
1-
Jel 2 and
det Me = 1 - [3 - 2 Re(c)]lcI 2 so the description of Q follows from Theorem 3.1.17. (i) is clear. (ii) If c = x + iy E Q then x ~ O,hence
E [
-1, 1] and det Me = 1 - (3 - 2x)(x 2 + y2)
2
y~
(1 - x)2(2x
3-2x
+
1)
'
which shows that x E [ -!, 1] and
IYI ~
(1 -
fiX+l x\j3=2X'
and the converse is also true. By differentiation of the function y = y(x) we get y'(I) = -)3. (iii) For c = te U) we have det Me = 1 - [3 - 2t cos 0]t 2 ~ 0 if and only if cos 0 ~ ~t-l - !t- 3. Since t~~t-l - !t- 3 maps [!, 1] onto [-1,1] the assertion follows. D
4.2. Lemma. Let qJ(tei(J) = tkeip(J, where k E r\Jo, P E 71... If' qJ E K then Ip I ~ k. If qJ E K we get by Lemma 4.1 that qJ(Q) ~ Q, in particular qJ(tei(J(t» = tkeip(J(t) E Q for! < t < 1, k where O(t) = arccos(~t-l - !t- 3). If t > ! and IpO(t) I < 1t, which is true for t sufficiently close to 1, we have IpO(t) I ~ O(t k), hence · O(tk) 1· kt k- 10'(tk) k I m - - = 1m = Ip I -~ 1t-l O(t) t-l O'(t) by I'Hopital's rule because PROOF.
lim O'(t) = t-l
-)3.
The direct evaluation of this limit is tedious but elementary. It is, however, no coincidence that this limit is y'(I) because
lim ~ (te i6 (1» = 1 + i lim (}'(t) t-l dt t-l is a tangent vector to y = y(x) at x = 1.
D
4.3. Lemma. Let qJ E C and ZED and define thefunctions qJl' qJ2' qJ3' qJ4: D -+ C by
qJl, 2(W) = qJ(w) qJ3,4(W) = qJ(w)
± ![qJ(zw) + qJ(zw)] ± (1/2i)[qJ(zw) - qJ(zw)].
Then qJl' ... , qJ4 all belong to C, too.
~4,
169
Schoenberg's Theorem for the Complex Hilbert Sphere
Let (ajk) be a fixed positive definite n x n matrix with entries in D, and let (c l' . . . , Cn) E C n also be fixed. Define the function <1>: D -+ C by
PROOF.
n
L CjCk ({J(zajk)·
(z):=
j, k= 1
We claim that E C. In fact, for any positive definite m x m matrix (b pq ) with entries in D and for all (db .. . , dm ) E cm we have m
L
p, q = 1
m
n
L
dpdq(b pq) =
L
p, q = 1 j, k = 1
dpcjdqCk ((J(bpqajk) ~ 0
because the tensor product (b pq ) ® (ajk) is positive definite by Corollary 3.1.13. Therefore n
I(z) I ~
L CjCk ((J(a jk ),
<1>(1) =
j,k= 1
hence, denoting G = Re(
IG(z) I ~ But G(z) =
IH(z) I ~
and
<1>(1)
<1>(1).
! L Cj~[({J(zajk) + ((J(za jk )] and H(z) = (1/2i)
L cjCk[({J(zajk) -
({J(zajk)]
and thus
L cjCk ({Jcx(ajk) ~ 0 for
It
= 1, 2, 3, 4, i.e., {({Jl' ({J2' ({J3'
({J4} ~
c.
D
4.4. Lemma. The set of continuous group characters on the torus lr is dense in the set o.f all group characters on lr, i.e. in Td • PROOF. The group G of all continuous group characters on lr is known to be canonically isomorphic to 71. Assume that G ~ ltd. Since the continuous characters on a locally compact abelian group separate points, there is a nontrivial continuous character ~ on ltd/G. Composing ~ with the canonical projection we obtain a nontrivial continuous character 11 on ltd' being 1 on G. By Pontryagin's duality theorem, cf. Rudin (1962), there is some z E lr such that 11(P) = p(z)forallpEltd • Thespecialp = idlrgives 1 = 11(P) = p(z) = z, so that 11 == 1. This contradiction proves our lemma. D
4.5. Theorem. The set K oj'normalized operatingfunctions is a Bauer simplex whose extreme points consist of the closure of {znzm In, mE r\J o} in the pointwise topology oj'C D• More precisely, denoting by Po the zero extension ofany group character P on the discrete torus lrd (i.e. Po = P on lr and Po Ib = 0), we have ex(K) = {znzm In, mE r\J o} u {Po Ip E ltd}.
170
5. Schoenberg-Type Results for Positive and Negative Definite Functions
Let qJ E ex(K) and ZED. By Lemma 4.3 there exist a = a(z) f3 = f3(z) ~ 0 such that
PROOF.
qJ(w) qJ(w)
+ t[qJ(zw) + qJ(zw)] = 1
+ 2i [qJ(zw)
This implies a = 1 further
aqJ(w)
for
wED,
- cp(zw)] = f3qJ(w)
for
WED.
+ t[qJ(z) + qJ(z)]
qJ(ZW)
+ qJ(zw) =
and f3 = 1
qJ(z)qJ(w)
+ (1/2i)[qJ(z)
~
0 and
- qJ(z)] and
+ cp(z)qJ(w)
as well as
qJ(ZW) - qJ(zw) = cp(z)qJ(w) - qJ(z)qJ(w), so that finally qJ(zw) = qJ(z)qJ(w) for all z, wED, that is, qJ is multiplicative. We conclude that p = cp IT is a character on the (discrete) torus, and from the proof of Theorem 3.4 we know that either qJ I] - 1, 1[ == 0 in which case the multiplicativity of qJ implies qJ Ib == 0, hence qJ = Po, or there is a kENo such that qJ(t) = t k for all t E [ - 1, 1], and then cp(te UJ ) = t k p(e ilJ ) for 0 ~ t ~ 1, 8 E lR. We will now show that p is continuous in the latter case. Since qJ(Q) ~ Q, cf. Lemma 4.1, we have for
t~t<
1,
I8 I ~ 8( t).
If k = 0 this implies that p(e itJ ) = 1 for 181 ~ 8(t), i.e. p == 1. Suppose k > 0 and let e > 0 be given. If t is chosen so close to 1 that t k > and 8(t k ) < e then
t
for
101
~
O(t)
by Lemma 4.1, and this shows that p is continuous at 1, hence continuous. Therefore there is p E 7L such that p(e ilJ ) = eiplJ , and from Lemma 4.2 we get Ipl ~ k. Now since cp( -1) = (_1)k = p(e iTt ) = eipTt , we see that k and pare of equal parity. Consequently, n:= (k + p)/2 and m:= (k - p)/2 are nonnegative integers and for each Z = te ilJ E D we get
We have now proved ex(K) ~ {znzm In, mE No} u {Po Ip E
lfd }·
To prove the reverse inclusion we first remark that Theorem 3.1.12 implies that (z ~--. znzm) E K for n, m E No. Furthermore, we see that 1lf(Z) = lim znzn n-+oo
belongs to K, so since K is closed under multiplication, the zero extension of every continuous group character belongs to K. Lemma 4.4 shows that the
171
§4. Schoenberg's Theorem for the Complex Hilbert Sphere
zero extension of arbitrary group characters on lr also belongs to K. This shows that where
r =
{qJ E K II qJ( z) 12 = qJ( I Z 12 ) for all zED}
is defined in accordance with Neumann's result for y(z) = Iz1 2 , cf. 2.5.11. By Corollary 2.5.12 we conclude that ex(K) = r. This shows that ex(K) is compact and that ex(K)
= {znzm I n, mE No} = {znzm I n, mE No} u {Po 1 P E Td }.
The fact that ex(K) is closed under multiplication together with the Stone-Weierstrass theorem shows that K is a Bauer simplex. D Let H denote a fixed infinite dimensional complex Hilbert space with scalar product <., .>and denote by S := {x E HI" x II = 1} the unit sphere of H. Restricted to S the scalar product is a mapping into the closed unit disc D and plainly for every qJ E K the composed map qJ(<x, y is a positive definite kernel. The next theorem shows that the converse also holds.
»
4.6. Theorem. Let qJ: D -+ C be afunction such that: (i) qJ(1) = 1; (ii) qJ(<x, y is a positive definite kernel on S x S.
»
Then qJ belongs to K. The proof is very similar to that of Theorem 3.6 and therefore omitted, cf. Christensen and Ressel (1982). 4.7. Corollary. A continuous complex-valued function qJ on D such that qJ(<x, y is positive definite on the unit sphere of some infinite dimensional complex Hilbert space, has the unique series representation
»
00
= L
qJ(z)
n,m=O
an,mznzm
forall
zED,
where all coefficients an,m are nonnegative and L~m=o an,m < PROOF.
00.
By Theorem 4.5 there is a uniquely determined finite Radon measure
J1 on ex(K) such that
qJ(z)
=
f.
p(z) dJl(p)
for all
ZED.
ex(K)
Let K be the countable set of functions z H znzm, n, m E No. Of course, measurable and with an,m:= Jl({znzm}) we have q>(z) -
f
n,m=O
an,mznzm
=
f.
_p(z) dJl(p)
ex(K)\K
K is
172
5. Schoenberg-Type Results for Positive and Negative Definite Functions
showing that the function on the right-hand side is also continuous on all of D. The description of ex(K) given in Theorem 4.5, however, shows that this function is identically zero in D; therefore Jl(ex(K)\K) = O. D
4.8. Corollary. Let qJl' qJ2' ... be a sequence of continuous functions of the type described in Corollary 4.7 and let them have the series representations 00
fn ..,.,k
(z)
= "i..J a(k) n, m znzm ,
ZED,
n,m=O
k = 1, 2, ....
If the pointwise limit qJ(z) = lim k
-+ oo qJk(Z) existsfor all ZED and ~lqJl [0,1] is continuous in 1, then qJ is continuous everywhere and for the coefficients an,m in
00
qJ(Z) = " f...J a n,m znzm n,m=O
PROOF. The cone C being closed we have of course qJ E C. Let qJ have the integral representation
qJ(Z)
=
f
p(z) dJl(p).
ex(K)
Then
o=
f
_P(1 - ~) dJl(p) -+
ex(K)\K
}
f
_p(l) dJl(p) = Jl(ex(K)\K),
ex(K)\K
so that Jl is in fact concentrated on K. The affine isomorphism between the two Bauer simplices K and M~(ex(K» and the fact that the functions Z H znzm are isolated points in ex(K), which is easily seen, imply the convergence of the corresponding coefficients. D 4.9. Exercise. Let qJ be a real-valued continuous function on D such that (qJ(a jk» is positive definite for each positive definite matrix (a jk) with entries in D. Show that in the representation of qJ, as stated in Corollary 4.7, we have an, m = am, n for all n, m. 4.10. Exercise. Let t/J: D -+ C have the property that for each posItIve definite matrix (ajk) with ajk E D the transformed matrix (t/J(ajk» is negative definite, and suppose t/J is normalized by the two conditions t/J(I) = 0, t/J(O) ~ 1. Then 1 - t/J E K. (Conversely, but this is trivial, if qJ E K, then t/J := 1 - qJ has the properties stated above.)
173
§5. The Real Infinite Dimensional Hyperbolic Space
§5. The Real Infinite Dimensional Hyperbolic Space In 1974 Faraut and Harzallah proved some results on positive and negative definite kernels defined on the real infinite dimensional hyperbolic space X and which depend only on a kind of natural distance on X. These results are very similar to Schoenberg's theorems discussed earlier in this chapter, and in fact their method of proof also uses the corresponding finite dimensional results to obtain the infinite dimensional version by a rather complicated limiting procedure. However, our approximation lemma (1.1) again gives easy and short direct access to these relations, slightly more general while not assuming continuity. The space X is formally defined by
X
:=
{x = (Xk)k~O
E
1~(No)lxo ~ 1, x~ -
f x~ = 1}.
k=l
The kernels L~= 1 XkYk and (xoYO)-l are both positive definite on X x X, hence so is 00
L XkYk
qJ(x, y):= _k=_l_ _ XoYo and the inequality 3.1.8 gives
qJ2(X, y) ~ cp(x, x)qJ(Y, y) =
x6 -
- - 2-
1
Xo
Y6 -
. - - 2-
1
Yo
so that
Hence 00
[x, y]
:=
xoYo -
L XkYk
~ 1
k=l
for all x, Y E X. The distance d(x, y) between x and Y is defined as the positive solution of cosh d(x, y) = [x, y]. From [x, y] = x oYo(l - cp(x, y» we conclude that for all t > 0
[x, y]-t = e-t1og[x,y] = (x oYo)-t(l - cp(x, y»-t. The right-hand side is positive definite by Corollary 3.1.14 so that t/f(x, y) := log[x, y] is a negative definite nonnegative kernel on X x X. The equation
174 cosh d
5. Schoenberg-Type Results for Positive and Negative Definite Functions
= exp(t/J) has for d, t/J
~
0 the unique solution
+ 10g(1 + Jl
d = t/J
- e- 2 t/1),
and therefore d is negative definite, too, by Corollary 3.2.10. Now we can state the above-mentioned result of Faraut and Harzallah (1974).
5.1. Theorem. Letj': [R+
~
[R be an arbitrary given junction. Then
f(log cosh d(x, y» is positive definite on X x X if and only iffE gpb([R+). The kernel f(log cosh d(x, y» is negative definite if and only iff E %'( [R +). PROOF. IffE gpb([R+), thenf(s) and then
=
flO,
00] e-J.s
dJ.l()") for some J.l EM +([0,00])
f(log cosh d(x, y» = f(log[x, y]) =
f
e-.
dJ1().)
= f[X,yr'
f(lOg[x, x]) f(log[x, y]») ( f(log[x, y]) f(log[y, y]) is then positive definite, hence
{f(log[x, y])}2
~
(f(0»2
and because of {[x, y] lx, y EX} = [1, 00[, f is bounded. Let S1' ••• , sp ~ 0 be given. We fix n E N and define for 1 ~ j ~ p and 1 ~ lJ.. ~ n the vectors xj~ E X as the columns in the matrix on the next page where empty places should be read as zeros. That is,
eSj
xj~«(J):=
Je {
2sj
o
for (J -
1
= 0,
for (J = (j - l)n otherwise.
+ lJ..,
Then if j =t= k, if j = k but ex =t= {3, if j = k and lJ.. = {3,
175
§5. The Real Infinite Dimensional Hyperbolic Space
~
c:
c:~
~
I I:l,
0
'1:u
0
0
000
"ll:l,
N
~
N
I:l,
c:~
'1:u
Il:~
~
~
~
0
~o
I:l,
0
c: c:N ~
~
c:N
'1:u
~
0
~
N c:~
~
~
c:~
~
'1:u
~
0
o~
0
~o
c: c:~
0
~
N c:N ~
o~o
0
0
0
o~
~~o
...
...
0
0
...
~
... o
0
0000
~ 0
000 0
0
000
0
0000
0000
176
5. Schoenberg-Type Results for Positive and Negative Definite Functions
so that Lemma 1.1 can again be applied and shows the positive definiteness of (f(sj + Sk»j,k=l, ... ,P. Hence fE&b(~+). The second statement follows D in the usual way.
Notes and Remarks The idea of using an approximation like that in Lemma 1.1 appears for the first time in Ressel (1976). The name Schoenberg triple-restricted, however, to the case where T is an abelian group and S is a semigroup with the identical involution-was introduced in Berg and Ressel (1978), where also the second part of Example 1.7 was given which may be considered as a discrete counterpart to Schoenberg's classical result (cf. Example 1.3). That (Z2' No, v) with v(O) = 0, v(l) = 1 is a Schoenberg triple was shown independently by Kahane (1979, 1981). The results of §1 have a very close connection to de Finetti-type theorems in probability theory; see Ressel (1983) where also Theorem 1.2 is sharpened in the sense that the measure representing qJ is shown to be concentrated on the closed subsemigroup {p E Sip V E &( T)} of S. Theorem 2.1 was published in Christensen and Ressel (1983). Dvoretzky's theorem on almost spherical sections has later found some simpler proofs, see for ex. Szankowski (1974) and Figiel (1976). The fact that negative definiteness of the norm square characterizes Hilbert spaces has a counterpart for LP-spaces, 1 ~ p ~ 2. If Ilxll P is negative definite on a Banach space B, then B can be linearly and isometrically imbedded into some LP-space. This was proved by Bretagnolle et al. (1966, Theoreme 2), the proof however is complicated and (ten pages) long, and a shorter one seems not to be available. Bretagnolle et al. also show in this paper that ~(IP) is degenerate if p > 2. For some related results see Kuelbs (1973) where in particular Theorem 2.3 is extended from CEO, 1] to Banach spaces Cb(X) which contain a sequence of functions with norm one and disjoint supports, X being an otherwise arbitrary topological space. There are many papers dealing with functions operating on positive definite matrices or positive definite functions on groups. Rudin (1959) showed that a function defined on the open interval] -1, 1[ and leaving invariant only the positive definite Toeplitz matrices can be represented as a power series with nonnegative coefficients. Horn (1969a) proved that a function defined on ]0, 00 [ which leaves invariant the cone of positive definite matrices with positive entries must be absolutely monotonic and therefore (Widder 1941, p. 146) analytic with nonnegative coefficients. Theorem 3.4 and some other results of §3 were published in Christensen and Ressel (1978). Rudin (1959) also conjectured that a complex-valued function defined on the open unit disc operating on positive definite Toeplitz matrices must be of the form Lan, m znzm with nonnegative coefficients an, m. The conjecture 0
Notes and Remarks
177
was proved by Herz (1963a, Theoreme 1) in even greater generality, replacing the group 7L (on which the positive definite functions define the Toeplitz matrices) by other abelian groups, called illimites. Although these operating functions are real analytic in the open unit disc, their boundary behaviour may be quite complicated as Graham (1977) has shown; see also the recent book of Graham and McGehee (1979, Chap. 9.6), where more references in this connection may be found. Harzallah (1969) determined the functions which operate on negative definite functions on abelian groups. Roughly speaking it is the Bernstein functions, cf. the remark following 4.4.3. Lukacs (1960) posed the question of determining the class of functions which operate on the set of characteristic functions, i.e. on the normalized continuous positive definite functions on the real line. Konheim and Weiss (1965) answered this question and proved (without assuming continuity of the operating function) that precisely functions of the form Lan. m znzm, where an,m ~ 0 and I an,m = 1, have this property. The main part of §4 was recently published in Christensen and Ressel (1982). Faraut and Harzallah (1974) proved Theorem 5.1 assuming continuity of f; they state that this result is originally due to Krein.
CHAPTER 6
Positive Definite Functions and Moment Functions
§1. Moment Functions Throughout this section, S = (S, +, *) will be an abelian semigroup with involution and zero element O. In Chapter 4 we introduced the convex cones f!j)b(S) S; f!/J" (S) S; f!/J(S) S; CS , and every function qJ E f!/Je (S) has a representation
cp(s)
=
r p(s) dJL(p)
Js *
with a uniquely determined measure J1 E MC+(S*). Our aim is to examine for which qJ E f!/J(S) there exists a representation as above with J1 E M + (S*) of not necessarily compact support. We start by giving a simple necessary and sufficient condition for a function qJ E f!/J(S) to belong to f!/Jb(S) in the case of which the above representation is possible with J1 supported by S.
1.1. Proposition. Let G be a generator setfor Sand qJ: S -.. C. Then if' and oniy if' qJ E f!j)(S) and (I - E a + a*)qJ E f!/J( S) jor all a E G. PROOF. If qJ E f!/Jb(S) has the integral representation
cp(s) = Lp(S) dJL(p), we find
s E S,
qJ E
f!/Jb(S)
179
§ 1. Moment Functions
so (1 - Ea+a*)qJ is positive definite (and bounded) for all a E S. If, conversely, qJ E f!J(S) and (1 - Ea+a*)qJ E f!J(S) for a E G we have qJ(s
+ s*)
~
0
(1 - Ea+a*)qJ(s
and
+ s*)
~
0
for
SE
S,
hence
o~
qJ(s
+ s* + a + a*)
~ qJ(s
+ s*)
a E G,
for
SE
S.
For aI' ... , an E G u {a* la E G} we then have
o~
+ aT + + an + a:) + a! + + an + a:) ~ qJ(a n + a:) ~ qJ(O),
qJ(a l
~ qJ(a2 ~
...
+ S*)qJ(O) we find qJ(a l + ... + an) 2 ~
and applying I qJ(S) 1 ~ qJ(S 2
I
hence IqJ(s) I ~ qJ(O) for all
1
SE
qJ(0)2,
D
S.
1.2. Remark. For the case of S = (~6, +, *) and G = {(1, O)}, cf. 4.4.11, we get that qJ: ~6 -.. C belongs to f!Jb(~6) if and only if qJ and (1 - Eo, l)qJ belong to f!J(~6), a result given by Atzmon (1975). For the case of S = (~o, +) and G = {1} we get that qJ: ~o -.. [R belongs to f!Jb(~o) if and only if qJ and (1 - E 2)qJ belong to f!J(~o), cf. Haviland (1936). More generally, for the case of S = (~~, +) and G = {e 1 , ••• , ek }, where ej = (0, ... ,0,1,0, ... ,0) with 1on thejth place, we get that qJ: ~~ -.. [R belongs to f!Jb(~~) if and only if qJ and (1 - E 2eJ )qJ belong to f!J(~~) for j = 1, ... , k. These results should also be compared with Propositions 4.4.12 and 4.4.9. Let E + (S*) denote the set of Radon measures /1 E M + (S*) such that
LIP(S)' dJ.l(p) Clearly MC+ (S*) E +(S*).
£;
E + (S*)
£;
<
00
for all
SE
S.
M~ (S*), and if /1, VEE + (S*) then /1
* vE
1.3. Definition. A function qJ: S -.. C is called a moment jilnction if there exists a measure /1 E E + (S*) such that
qJ(S) =
r p(s) dJ.l(p)
Js*
for
SE
S.
(1)
The set of moment functions is a convex cone denoted Jt(S) and it is clear that f!Jl'(S) £; Jt(S) £; f!J(S), cf. 4.2.5. Furthermore, Jt(S) is stable under products. In fact, if qJ1 and qJ2 in Jt(S) are represented via (1) by /11 and /12 then qJ1 . qJ2 is represented by /11 * /12'
180
6. Positive Definite Functions and Moment Functions
1.4. Two interesting problems turn up: (a) For which semigroups S is it true that Jt(S) = &>(S), and if Jt(S) =t= &>(S) find the necessary and sufficient conditions for cp E &>(S) to belong to Jt(S). (b) For cP E Jt(S) describe the set E+(S*, cp) of representing measures for cp, i.e. the set of J.l E E + (S*) such that (1) holds. These questions seem to be very difficult due to their generality. Theorem 1.11 below gives a necessary and sufficient condition for Jt(S) = &J(S) under fairly general assumptions, and this theorem is used for special semi-groups in the following sections. Clearly E+(S*, cp) is a convex set in M +(S*). The moment function cp E Jt(S) will be called determinate if E + (S*, cp) is a one-point set, and indeterminate if it consists of more than one point. Similarly J.l E E + (S*) will be called determinate or indeterminate if the corresponding moment function cp given by (1) is so. By Theorem 4.2.5 we have
1.5. Theorem. Every exponentially bounded positive definite function is a determinate moment function. In general S* is not locally compact, however we have the following sufficient condition for this to be true:
1.6. Proposition. If· S is finitely generated then S* is homeomorphic and isomorphic with a closed subsemigroup of(C n , • )for a suitable n E ~, in particular S* is locally compact with a countable base for the topology. PROOF. By definition there exist finitely many elements e l ' ... , en E S such that every s E S has a representation of the form
(2)
en
n
For P E S* we define I(p) E by I(p) = (p(e 1 ), ••• , p(e » and it is clear that I: S* -.. Cn is a continuous homomorphism and one-to-one. To see that I is a homeomorphism and that I(S*) is closed in we consider a net (PcJe<EA from S* such that lime< I(pe<) exists in i.e. lime< pe«e j) exists for each j = 1, ... , n. If s E S is given by (2) we find for P E S* that
en
en,
np(ej)k n
p(s) =
j
,
j= 1
so lime< Pe«s) exists for each s E S. Defining p(s) := lime< Pe«s), s E S, it is clear that P E S*, I(p) = lime< I(pe<) and Pe< -.. P in S*. D For s E S we denote by Xs the function of S* into
Xs(p) = p(s)
for
pES*,
e given by (3)
~ 1.
181
Moment Functions
i.e. the point evaluation at s, and we denote by V(C) the complex vector space spanned by the functions Xs' s E S, i.e.
{'±ajXsJln ~ 1, ajEC, SjES,j = 1, ... , n}.
V(C):=
)=1
Notice that Xo == 1. Since Xs* = Xs we have that Vee) is stable by complex conjugation and V(C) = V(lR) + iV(lR) where V(lR) is the real vector space of real-valued functions in V(C). If the involution on S is the identical it is, of course, sufficient to consider the real vector space V(IR) spanned by the real functions Xs' s E S.
1.7. Proposition. Ij· S is finitely generated then V(lR) is an adapted space and Jt(S) is closed in Cs . The set E+(S*, qJ) oj'representing measuresfor qJ E .Yt(S) is a metrizable compact convex set in M +(S*) with the vague topology. PROOF. Let e 1, ... , en be elements of S such that every s E S has a representation (2). To show that V(lR) = V(lR) + - V(lR) + it suffices to verify that Re Xs' 1m Xs E V(lR)+ - V(lR)+ for s E S. But this follows from the equation
Xs
=
Xs
+ 1 + Xs+s*
- (1
+
Xs+s*),
since IXs I ~ 1 + Xs+s*· Next let s E S be given. We define n
j' = 1
+ Xs+s* +
L Xej+t"
(4)
j= 1
and will show that Xs/j' tends to zero at infinity on the locally compact space S*. Having proved this, it follows easily that V(IR) is an adapted space. Given c; > 0 there exists N E N such that 0 < xj(1 + x 2 ) ~ c; for x ~ N. The set K = {p
E
S*llp(s)1 ~ N, I p(ej) I ~ N,j = 1, ... , n}
is compact, cf. the proof of Proposition 1.6, and for p E S*\K we find
I~.(~;I = IP(S)I{ 1 + Ip(s)1 Ip(s)1
~
1 + Ip(s)1 2
N
1
In fact, if Ipes) I ~ Ip(ej) I > N.
+
N2 ~ c;
r
+ jtl IP(e)1 2 iflp(s)1 >
<
=
2
1
N,
c;
iflp(s)1 ~ N.
Nand p E S*\K there exists j
E
{1, ... , n} such that
Since S is finitely generated S is countable, so C S is metrizable. To prove that Jt(S) is closed we consider a sequence (qJn)n~ 1 from .Yt(S) converging
182
6, Positive Definite Functions and Moment Functions
pointwise to qJ E f!JJ(S). If /In E E + (S*) is a representing measure for have for S E S lim f 1p(s) 12 d/ln(p) n-oo
=
lim ({In(S
+ s*) =
({J(S
n-oo
({In
we
+ ~s*),
so the measures in the sequence (I p(s) 12 /In) have bounded total mass. Putting S = 0 we get by 2.4.6 and 2.4.10 that (/In) has a vaguely convergent subsequence (/lnp)p~ 1 with limit /l E M~ (S*), and therefore
f Ip(sW dJl(p)
~ li~~f f
2
Ip(s)1 dJlnp(P)
= cp(s + s*),
hence
(fIP(S)ldJl(P)Y
~Jl(S*) fIP(SWdJl(P) < 00,
so /l E E +(S*). Let S E S be fixed. Withj'as in (4) to jit, and since
(jitnp)p~ 1
converges vaguely
we even have lim fgj'd/l np
=
fgf dJl
p-oo
for g E CO(S*), cf. 2.4.4. If this is applied to the function g = Xs/fwe get
f
p(S) d/l(p)
=
lim fp(S) d/lnp(p)
=
p-oo
lim ({Jnp(s)
=
({J(s),
p-oo
hence ({J E .Yt(S). For ({JE.Yt(S) and /lEE+(S*, ((J) the linear functional L: V(C) -.. C defined by L(f) = j'd/l is independent of /l E E+(S*, ((J) and we have
J
L(Xs) = f p(s) dJl(p) = cp(s),
SE
S.
It follows that E + (S*, ((J) is the set of representing measures for the positive linear functional L restricted to the adapted space V(IR), hence vaguely compact by Proposition 2.4.8. We have already remarked that M +(S*) is D metrizable in the vague topology. In general the functions Xs' S E S need not be linearly independent in C S *, cf. Remark 1 in 4.6.9. The following result is inspired by an argument in Hewitt and Zuckermann (1956).
183
§1. Moment Functions
1.8. Proposition. For an abelian semigroup with involution S the following conditions are equivalent: (i) S* separates the points of S. (ii) The junctions Xs, S E S are linearly independent in C S *.
"(i) => (ii)" Suppose ajXs j = 0 on S*, i.e.
PROOF.
I
S l'
... ,
Sn E
S are distinct points and that
n
I
ajp(Sj)
=
pES*.
for
0
j= 1
In order to prove that a 1 = 0 we choose Pj E s* for j Pj(Sl) =f Pj(Sj), which is possible by (i), and define
=
2, ... , n such that
n
j'(s)
Il (Pj(s) -
=
Pj(Sj»(Pj(Sl) - Pj(Sj»-l,
SE
S.
j= 2
Thenj'belongs to the linear span of S* in CS andf(sl) j = 2, ... ,n, hence
= 1,f(sj) =
0 for
n
a1 =
I
ajf(sj) = O.
j= 1
D
The implication" (ii) => (i)" is trivial.
1.9. Suppose that the equivalent conditions of Proposition 1.8 are verified. Then C S and V(C) form a dual pair under the bilinear mapping
\ cP,
Jl
aj xSj )
=
Jl
ajcp(s),
(5)
which is well-defined because of 1.8(ii). The topology of pointwise convergence on C S is compatible with this duality. In fact, if L: C S -.. C is a continuous linear functional there exist a finite set D £; Sand c; > 0 such that IL(
OU(D,c;)
=
{
Then clearly n
=
0,
n
L(
= L(
I
aj
j= 1
where D = {Sl' ... , sn} and aj = L(1{sj})' Similarly the finest locally convex topology on V(C), cf. 1.1.9, is compatible with the duality. In V(C) we consider the convex cone L generated by the absolute squares of elements of V(C), i.e. L
=
{Ij~
2
1
+ ... + Ihl 2
1
n ~ 1,jjE V(C),j
= 1, ... , n}.
(6)
184
6, Positive Definite Functions and Moment Functions
For f = j~ + if~ E V(C) with j~, j~ E V(IR) we have I j'I 2 = j't + f~, so l: is equal to the convex cone of finite sums of squares of elements in V(IR). It is clear that l: £; V(IR) + . For a convex cone C £; V(C) we have introduced the closed convex cone Cl- £; C S by
cf. Remark 1.3.7. With this notation we can describe Jt(S) and [!P(S) as the following theorem shows.
1.10. Theorem. Let S be afinitely generated abelian semigroup with involution in which S* separates the points. Under the duality between CS and V(C) given by (5) we have:
(i) &(S) = l:l-; (ii) Jt(S) = (V(IR)+)l-.
PROOF. For
qJ E
CS andj' = Lj= 1
V(C) we have
CjXS j E n
Ij'1 2 >=
L
CjCk qJ(Sj
+ st),
j, k= 1
which shows that qJ E [!P(S) if and only if ~ 0 for all j'E V(C), and this establishes (i). For qJ E Jt(S) there exists a measure J1 E E + (S*) such that cp(s)
= f p(s) dJl(p)
for
s E S,
so for j'E V(IR)+ we get
hence
qJ E
=
f f(p) dJl(p)
~ 0,
(V(IR)+)l-. Conversely, if qJ E (V(IR)+)l- then the mapping j'H
is a C-linear functional on V(C) and a positive IR-linear functional on V(IR) which is adapted by Proposition 1.7. By Theorem 2.2.7 there exists J1 E M +(S*) such that V(IR) £; 2 1 (J1) and
for j'E V(IR), and it follows easily that J1 E E + (S*) and cp(s) =
i.e.
qJ E
Jt(S).
for
s E S,
D
185
§2. The One-Dimensional Moment Problem
1.11. Theorem. Let S be afinitely generated abelian semigroup with involution in which S* separates the points. The jollowing conditions are equivalent: (i) Jf(S) = &>(S); (ii) ! = V([R) +. Here the closure refers to any locally convex topology on V(C) compatible with the duality between CS and V(C) given by (5).
By Proposition 1.3.8 the closure! of the convex set I: is the same for all locally convex topologies on V(C) compatible with the duality. We may therefore assume that V(C) is equipped with the finest locally convex topology. Since this topology is finer than the topology of pointwise convergence on S*, we have that V([R) and V([R)+ are closed in V(C), hence PROOF.
!
~ V([R)+.
First assume that there existsj~ E V([R)+ \!. By the theorem of separation (1.2.3) applied to V([R) there exists a [R-linear functional L: V([R) -.. [R such that L(j~) < 0 ~ L(j') for allj'E!. Defining L: V(C) -.. C by L(j~ + if2) = L(j~) + iL(j~) for j'1' j~ E V([R) and qJ(s) = L(Xs) for SES, we see that L is a C-linear functional on V(C) such that L(f) = qJ,j') for j'E V(C). It follows that (qJ, Ij'1 2 ) = L(lj'1 2 ) ~ 0 for j'E V(C), hence qJEf!lJ(S). On the other hand, the inequality (qJ,j~) < 0 shows that qJ ¢ .Yf(S), and we have proved that "(i) => (ii)." Finally, if ! = V([R)+ then I:l- = (!)l- = (V([R)+)l-, and &>(S) = Jf(S) follows from Theorem 1.10. D
<
In the following sections, we shall see examples where Jf(S) = &>(S) and also where Jf(S) =t= &>(S).
1.12. Exercise (The Method of Moments). Let S be a finitely generated abelian semigroup with involution and let (Iln) be a sequence from E + (S*) with corresponding sequence of moment functions (qJn)' Prove that if qJ E Jf(S) is a determinate moment function with representing measure Il E E +(S*) and if lim n oo qJn(s) = qJ(s) for each s E S then lim n oo Iln = Il weakly. -+
-+
§2. The One-Dimensional Moment Problem In this section, we will apply the results of §1 to the semigroup S = (No, + ), the involution being the identical. For x E [R the mapping Px: No -.. [R given by px(n) = x n , n ~ 0, is an element of N6 and it is easy to see that the mapping x H Px is a topological semigroup isomorphism of (IR, .) onto N 6. The functions Xn' n E No, cf. (3) in §1, will be identified with the monomials x H x n, which are linearly independent, so V([R) is the vector space of polynomials in one variable with real coefficients. The following lemma shows that the cone I: (see (6) in §1) is equal to V(IR) + .
186
6. Positive Definite Functions and Moment Functions
°
2.1. Lemma. Let P be a polynomial with real coefficients and suppose p(x) ~ jor x E [R. Then there exist two polynomials with real coefficients PI' P2 such that p(x) = PI (X)2 + P2(X)2 jor x E [R. PROOF. Complex roots of P appear in conjugate pairs and real roots have an even multiplicity. Therefore we may write p(x) = r(x)r(x), x E [R for some polynomial r with complex coefficients. Writing r = PI + iP2' where PI' P2 are real polynomials, we get p(x) = PI (X)2 + P2(X)2 for x E [R. 0
By Theorem 1.11 it follows that &>(~o) = Jr(~o), so we have proved the following theorem of Hamburger, called the solution of Hamburger's moment problem, cf. Akhiezer (1965).
2.2. Theorem. A sequence s = (sn)n~o oj'real numbers is the sequence oj' moments oj' a measure /l E E + ([R) where E+(IR)
=
{JlE M +(IR)I flxnl dJl(x) < 00 for all n ~ o}
i.e. is of the form Sn =
f
x n dJl(x),
n
(1)
= 0, 1, ... ,
if'and only if's E &>(~o). The set E+([R, s) oj'measures /l E E+([R) satisjying (1) is a nonempty compact convex set in M +([R) in the vague topology. 2.3. Remark. A sequence s = (sn)n~O belongs to &>(~o) if and only if for every n ~ the matrix ((Sj+k)o~j,k~n) is positive definite. Using Theorem 3.1.16 it follows that s is strictly positive definite if and only if
°
n = 0, 1, 2, ....
for Let SE&>(~o) and /lEE+([R, s). For {co, c l , k L~=o CkX and have
j.~/jCkSj+k =
f
... ,
p(X)2
cn}
~
IR we define p(x)
=
dJl(x),
so s is strictly positive definite precisely when supp(/l) is an infinite set.
2.4. Let F be a closed subset of [R. The F -moment problem consists in characterizing the F -moment sequences, i.e. the real sequences s = (sn)n ~ 0 of the furm Sn
=
L
n
x dJl(x),
n
~
where /l E E + ([R) is supported by the closed set F.
0,
187
§2. The One-Dimensional Moment Problem
Hamburger's moment problem corresponds to F
°
= IR. The case of
F = [ -a, a], a ~ has been solved in Proposition 4.4.9, and in the special case a = 1 which we will call Haviland's moment problem, two different characterizations of [-1, 1]-moment sequences s have been found, (a) SE&,b(N o), (b) sE&'(N o) and s - E 2 sE&,(N o), the latter going back to Haviland (1936), cf. 1.2. The case of F = [0, 1] is Hausdorff's moment problem and the [0, 1]-moment sequences are precisely the completely monotone functions on the semigroup (No, +), see 4.6.11. We shall now consider the case F = [0, oo[ which is called Stieltjes' moment problem, solved by Stieltjes in 1894. The [0, 00 [-moment sequences are also called Stieltjes moment sequences.
2.5. Theorem. For a sequence S tions are equivalent:
=
(sn)n
~0
of'real numbers the following condi-
(i) s, E 1 SE &'(N o). (ii) t = (So, 0, Sl' 0, S2'· ..) E &'(N o). (iii) There exists J1 EM +([0, oo[) such that Sn
PROOF.
"(i) ~ (ii)" Let
tn
=
= sn/2
f"
n
n
x d/l(x),
for n even and t n
~
=
0.
°
for n odd. For
we then have n
I
n
n CjCktj+k
=
j,k= 0
I
CjCktj+k
+ L
[(n - 1 )/2]
[nI2]
=
I
CjCkt j + k
j,k= 0 j, k odd
j,k=O j, k even
C2pC2qSp+q
I
+
p,q=O
C 2p + 1 C 2q + 1 Sp+q+ 1
p,q=O
showing that t E 9(N o). "(ii) ~ (iii)" By Hamburger's theorem (2.2) there is t 2n + 1
=
f
x 2n + 1 d(J(x)
(J
E
~
°
E + (IR) such that
= 0,
n
~
0,
hence
n
~
0,
where J1 is the image measure of (J under the continuous mapping x ~ x 2 of IR into [0, 00 [. "(iii) ~ (i)" is clear since E 1 S is represented by the positive Radon measure xJ1. D
188
6, Positive Definite Functions and Moment Functions
In analogy with Theorem 2.2 we can give an integral representation of all functions ljJ E JV(N o) with a representing measure which is not necessarily unique.
2.6. Theorem. Ajilnction ljJ: No ~ lR is negative definite if'and only if'it has a representation oj' the form ljJ(n)
where a,
r
= a + bn - cn 2 + bE
(1 - x n
n(1 - x)) d/l(x),
-
J~\{l}
lR, c ~ 0 and Il E M +(lR\ {I}) satisfies
f.
(1 - X)2 dll(X) <
00,
0< Ix- 11 < 1
t-ll <; llx In d/l(x) < PROOF.
for all
00
n E No·
We note that
. 1- x lIm
n
n( 1 - x) 2 (1 - x)
x-1
=
_
(n )
for
2
n
~
2,
and it follows easily that any function ljJ of the above form is negative definite. If ljJ E JV(N o) we put a = ljJ(O) and ljJ - a is again negative definite. Without restriction we may therefore assume that ljJ(O) = O. By Theorem 3.2.2 there exist (possibly many) Ilt E E +(lR) such that
n E No,
for
t
> 0,
and Ilt is a probability measure. We find -1 t
foo
(1 - x n
n(1 - x)) dllt(X)
-
= -1 (1
- e-tt/J(n») - -n (1 - e-tt/l(l))
t
-00
t
and
o ~ t1 foo_ (l 00
-
n X )2
d/ll(x)
= t1 (1
- 2e- I I/J(n)
+ e- I/J(2n»), I
which shows that lim -1 t-O
t
foo
(1 - x n
-
n(1 - x)) dllt(X)
=
ljJ(n) - nljJ(l)
-00
and
o ~ lim -1 foo t-O
t
-00
(1 - Xn )2 dllrCX)
=
2ljJ(n) - ljJ(2n).
189
§2. The One-Dimensional Moment Problem
The last equation shows that there is a constant An depending on n E No such that for so by Proposition 2.4.6 there is a measure tending to zero such that .
1
2
lIm - (1 - x) Ilt- = tj
j-HXJ
(J
0 < t
~
1,
EM +(lR) and a sequence (t j )
vaguely.
(J
J
Since (1 - Xn )2( 1 - x) - 2 is continuous on IR we get
for allj'E CC(lR), and by Proposition 2.4.4 this holds even for j'E CO(lR). Choosing a function qJ E CC+(lR) with qJ(x) = 1 for x E [0,2] we now find for n ~ 1 -1 foo (1 - x n tj
-
n(1 - x)) dlltix)
-
00
where j(x)
Letting j
--+ 00
1 - xn :=
-
n( 1 - x)
(1 - x n )
2
(1 - qJ(X)) E CO(lR).
we therefore get
ljJ(n) - nljJ(1)
OOl - x n - n( 1 - x) 00 (l _ X)2
f
= _
f
+ oo -
00
(1-_- X )2 n
oo (1 - x n
f
= _ = -
00
x
1
-
(l -
j'(x) d(J(x)
n( 1 - x)) X)2
(~)a( {1}) + r
J~\{l}
da(x) (1 - x
n
-
n(1 - x)) d/l(x),
190
6. Positive Definite Functions and Moment Functions
with J1 = (1 - x) ljJ(n) = n(ljJ(l)
2(0-
IfR\ {1}), hence,
+ ta( {l}))
- tn 2 a( {l})
+
r
J~\{l}
(1 - x" - n(1 - x)) dJ1(x).
By Proposition 2.4.6 we have
n E N,
for all
D
which shows that J1 has the asserted properties.
2.7. Exercise. Show that if J1 E M +(fR) satisfies Jexp(ax 2 ) dJ1(x) < 00 for some a > 0 then J1 E E + (fR) and J1 is determinate. Hint: Show that if vEE + (fR) has the same moments as J1 then Jexp(ax 2 ) dv(x) < 00 and Jexp(ixy)dJ1(x) =
Jexp(ixy) dv(x) for all y E fR.
2.8. Exercise. Show that the normal distribution J1 = (21t) - 1/2 exp( - tx 2 ) dx belongs to E + (fR) and is determinate. The corresponding moment sequence is
So
= 1, S2n = 1 ·3 . 5 ... (2n - 1), S2n-l = 0 for n
~
1.
§3. The Multi-Dimensional Moment Problem In this section, we will apply the results of §1 to the semigroup (Nt, +), the involution being the identical. We use the notation from 4.4.8 so (fRk, .) ~ (Nt)* via the mapping x ~ Px' where px(n) = x n. The functions Xn' n E Nt, cf. (3) in §1, will be identified with the monomials x ~ x n , which are linearly independent, so V(fR) is the vector space of polynomials in k variables with real coefficients, which from now on will be denoted A(k). The convex cone L of finite sums of squares of polynomials, cf. (6) in §1, will be denoted L(k). In contrast to the one-dimensional case we have L(k) =t= A~) for k ~ 2 as the following shows:
3.1. Lemma. The polynomial p(x, y) = X2y2(X 2
+ y2
- 1)
+
1
belongs to A<;)\L(2). PROOF. It is easy to see that p ~ 0 and in fact its greatest lower bound is 26/27. Suppose that p = qi + ... + q~ for certain polynomials q 1, ... , qn' Since p(x,O) = p(O, y) = 1 we get that qi(X, O) and qi(O, y) must be constant for i = 1, ... , n. Therefore each qi can be written
191
§3. The Multi-Dimensional Moment Problem
where ai is a constant and hi is of degree at most 1. By comparing terms of the same degree we find n
I
i= I
ar =
n
2xy I aihi(x, y) = 0
1,
i= I
and
X2y2(X 2 + y2 - 1) = X2y2
n
I
hr(x, y),
i= I
hence n
x2
+ y2
- 1=
I
hr(x, y)
i= I
D
which is a contradiction.
The space A(k) is equipped with its finest locally convex topology. For d = 0, 1, ... let A~k) denote the subspace of polynomials of degree ~ d. Being a finite dimensional space A~k) has a canonical topology, cf. 1.3.9. 3.2. Theorem. The convex cone
L(k)
is closed in
A(k)
for k
~
1.
PROOF. First we show that L(k) n A~k) is closed in A~k) for any fixed d ~ O. Let us choose d + 1 different points ai' i = 1, ... , d + 1 in fR. The points a = (ail' ... , a ik ) E fRk, where iI' ... , i k E {I, ... , d + I} form a set H of (d + l)k points in fRk and they separate A~k): If two polynomials PI' P2 in A~k) agree in every point of H they are identical. Therefore the topology on A~k) of pointwise convergence on the set H is a Hausdorff topology compatible with the vector space structure, hence equal to the canonical topology. In particular, if a sequence (Pn) of polynomials from A~k) converges at each point a E H, then there is P E A~k) such that Pn converges to P in the canonical topology of A~k), which is equivalent to convergence of the coefficients. Suppose that P E L(k) n A~k) is written as
P=
pi + ... + P;,
where PI' ••• , P, E A(k). Writing a polynomial as the sum of its homogeneous parts, it is easy to see that the degree of P is an even number 2q ~ d and that the degree of each Pi is at most q. We can always assume that P is written as a sum of N squares where N = dim A~k). For ifr < N then zeros can be added, and if r > N there is a nonzero set of real numbers c l , ... , c, such that ciPr = O. By rearranging, if necessary, we may assume that Icll ~ Ic 2 1 ~ ••• ~ Ic,l. Then
Ir=l
i 2 P = I, ( 1 - -c ) Pi' i=2 c1
192
6. Positive Definite Functions and Moment Functions
which is a sum of r - 1 squares. Suppose now that (Pn) is a sequence from A~k) n L(k) converging in A~k) to P E A~k). We can find polynomials Pni E A~~~2]' n = 1,2, ... , i = 1, ... , N, such that N
2 Pn -_ "i...J Pni· i= 1
Since Pn(a) converges to pea) for every a in the finite set H there exists a constant C > 0 such that for
a E H,
n E N,
i
=
1, ... , N.
Consequently, there is an increasing sequence (n k ) in N such that lim Pnki(a) k-oo
exists for all a E Hand i = 1, ... , N. This implies the existence of polynomials Pt, ... , PN E A~~~2] such that lim k _ oo Pnk i = Pb i = 1, ... , N, and hence N
P=
Lt pr·
i=
To finish the proof we may either appeal to the theorem of Krein-Smulian as in Berg et al. (1979) or more simply make use of the following lemma:
3.3. Lemma. Let (En)n~ 1 be a strictly increasing sequence ojfinite dimensional subspaces oj' a vector space E such that E = u~ 1 En- A convex subset C ~ E is closed in the finest locally convex topology on E ~I'and only if'C n En is closed in the canonical topology on Enjor each n. PROOF. The "only if" part is clear. Next suppose that C is a convex set such that C n En is closed in En for each n and let a ~ C. We shall construct a neighbourhood of a disjoint with C. Without loss of generality we may assume a = O. Since C n E 1 is a closed convex subset of E 1 not containing 0 there is an absolutely convex and compact neighbourhood VI of 0 in E 1 such that VI n C = 0. Inductively, we now construct a sequence (~)n~2 where ~ is an absolutely convex and compact neighbourhood of 0 in En such that
~
n C = 0,
~
n En -
1
=
~-1
for
n
~
2.
(1)
Suppose that Vt, ... , ~-1 are constructed. There exists a closed and absolutely convex neighbourhood V~ of 0 in En such that V~ n En- 1 = ~- l' and by the separation theorem (1.2.3) applied to the disjoint sets ~-1 and C n En in En there exists a compact, absolutely convex neighbourhood V~ of ~-1 in En such that V~ n (C n En) = 0. Then ~ = V~ n V~ satisfies (1). Finally, V = 1 ~ is absolutely convex and absorbing in E, hence a neighbourhood of 0 in the finest locally convex topology on E, and V n
U:'=
C=0. Since 1.11 :
D L(k)
= ~(k) 9=
A~)
for k ~ 2 we get by specialization of Theorem
193
§3. The Multi-Dimensional Moment Problem
3.4. Theorem. For k ~ 2 there exist positive definitefunctions on Nt which are
not moment filnctions. For the case of the complex moment problem, cf. 4.4.11, where S = +, *) and (n, m)* = (m, n) the situation is analogous to the two-dimensional moment problem.
(~6,
3.5. Theorem. There exist positive definitefunctions on (N6, +, *) which are
not moment functions. PROOF. If S* is identified with C as in 4.4.11 then X(n, m) is identified with the function z ~ znzm and V(C) is the complex vector space spanned by these functions. Putting z = x + iy, (x, y) E ~2 we can identify V(C) (resp. V(~)) with the complex (resp. real) vector space of polynomials in x, y with complex (resp. real) coefficients. Under this identification the convex cone L, cf. (6) in §1, is equal to the cone L(2) above. By 3.2 and 3.1 we know that L(2) is closed and different from A<;) = V(~)+, hence Jf(S) =t= PJJ(S) by 1.11. D
°
If s = (sn)n ~ is a Stieltjes moment sequence then Eas is obviously positive definite for alfa E No, and this property implies on the other hand, that s is a Stieltjes moment sequence, cf. Theorem 2.5. It is natural to examine if this result extends to k dimensions.
3.6. Definition. A function qJ: Nt ~ ~ is called completely positive definite if EaqJ is positive definite for all a E Nt, and qJ is called a Stieltjes moment filnction if there exists J1 E M + ([0, 00 [k) such that
fx
=
n
dJ1(x)
nE
for
I\J~.
A Stieltjes moment function is completely positive definite. The converse is, however, false if k ~ 2 because the following generalization of Theorem 3.4 holds. 3.7. Theorem. For k ~ 2 there exist completely positive definite functions on Nt which are not moment functions. The proof of Theorem 3.7 depends on several lemmas. We introduce the convex cone C(k) ~ A(k) generated by the polynomials of the form Xap(X)2 where a E r\Jt, p E A(k). Since we can put even powers of the coordinates into the square we have C(k)
=
L
XaL(k).
ae{O,l}k
3.8. Lemma. Afunction qJ: Nt ~ ~ is completely positive definite
~
0
for all cnx n.
L qJ(n)cn if p(x) = L
p E C(k),
if and only if
194 PROOF.
6. Positive Definite Functions and Moment Functions
For a E N~, P E A(k) we have
<E qJ, p 2 >= Q
Q
and the assertion follows since E qJ E fJJ( N~) if and only if <E qJ, p 2 > ~ 0 for all P E A(k), cf. 1.10. D Q
3.9. Lemma. The convex cone
Q
C(k)
is closed in
A(k).
PROOF. We use the same idea of proof as in Theorem 3.2, so it suffices to prove that C(k) n A~k) is closed in A~k) for every d ~ O. The point set H is chosen so that each ai ~ 1. Every polynomial P E C(k) n A~k) can be written
p(x) =
L
xQp(a, x),
Qe{O, l}k
where pea, .) E L(k), and we see that the degree of pea, .) is at most d (a l + ... + ak ). If (Pn) is a sequence from C(k) n A~k) converging in A~k) to P E A~k), the choice of H implies that each sequence (Pn(a, . )) is bounded on H, and we conclude as in 3.2 that a subsequence Pnk(a, .) converges to pea, .) E L(k) for each a E {O, l}k and finally that P = L xap(a, .) E C(k). D We will next show that A
~ 0
for
P E C(k)
and
showing by Lemma 3.8 that qJ is completely positive definite. The inequality < 0 is inconsistent with qJ being a moment sequence. It is plausible that the polynomial P of Lemma 3.1 does not belong to C(2), but we have not been able to decide it. We will instead modify an example of Robinson (1969) in order to construct a polynomial Po E A<;)\C(2).
3.10. Lemma. If a polynomial P E A~2) is zero at the eight points of {-1, 0, 1}2\{(O, O)}, then automatically p(O, 0) PROOF.
= O.
We may write 3
p(x, y) =
L Pi(y)X i, i=O
where Pi E A~l~i' By assumption the polynomials p(l, y)
= Po(y) + Pl (y) + P2(Y) + P3(Y),
p( -1, y)
= Po(y) - Pl (y) + P2(Y) - P3(Y),
195
§3. The Multi-Dimensional Moment Problem
have zeros at y = -1, 0, 1, and so have half of their sum Po(Y) + P2(y) and half of their difference PI (y) + P3(Y). Since PI + P3 E A~l) we must have PI + P3 = 0, and we may write Po(Y) + P2(Y) = ay(y 2 - 1). However, Po(Y) = p(O,y) has zeros at y = ±1, hence P2(±I) = o. Since P2EAil) we get P2 = O. The polynomial P3 is a constant b E [R, hence PI (y) = - band finally p(x, y) = ay(y2 - 1) + bx(x 2 - 1), which shows p(O, 0) =
o.
D
3.11. Lemma. Let p(x, y) be the polynomial of'Robinson p(x, y) = X2(X 2 - 1)2
+ y2(y2
- 1)2 _ (x 2 - 1)(y2 - 1)(x 2
+ y2
- 1).
Then q(x, y):= p(x - 2, y - 2) E A<;)\C(2). PROOF. To see that p(x, y) ~ 0 it suffices to consider x ~ 0, y ~ O. This domain is divided by the unit circle and the lines x = 1 and y = 1 in five domains denoted D 1 , ... , D s in Figure 3. The expression for P shows that p(x, y) ~ 0 in D 1 , D2 and D3 , but P can be written
p(x, y)
=
(x 2 - y2)2(X 2 + y2 - 1)
+ (x 2 -
1)(y2 - 1)
showing that p(x, y) ~ 0 in D4 and D s . Assume that q E C(2), i.e. that q may be written q(x, y)
= ql(X, y) + xq2(X, y) + yq3(X, y) + Xyq4(X, y),
where qi E L(2), hence
+ (x + 2)P2(X, y) + (y + 2)P3(X, y) + (x + 2)(y + 2)P4(X, y), qi(X + 2, y + 2) E L(2). Robinson's polynomial P has zeros
p(x, y) = PI(X, y)
where Pi(X, y) = at {I, 0, -1}2\{(0, O)} and so have Pi' i = 1, ... ,4. However, PI is of degree of most 6, and P2' P3 and P4 are of degree at most 4. Furthermore, each Pi is of the form Pi = prj so each Pij is of degree at most 3 and has zeros at {1,0, 1}2\{(0, O)}. By Lemma 3.10 we see that Pij(O, 0) = 0 for all i,j, hence p(O, 0) = 0, which is a contradiction since p(O, 0) = 1. D
I
y
x
Figure 3
196
6. Positive Definite Functions and Moment Functions
It is worth noticing that even if we replace the cone &>(Ni) by the smaller cone ~(N~) of moment functions we have no generalization of Theorem 2.5.
3.12. Theorem. For k ~ 2 there exists a jilnction qJ: N~ ~ fR such that E aqJ E ~(N~) jor all a E N~, but qJ is not a Stieltjes moment function.
Let F = [0, OO[k and let A~)(F) be the set of polynomials P E A(k) which are nonnegative on F. Let D(k) be the convex cone in A(k) generated by the polynomials xap(x) where a E N~, p E A~). We have PROOF.
D(k)
L
=
xaA~)
ae{O, l}k
and C(k) £; D(k) £; A~)(F). Using the same idea of proof as in Lemma 3.9 it follows that D(k) is closed in A(k). Once we have constructed a polynomial Po E A~)(F)\D(k), it can be separated from D(k) by a (continuous) linear functional L: A (k) ~ fR. There is consequently a function qJ: N~ ~ fR such that ~ 0
for all
p E D(k)
and
in particular <EaqJ,P>~O
forall
pEA~),
showing by Theorem 1.10 that EaqJE~(N~) for all aEN~. However, < 0 is inconsistent with qJ being a Stieltjes moment function. The existence of Po E A~)(F)\D(k) is established in the following lemma for k = 2.
D 3.13. Lemma. The polynomial
= xy(x + y A<';)(F)\D(2) where F = [0, 00[2. Po(x, y)
belongs to
PROOF. It is easy to see that Po(x, y) assume that Po E D(2) we have
Po(x, y)
~
- 1)
0 for x, y
+ ~
1
0, cf. Lemma 3.1. If we
= so(x, y) + xs 1 (x, y) + y S 2(X, y) + Xy S 3(X, y),
where Si E A<';). Each Si must be of even degree ~ 2 and S3 must be a nonnegative constant. A nonnegative polynomial of degree 2 in any number of variables is a sum of squares of polynomials as is known from the theory of quadratic forms. It follows that Si E l:(2), hence Si(X 2 , y2) E l:(2) for i = 0, 1,2,3, but then Po(x 2, y2) E l:(2), in contradiction with Lemma 3.1. D
3.14. Exercise. Show that a moment function
is strictly positive definite if and only if J1 is not supported by an algebraic variety, i.e. a set of the form p - 1 ( {O}) for pEA(k)\ {O}. Show that the set of
197
§3. The Multi-Dimensional Moment Problem
strictly positive definite functions on N~ is dense in 9(N~), and conclude that there are strictly positive definite functions which are not moment functions, when k ~ 2.
3.15. Exercise. Let qJ: N~
-+
IR be strictly positive definite. Show that
:=
is a scalar product on the space A~) of polynomials with complex coefficients. Let H be the Hilbert space completion of the pre-Hilbert space (A~), )lp) and let T;: A~) -+ A~) £; H be the densely defined operator T;p(x) = xip(x), P E A~), i = 1, ... , k. Show that each T; is a symmetric operator and that T; 1] = 1] T;. However, if qJ ¢ ..tf(N~), then it is impossible to find self-adjoint extensions ~ of T; such that f 1 , .•. , f" commute in the sense of spectral theory. Hint: If' P is the spectral measure on IRk of a commuting family f 1 , ... , t of self-adjoint extensions, then J1 =
<.,.
cp(n)
=
f
n
x df.l(x),
cf. Fuglede (1983).
3.16. Exercise (Haviland 1935, 1936). Let qJ: N~ -+ IR and let L: A(k) -+ IR be the linear functional determined by L(xn) = qJ(n), n E N~. Show that qJ is the moment function for some J1 E M + (IRk) supported by a closed subset F £; IRk if and only if L(p) ~ for all p E A(k) which are nonnegative on F. Hint: Use Exercise 2.2.12.
°
3.17. Exercise. Let qJ: N~
-+
IR be given. Show that qJ E 9(N~) and
(I - (Ei
+ ... + E~))qJ E 9(N~)
if and only if qJ is a moment function for a measure J1 E M + (IRk) supported by the unit ball {XE IRk IIlxll ~ I} where E j = E ej and e 1 = (l,O, ... ,O), ... ,ek = (0, ... ,0,1). Hint: Let H be the RKHS associated with qJ and H o the dense subspace spanned by the functions qJn, n E N~, defined by qJn(m) = qJ(n + m)~ Show that each E j is a bounded self-adjoint operator on Hand that E 1 , ••• , E k commute. Show that J1 =
3.18. Exercise. Let S = (NbOO ), +) be the countable direct sum of No consisting of all sequences n = (n 1 , n2 , •••) of nonnegative integers which become eventually zero. Show that S* is isomorphic and homeomorphic with (IR~, .). Show that, in the terminology of §1, L is closed in V(IR) in its finest locally convex topology and a proper subset of V(IR)+. Conclude that 9(S)
=f
..tf(S).
Show that (N, .) is isomorphic with (NbOO ), ..tf(N, .).
+) and conclude that 9(N, .) =f
198
6, Positive Definite Functions and Moment Functions
§4. The Two-Sided Moment Problem This problem consists in characterizing the two-sided moment sequences, i.e. the functions qJ: 7L -+ lR of the form
qJ(n) =
f.
x n dj1(x)
for
n E 7L,
(1)
~\{O}
where j1 E M + (lR\ {O}) is such that x n is j1-integrable for all n E 7L. The problem is called the strong Hamburger moment problem in Jones et al. (1984). We shall solve it by applying the preceding theory to the semigroup (7L, +) with the identical involution. For x E lR\ {OJ the function Px: 7L -+ lR given by px(n) = x n, n E 7L, is an element of 7L*, and it is easy to see that the mapping x 1-+ Px is a topological semigroup isomorphism of (lR\ {OJ, .) onto 7L*. Under this isomorphism Z corresponds to ({ -1, I}, .) so
9 b (7L)
{n 1-+ a( _1)n
=
+ bla, b ~ OJ.
The semigroup (7L, +) is generated by { -1, I}. The functions Xn' n E 7L, cf. (3) in §1, are the functions x 1-+ x n on lR\ {OJ, which are linearly independent, so V(lR) is the subspace of C(lR\ {O}) spanned by x n , n E 7L. It follows that V(lR) is the set of functions on lR\ {OJ of the form p(x)/x n, where n ~ 0 and p is a polynomial with real coefficients. A representation j'(x) = p(x)/x n of j'E V(lR) is of course not unique, and we may always assume that n is even. If j'E V(lR)+ we may write f(x) = p(x)/x 2n , where p(x) ~ 0 for x E lR\ {OJ, hence for all x E lR, and by Lemma 2.1 we may write p = pi + p~ so that
=
f(x)
(p~~X)r + (p~~X)r·
With the terminology of (6) in §1 we then have L = V(lR)+, hence 9(7L) =
..tf(7L), and we have proved the following result: 4.1. Theorem. Ajunction qJ: 7L -+ lR is a two-sided momentfunction if'and only if' qJE9(7L). The set E+(lR\{O}, qJ) of representing measures is a nonempty compact convex set in M + (lR\ {O}) with the vague topology. 4.2. Remark. A function qJ E lR~ belongs to 9(7L) if and only if for every n ~ 0 the (2n + 1) x (2n + 1) matrix (qJ(j
+ k»,
j,kE{-n, -n
+
1, ... ,0, 1, ... ,n}
is positive definite, and qJ is strictly positive definite if and only if these matrices are strictly positive definite. Introducing the determinants
H k(n)
-
qJ(n
+
qJ(n
+ k)
1)
qJ(n qJ(n
+ 1) + 2)
qJ(n + k) qJ(n + k + 1)
-
qJ(n + k
+ 1)
qJ(n
+ 2k)
199
§4. The Two-Sided Moment Problem
for n E 7L, k = 0, 1, ... it follows by an easy modification of Theorem 3.1.16 that qJ is strictly positive definite if and only if H~~2n)
> 0,
for
n
Let qJ E 9(7L) and p, E E+(~\ {O}, qJ). For {c- n , we define
= 0,
1, ....
C- n +1' . . . ' Co, C 1 , ••• , Cn }
£; ~
n
p(x) =
L
CkX
k
k= -n
and have j,kt_nCjCkCP(j
+
k)
=
f
p(X)2 dj1(x),
and it follows that qJ is strictly positive definite precisely when supp(p,) is an infinite set. We shall now discuss the possibility of extending a positive definite sequence s: No -+ ~ to a positive definite function qJ: 7L --+ ~. 4.3. Theorem. Let s: No -+ ~ be a positive definite sequence and let E+(~, s) be the set oj' representing measures. Then s can be extended to a positive definite function qJ: 7L -+ ~ if and only if there exists a measure p, E E+(~, s) satisfying
p,({0}) = 0
r -,l,n dp,(x) <
and
J~\{O}
00
for
n
~
(2)
1.
X
If such a measure p, exists then cp(n)
r
=
J~\{O}
x n dj1(x),
(3)
nElL
is a positive definite extension oj's. PROOF. If qJ is a positive definite extension of s there is a measure 1L E E + (~\ {O}, qJ) such that
cp(n)
Since 1L(~\ {O}) setting
=
= qJ(O) <
r
J~\{O}
00
x n dj1(x)
for
n E 7L.
we can extend p, to a Radon measure fi on
fi(B) = lL(B\ {O})
for
~
by
B E c1l(~),
and it is clear that fi E E + (~, s) and that fi satisfies (2). On the other hand if there exists a measure 1L E E + (~, s) satisfying (2), then qJ defined by (3) is a positive definite extension of s. 0 If s is determinate we at once get the following:
200
6, Positive Definite Functions and Moment Functions
4.4. Corollary. Let s: No -+ lR be a determinate positive definite sequence with representing measure J1. Then s can be extended to a positive definite function qJ: 7L. -+ lR if and only if J1 satisfies (2). If this is the case then the extension is uniquely determined, it is given by (3) and is determinate. Using well-known results from the theory of the one-dimensional moment problem in the indeterminate case we can prove:
4.5. Theorem. Let s: No -+ lR be an indeterminate positive definite sequence. The set F of'positive definitefunctions qJ: 7L. -+ lR extending s is a closed convex set in 9(7L.), and there is a continuous injection t 1-+ qJt of lR into F such that qJt is determinate for each t E lR. PROOF.
Without loss of generality we may assume that s(O) = 1. It is clear that
F is a closed convex set. To see the last assertion we remark that any representing measure J1 E E + (lR, s) for which 0 ¢ supp(J1), satisfies (2) and gives an extension of s by the formula (3). That there are such measures in E + (lR, s) follows from the Nevanlinna parametrization of E+(lR, s), cf. Akhiezer (1965), according to which the Nevanlinna extremal measures at E E+(lR, s) for t E lR u {oo} satisfy
dac<X) = _ A(z)t - C(z) z B(z)t - D(z)
fx -
for
Z E
C\lR,
(4)
where A, B, C, D are certain entire functions and a l is a discrete measure with mass at the zeros of the entire function Z 1-+ B(z)t - D(z). (For t = 00 this function shall be interpreted as B(z).) It is known that A(O) = D(O) = 0 and - B(O) = C(O) = 1. It follows that a l has mass at 0 if and only if t = 0, hence that 0 ¢ supp(a t ) for t E (lR u {oo })\{O}. If we let z tend to zero through imaginary values we get from (4) that
daC<X) =
f
_!
x
t'
t E(lR u {00})\{0},
and it follows that
cpln):=
r
J~\{O}
x" dO'l/lx),
n E 7L,
belongs to F for each t E lR. Since qJt( - 1) = - t the mapping t 1-+ qJl is one-to-one. The continuity of the mapping t 1-+ qJln) is clear for n ~ 0, simply because the function is constant s(n), and follows for n ~ -1 by Proposition 2.4.4. In fact, for fixed to E lR there exists 8 > 0 such that supp(a l/J £; lR\] - 8,8[ for t E ]t o - 8, to + 8[, and then
x n E CO(lR\] -
8,
e[)
whenn ~ -1. We shall finally prove that qJt E 9(7L.) is determinate. We first establish that (1jx 2 )a l/l generates a determinate Hamburger moment sequence, and for
201
§4. The Two-Sided Moment Problem
this it suffices to see that
inf{f p(X)2
:2
= I} = 0,
dO"l/t(x)!p polynomial, p(O)
(5)
cf. Akhiezer (1965, pp. 49 and 60). Now, l/x E 2 2(0'1/t) and the set of polynomials is dense in 2 2(0' lit) because a 1/t is Nevanlinna extremal, so for e > 0 there is a polynomial q such that e>
f(~ -
q(x)
r
dO" l/t(X)
=
f
(l - xq(X»2 :2 dO" l/t(X),
and (5) follows because p(x) = 1 - xq(x) is a polynomial verifying p(O) = 1. To see that CP, is determinate we consider Jl E E + (lR\ {O}, CPt) and will prove that Jl = a Ill' By assumption
r
J~\{O}
x n d/1(x) =
r
J~\{O}
x n dO" l/t(X)
for
n E"Z,
in particular for
n
~
and we conclude that (1/x 2)Jl = (1/X 2)0'1/n whence Jl = a lit.
0,
D
4.6. Example. We will give an example of an indeterminate two-sided moment sequence. The log-normal distribution has the following density function on ]0, 00 [
which generates cP E 9("Z) given as q>(k)
= f'xo_1_xk-le-O/2)(IOgx)2dx,
Jo
kE"Z.
Jhc
Substituting t = log x yields cp(k) = _1_
Jhc
foo
e-O/2)t2+kt dt = e(/2 )k 2 _1_
Jhc
-00
foo
e- O/2 )(t-k)2 dt = e(1/ 2)k 2.
-00
(This should be compared with the fact that ljJ(k) = - k 2 is negative definite on ("Z, +) in the semigroup sense.) That cP is indeterminate follows from the fact that all the density functions da(x) = d(x)(l
+ a sin(2n
log x)),
XE
]0,
00[,
202
6. Positive Definite Functions and Moment Functions
where a E [ -1, 1], generate the same moment sequence
rooxk sin(2n log x) d(x) dx = _1_ Jo ~
= e(l/2)k qJ E lR~
4.7. Exercise. A function
foo
2
e-(1/2)t +kt
qJ
because for k E 7L
sin(2nt) dt
-00
2
_1_
~
foo
e-(1/2)y2
sin(2ny) dy
=
O.
-00
is called a two-sided Stieltjes moment 00 [) such that
sequence if there is a measure J1 E M + (]O,
cp(k) =
1 00
x k df.l(x)
for
k E 7L.
Show that the following conditions are equivalent for
qJ E lR~:
(i) qJ is a two-sided Stieltjes moment sequence. (ii) f: 7L -+ lR defined by
f(k) = {cp(~).
keven,
k odd,
0, (iii)
belongs to 9(7L). qJ, E 1 qJ E 9(7L).
4.8. Exercise. Let S = (7L k , +). Show that S* is isomorphic and homeomorphic with «lR\ {O})k, .) under the mapping x ~ Px where px(n) = x" = x = (Xl' ... ,xk) E (lR\ {O})k and n = (n l , ••. ,nk) E 7L k. Show that, in the terminology of §1, L is closed in V(lR) and a proper subset of V(lR)+ when k ~ 2. Conclude that there exist functions qJ E 9(7L k )\Yf(7L k ) when k ~ 2. X~l ••• X~k for
4.9. Exercise. Show that a function 1jJ: 7L semigroup sense on (7L,
r/J(n)
=
a
+ bn
-+
lR is negative definite in the representation of the form
+) if and only if it has a
- cn 2
+
r
J~\{O, l}
(1 - x n
n(1 - x)) df.l(x),
-
nEZ,
where a, b E lR, c ~ 0 and J1 E M + (lR\ {O, 1}) satisfies
r J
O
for all n ~ O.
(1 -
X)2
df.l(x) <
00,
f.
-
1
0
dJ1(x) <
00,
203
§5. Perfect Semigroups
4.10. Exercise. Let S = (Z(OO), +) be the countable direct sum of Z. Show that 9(S) =F £(S). Show that S is isomorphic with (0+ \ {O}, .). Hint: See Exercise 3.18.
§5. Perfect Semigroups 5.1. Definition. An abelian semigroup (S, +, *) with involution is called perfect if every qJ E 9(S) is a determinate moment function. Every semigroup with involution for which 9(S) = 9 b(S) is perfect, cf. 1.5, so, in particular, finite semigroups, idempotent semigroups and abelian groups with the involution x* = -x are perfect semigroups. Abelian groups with the identical involution need not be perfect as the example of (Z, +) shows, cf. 4.6. Although 9(N o) = £(N o) the semigroup (No, +) is not perfect since there exist indeterminate moment sequences. By the results of §§3 and 4 (N~, +) and (Zk, +) are not perfect. Further examples of perfect semigroups are given below. Let S be a perfect semigroup. The set of signed measures of the form J11 - J12 + i(J13 - J14) with J1 j E E + (S*) will be denoted E(S*). Extending the definition of convolution of measures in E + (S*) to E(S*) by bilinearity, £(S*) becomes an algebra. The generalized Laplace transformation defined in 4.2.10 can be extended from MC(S*) to E(S*) in the following way: For J1 E £(S*) we denote by fi, the function
(1(s) =
f
p(s) dJ1(p),
s E S,
s*
and the properties (i), (ii) and (iii) of 4.2.10 remain valid: 5.2. Proposition. Let S be a perfect semigroup. The generalized Laplace transformation J1 r--+ (1 is an injective algebra homomorphism of' E(S*) into C S . We restrict attention to the injectivity, the rest of the proposition being obvious. Suppose that
PROOF.
for all
s E S,
where J1j E E +(S*) for j = 1, ... , 4. Since (1j(s*) = (1j(s), j = 1, ... , 4 we also have that for all
SE
S,
for
SE
S.
hence and
Since S is perfect we conclude that J11 = J12 because they represent the same D moment function. Similarly J13 = J14.
204
6. Positive Definite Functions and Moment Functions
5.3. Remark. Let (CP(%)(%EA be a net from fdIJ(S) converging pointwise to a function cp E 9(S) on a perfect semigroup, and let (J1(%)(%EA and J1 be the uniquely determined measures in £+(S*) such that {i(% = qJ(%, {i = cp. Then lim (%
f. f
dJ1(%
S·
=
f. f
dJ1
S·
for all j'E V(C), the subspace of C s * spanned by the functions Xs' S E S. It is natural to believe that lim(% J1(% = J1 weakly, but we have not been able to settle this question. However, if S in addition is finitely generated, or if more generally S* is locally compact and V(lR) is an adapted space, then lim(% J1(% = J1 vaguely, and hence weakly, by 2.4.2. In fact, a modification of the proof in Proposition 1.7 will show that any vague accumulation point (J of the net (J1(%)(%EA satisfies 8 = cp, and since cp has a unique representing measure by definition, every vague accumulation point of (J1(%)(% E A is equal to J1, hence lim(% J1(% = J1 vaguely. This remark should be compared with Exercise 1.12. As an application of Theorem 2.1.10 on Radon bimeasures we can prove
5.4. Theorem. The product S x T oj'two perject semigroups Sand T is perject. PROOF. For (p, ,) E S* x T* the mapping (s, t) ~ p(s)'(t) is a semicharacter on S x T, and every semicharacter ~ on S x T is of this form with p(s) = ~(s, 0) and '(t) = ~(O, t). This makes it possible to identify (S x T)* with S* x T*, cf. 4.2.13. Let cp E 9(S x T). For each t E T we have
cp(., t*
+ t) E 9(S),
(1)
cp(., t* + t) + cp(., 0) - cp(., t*) - cp(., t) E 9(S), cp( ., t*
+ t) + cp(., 0)
(2)
+ icp(·, t) E fdIJ(S). + S2' t* + t) = (Sl' t)* + (S2' t).
- icp(·, t*)
(3)
Here (1) follows simply because (sT To see (2) and (3) let {Sl' .. " sn} 5; Sand {c l , ... , cn} 5; C be given. By expressing the defining property of cp E fdIJ(S x T) for the sets {(Sl' t), ... , (sn, t), (Sl' 0), ... , (sn, O)} and find
{Cl' ... ,
Cn'
-C l , ... ,
-cn}
5;
5;
S x T
C (resp. {c l , ... , Cn' ic l , ... , ic n}
5;
n
L CjCk(CP(sj + Sk' t* + t) + cp(sj + Sk' 0) -
cp(sj
+ Sk' t*)
icp(sj
+ Sk' t*)
j,k= 1
(resp. n
L CjCk(CP(sj + Sk' t* + t) + cp(sj + Sk' 0) j,k= 1
+ icp(sj + Sk' t»
~ 0),
C) we
205
§5. Perfect Semigroups
which shows (2) and (3). For t E T we denote by at, Lt and Vt the uniquely determined representing measures in E + (8*) for the functions in (1), (2) and (3), and we define J1t := !«at + ao - L t ) + i(at + ao - vt))· Then /It is the unique signed measure in E(8*) such that
cp(s, t}
=
r p(s} d/l,(p}
(s, t) E 8 x T.
for
Js *
t n}
~
T
8 and {d1, ... ,dm}
~
C
The mapping t r--+ /It of T into E(8*) is positive definite, i.e. for {t l' and {c 1 , ••• , cn} ~ C, we have
... ,
n
L
CjCk/lt'+tk E E+(S*).
j,k= 1
To see this we consider the function F: 8
F(s} =
Jrs*P(S}d(
f
j, k
=1
CjCk/lt,+lk)
-+
C defined by
=
i
j, k
=1
and claim that FE9(8).ln fact, if {S1, ... ,sm} we find m
L
m
dpdqF(s:
+ Sq) = L
p, q = 1
CjCkCP(S, tj
~
+ tk }
n
L
dpcjdqc k qJ(sp, t j)*
+ (Sq,
t k )) ~
o.
p, q = 1 j, k = 1
Since 8 is perfect there is a measure /l E E + (8*) such that
F(s}
=
r p(s} d/l(p}
Js *
for
S
E8
and by Proposition 5.2 it follows that n
L
CjCk/lt'+tk
= /l E E + (8*).
j,k= 1
In particular, for any set A E 81(8*) the function t and is therefore of the form for
r--+
t
/It(A) belongs to f!JJ(T)
ET
for a uniquely determined measure LA E E + (T*). The mapping A f1l(8*) into E+(T*) is a "Radon vector measure", i.e.
r--+ LA
(i) Le> = 0; (ii) L uf An = 1 LAn when (An) is a sequence of disjoint sets in 81(8*); (iii) LA = SUP{LK IKE $'(8*), K ~ A} for A E 81(8*).
L:'=
Here (i) is clear. Concerning (ii) we have for t E T:
Lyt} dr ur An(O /It(QAn) = n~/t(An} n~l Lyt} drAn(O, =
=
of
206
6. Positive Definite Functions and Moment Functions
in particular for t = 0
n~lrAn(T*) = 1l0(Q An) < and for t replaced by t*
(4)
00
+t
Jl {.IW)1 2drAn(O = 111·+,(9 An) <
(5)
00.
L::=
Equation (4) shows that I LAn is a Radon measure, cf. Exercise 2.1.28, and since "(t)1 ~ 1 + 1'(t)1 2 for 'E T*, (4) and (5) imply that LAnE E +(T*) and (ii) follows by Proposition 5.2. From (i) and (ii) we see that L is increasing: If AI' A 2 Ef1l(S*), Al £; A 2 then LAl ~ LA2. For each A E PA(S*) the net {LK IKE ff(S*), K £; A} is increasing if the index set is ordered by inclusion, so by Exercise 2.1.29
L::=I
!A:=SUP{LKIKEf(S*),K is a Radon measure on T* and ! A t E T we find
f
'(t) d!A(') = lim
T*
K
=
f
f
~
A}
£;
LA so in particular ! A E E + ( T*). For
'(t) dLK(() = lim J-lr(K) = J-l,(A)
T*
K
'(t) dL A('),
T*
where the limit is along the index set {K E ff(S*), K £; A}, showing by 5.2 that LA = !A for every A E ~(S*). The function <1>: PA(S*) x gB(T*) -+ [0, oo[ defined by (A, B) = LA(B)
for
A E PA(S*),
BE f1l(T*)
is a Radon bimeasure by (i), (ii) and (iii), so Theorem 2.1.10 implies the existence of a Radon measure K on S* x T* such that A E f1l(S*),
for
BE f1l(T*),
where for functions j': S* -+ C, g: T* -+ C the tensor product j' ® g denotes the function on S* x T* defined by j' ® g(p, ,) = j'(p)g(') for (p, () E S* x T*. By standard arguments from integration theory we then get
r
Js*
lA®hdK= x
T*
J
hdLA
T*
for A E PA(S*) and any LA-integrable function h: T*
-+
C, in particular for
t E T.
207
§5. Perfect Semigroups
Replacing t by t*
+ t and using similar standard arguments we find
fs*g dJ1.,*+1 = fs*
g(p) I'(t) /2 dK(p, ,) x
T*
for t E T and any Borel measurable function g: s*
---. [0, 00], in particular
showing that K E E +(S* x T*). Finally we get
fs*
p(s)'(t) dK(p, ,) = x
T*
f
p(s) dJ1.I(P) = qJ(s, t)
for
(s, t)
E
S x T
s*
showing that qJ is a moment function with representing measure K. We still have to show the uniqueness of the representing measure. Suppose K 1, K 2 E E+(S* x T*) satisfy for where X(s,n:
s*
(s, t) E S x T,
x T* ---. C is given by for
pES*,
'ET*.
Denoting 1r 1 and 1r2 the projections of s* x T* onto s* and T*, the image measures (X(O,nKlyrl and (X(O,nK2yrl belong to E(S*) and have the same generalized Laplace transform. (The notion of image measures is extended in an obvious way from positive finite Radon measures to finite signed Radon measures.) They are therefore equal and we find for
tET
and
A E fJ6(S*),
showing that the image measures (1 A x T* K 1)1t 2 and (1 A x T* K 2)1t 2 belong to E + (T*) and represent the same function in f!J>(T). This yields K1 (A x B) = K2 (A x B) for all A E fJ6(S*), BE fJ6(T*) and hence K1 = K2' 0 Another way to preserve perfectness is given by the following result.
5.5. Theorem. Let Sand T be *-semigroups and let h: S ---. T be a surjective *-homomorphism. Then if'S is perfect, so is T. PROOF. The dual homomorphism h*: T* ---. S*, defined by h*(r):= r h is injective and its image F = h*(T*), consisting of all p E S* which can be factorized over h, is a closed subset of S*. Furthermore h* is a homeomorphism from T* to F. 0
208
6. Positive Definite Functions and Moment Functions
Let (T) be given, then
(S) has the unique integral representation cp(h(s))
f p(s) dj1.(p), Js *
=
s E S.
We shall prove that J-l is concentrated on F. For a, bE S we put
Ga,b:= {pES*lp(a) =F pCb)}; these sets Ga, b are open and
Fe =
U
Ga,b'
a,bES
h(a) = h(b) hence it suffices to show that J-l(Ga,b) = 0 if h(a) = h(b). But under this assumption we get for s E S
f
p(s)p(a) dj1.(p)
=
f
p(s
+ a) dj1.(p) =
=
=
f
+ h(a)) =
cp(h(s
+ a))
+ h(b)) =
qJ(h(s
+ b))
p(s)p(b) dj1.(p),
and therefore
p(a)J-l = p(b)J-l by Proposition 5.2, i.e. Re p(a)J-l = Re p(b)J-l and 1m p(a)J-l = 1m p(b)J-l. This implies
f
[Re p(b) - Re p(a)] dj1.
J
=0
{Re p(b) > Re p(a)}
so that J-l( {p IRe pCb) > Re p(a)}) = 0 and similarly J-l( {p IRe pCb) < Re p(a)}) = 0 as well as the corresponding result for the imaginary part, i.e. we have J-l(Ga,b) = O. Let v E M:(T*) be the image of J-l under (h*)-l, then J-l = vh* and for t = h(S)E T we get
=
{.(h*(r))(s) dv(r)
=
f
=
f
Jh*(T*)
pes) dvh*(p)
r(h(s)) dv(r)
r(t) dv( r),
which shows existence of a representing measure for
209
§5, Perfect Semigroups
We will now give an important example of a perfect semigroup.
5.6. Proposition. The semigroup S = (0+, +) oj' nonnegative rational numbers is perject. Every qJ E (!jJ(S) has a unique representation q>(r) where a PROOF.
=
+
al{o}(r)
f:oo e
rx
dJ-l(x)
0, and J-l E M + (IR) is such that erx is J-l-integrable for each r E Q + . For x E IR the function px: 0+ -+> IR defined by px(r) = erx is a semi~
character and so is p_ 00 := l{o}. Conversely, if p E S* then p(l) ~ 0 and defining x = log p(l) E [R u {- oo} we have p = Px. It is easy to see that x ~ px is a topological semigroup isomorphism of ([ - 00, 00[, +) onto S*, the topology on [ - 00, oo[ being the obvious with {[ - 00, -n[ln E N} as a base for the filter of neighbourhoods of - 00. Let qJE(!jJ(O+) and assume cp(O) = 1. For each nEN we have that (qJ(k/n))k?:O is a Stieltjes moment sequence because
k~
and
qJ
k + 1) = qJ (k;; + 2 2n1) (-n-
are positive definite on (No, +), cf. Theorem 2.5. Therefore there exists J-ln E M~([O, oo[) such that
q>(~) = Loox The
mappingj~: [
-
k
00, 00 [ -+>
j~(x) =
[0,
00 [
ex / n
n
~
0,
n E N.
defined by
for
is a homeomorphism, so there exists (cf. 2.1.14), hence
qJ(~)
k
for
dJ-lix)
xE[-
Tn E
= [
00,
oo[
M~([ - 00, oo[) such that T~"
= J-ln
e(k/n)x dTn(X).
J[-oo.OO[
By 2.4.6 and 2.4.10 there is an increasing sequence nt < n2 < ... in N such that T(nk)! converges vaguely to a measure T E M +([ - 00, oo[) of total mass ~ 1. Let r = p/q E 0+ be fixed and assume that p, q E No have no common prime factors, q =t= O. For kEN such that nk ~ q we find k )!) qJ(r) -- qJ (r(n -( )' nk •
f
erx dT(nk)!(x),
[-00,00[
and using that for each nonnegative continuous functionj'on a locally compact space the integral j'dJ-l is lower semicontinuous in J-l with respect to the vague topology (cf. 2.4.1), we find
J
r
J[ - 00, oo[
erx dr(x)
~
q>(r),
210
6. Positive Definite Functions and Moment Functions
which shows that erx is r-integrable for all r E Q+. Let hE CC+([ - 00, oo[) satisfy 0 ~ h ~ 1, hex) = 1 for x E [ - 00, 0], hex) = 0 for x ~ 1. Then cp(r) =
r
J[ -
+
erxh(x) dr(nk)!(x) 00,
oo[
r
J[ -
e(r+ l)x(l - h(x»e - x dr(nk)!(x), 00,
oo[
(6)
and since erxh(x) has compact support in [ -
f J[ Furthermore, (1 - h(x))e- X
E
00,
oo[ the first term converges to
erxh(x) dr(x). 00,
oo[
CO([ -
00,
oo[) and the sequence
converges vaguely to e(r+ l)x r with the total masses bounded by cp(r + 1). By Proposition 2.4.4 it follows that the second term of (6) converges to
r
J[-
e(r+ l)x(l - h(x»e- X dr(x) 00.
=
oo[
r
J[ -
erx(l - h(x» dr(x), 00,
oo[
hence cp(r) showing that
=
r
J[ -
qJ E Jff( Q +).
erx dr(x) 00,
oo[
Putting
cp(r)
Q
=
r({ -
= al{o}(r) +
00 })
and J-l
= r I~ we have
{'' oo erx d/-l(x),
hence lim cp(r) =
J-l(~)
= cp(O) -
Q.
r-+O
The function
is well defined and continuous on the closed half-plane {z Eel Re z ~ O} and holomorphic in its interior, hence uniquely determined by cp. On the imaginary axis is the Fourier transform of J-l which shows that J-l is uniquely D determined by cp. By Proposition 5.6 and Theorem 5.4 we get that (Q~, +) is a perfect semigroup for k ~ 1. For x = (Xl' ••• , x k ) E [ - 00, OO[k and r = (r 1 , ••• , rk ) E (p~ we define <x, r) = 1 xjr j and px(r) = e<x,r) and so we have:
LY=
211
§5. Perfect Semigroups
5.7. Proposition. For every qJ E .?J>(O~) there is a unique measure J-l E M + ([ - 00, 00 [k) such that
qJ(r) =
i
e<x, r) dJ-l(x)
r E O~.
jor
[- 00, oo[k
As an application of this result we prove:
5.8. Theorem. A necessary and sufficient condition that a jilnction qJ: IR~
-+>
IR
can be represented as q>(y)
=
r e<x,y) dJ1(x)
for
J~k
y
E
IR~
(7)
with J-l E M + (IRk), is that qJ is continuous and positive definite on (IR~, +). The measure J-l is uniquely determined by qJ. PROOF. It is clear that qJ given by (7) is continuous on IR~ and positive definite on (IR~, +). Conversely, if qJ has these properties the restriction of qJ to O~ belongs to [!J(O~) and is hence of the form
qJ(r) =
i
e<x, r) dJ-l(x)
for
rE
Q~
[- 00, oo[k
with J-l E M +([ - 00, OO[k). As in the proof of Proposition 4.4.7 the continuity of qJ at (0, ... , 0) implies that J-l is concentrated on IRk. The two continuous functions qJ and y
~
r e<x,
J~k
y)
dJ1(x)
agree on O~ and (7) follows. The uniqueness follows from Proposition 5.7.
o
5.9. Remark. The function qJ in Theorem 5.8 has a unique continuous extension iP to {z E C k IRe zj ~ 0, j = 1, ... , k} which is holomorphic in the interior. It is given as iP(z) =
and iP(iy), y E
f
e<x,z) dJ-l(X) IRk IRk, is the Fourier transform of J-l.
The results about 0+ can be extended to (0, +) with the identical involution. 5.10. Proposition. The semigroup (0, +) with the identical involution is perject. Every qJ E [!J(Q) has a unique representation as
jor where J-l E M + (IR).
r E Q,
212
6. Positive Definite Functions and Moment Functions
Since h: 0 + x Q + ---. Q defined by h(s, t) = s - t is a homomorphism and onto, it follows from 5.4, 5.5 and 5.6 that (0, +) is perfect, too. Furthermore, the proof of Theorem 5.5 shows that Q* is isomorphic (and homeomorphic) to h*(Q*). Noting that (Q~)* ~ ([ - 00, 00[2, +), cf. Proposition 5.7, we get for r E Q*, h*(r) = (x, y) E [ - 00,00[2, and s, t E 0+ PROOF.
e
XS
+ yr
= (h*( r))(s, t) = r(h(s, t)) = r(s - t),
implying x > - 00, Y > - 00 and y = -x, i.e. r(s) = eXs • Hence 0* is isomorphic with (IR, +) and a given qJ E &>(0) has the unique integral representation for where J1
E
r E Q,
M~(IR).
D
Also (Qk, +) is perfect and in analogy with 5.8 we find:
5.11. Theorem. A necessary and sufficient condition that afunction qJ: IRk ~ IR can be represented as cp(y)
=
r
J~k
e(x,y)
dJ-l(x)
with J1 E M + (IRk) is that qJ is continuous and positive definite in the semigroup sense on (IRk, +). The measure J1 is uniquely determined and qJ can be extended to an entire holomorphic jilnction on C k • Theorems 5.8 and 5.11 can be generalized to a non-semigroup setting, cf., Widder (1941, p. 273) and Akhiezer (1965, pp. 211 and 229).
5.12. Theorem. Let H £; IRk be an interval. A necessary and sufficient condition that a jilnction qJ: H ---. IR can be represented as cp(y)
=
r
J~k
e(x,y)
dJ-l(x)
jor
yEH
with J1 E M + (IRk) is that qJ is continuous and the kernel (x, y) H qJ(x positive definite on (tH) x (tH).
+ y)
is
The proof will not be given here, but let us indicate how a proof may run. The result is first established for k = 1 using the truncated moment problem, cf. Akhiezer (1965), and then extended to k dimensions by the technique of Theorem 5.4. We will now give an integral representation of the cone %(Q +) of negative definite functions on the semigroup Q+ .
~5.
213
Perfect Semigroups
5.13. Proposition. A jilnction l/J: 0 has a representation oj' the jorm
+ --+ ~
r
tjJ(r) = a + br - cr 2 + d1<1:l!+\IO}(r) +
jor r E Q+ , where a, b E
f
dJ-l(x) <
X2
~,
c, d
~
J~\{O}
(1 - erx - r(l - eX)) dJ-l(x)
0 and J-l E M +(~\ {O}) satisfies
f
00,
0< Ixl ~ 1
is negative definite if· and only if· it
1<
erx dJ-l(x) <
The quintuple (a, b, c, d, J-l) is uniquely determined by PROOF.
jor all
00
r E Q+ .
Ixl
l/J.
We note that
and it follows easily that any function of the above form is negative definite. Conversely, it suffices to prove the above integral representation with a = 0 for l/J E %(0+) such that l/J(O) = O. By Theorem 3.2.2 and Proposition 5.6 there exists a unique measure J-l1 E E+(~) on the locally compact space ~ := [ - 00, oo[ such that
e-'ojJ(r) =
f~erx dJ-l,(x)
for
r E 0+,
t
> 0,
and J-lr is a probability measure. We have
and
o ~ -1 f (1 t
- erx )2 dJ-lr(x)
~
= -1 (1 - 2e- 1l/!(r) + e- 1l/!(2r»), t
which show that
lim ~ 1-+0
t
f (1 ~
erx
-
r(1 - eX)) dJ-l1(X) = l/J(r) - rl/J(I)
and
The last equation implies that there is a constant A r depending on r E Q + such that for
0 < t
~
1,
214
6. Positive Definite Functions and Moment Functions
so by Proposition 2.4.6 there is a measure a E tending to zero such that
M+C~)
vaguely. SInce (1 - e'X)2(1 - eX) - 2 is continuous on
1i (1 -
~im ~ t)
)-00
e,x)2j'(x) dj.1tj
~
=
i (1 ~
rX
and a sequence (t j )
!!! we get
)2
e X j'(x) da(x) -_1 e
for allj'E CC(~), and by Proposition 2.4.4 this holds even for j'E Co(~). Choosing a function qJ E Cc+(~) with qJ(x) = 1 for x E [ -1, 1] we find for rE 0+ \{O}
~
L
)
(1 - e
-
1
=-
rx
i
tj
~
- r(l - eX» dJ1.ri x )
1 - e'x - r( 1 - eX) X2 (x)2 qJ(x)(1 - e) dJ1.r/ x ) 1- e
+~
rf(x)(1 -
e(r+ llx)2 dJ1.rJ(x),
t j J~
where ,
j (x) Letting j ---.
00
:=
1 - e'x - r( 1 - eX) (1 _ e(r+ llx)2 (1 - qJ(x»
E
COm).
we therefore get
+
f
=
J~
i( ~
1 - e('+ 1)X)2 X j'(x) da(x) 1- e
1 - e'X - r( 1 - eX) (1 - e X )2 da(x)
= !(r - r 2)a({O}) + (1 - r)a({ - oo})
+ f
J~\{O}
where j.1
ljJ(r) with b
= (1 -
(1 - e'X - r(1 - eX» dj.1(x),
eX )-2(al ~\{O}), hence
= br - cr 2 + d1ilJ!+ I{O} (r) +
= l/J(I) + !a({O})
-
r
(1 - erx - r(1 - eX» dJ1.(x),
J~\{O} a({ -oo}), c = !a({O}), d =
a({ -oo}).
215
§5, Perfect Semigroups
By Proposition 2.4.6
f~ C1 _~:)2 du(x) ~ A
r
and it follows that J-l has the asserted properties. To see unicity of the representation we first remark that a = t/J(O) and a + d = t/J(O+), which show that a, d are uniquely determined. For t/J E AI( Q+) of the form ljJ(r) = br - cr 2 +
r
J~\{O}
(1 - erx
r(l - eX» dJ-l(x)
-
we find 21jJ(r
+
l} - ljJ(r) - ljJ(r
+ 2) = =
2c
+
{e
rx
r
J~\{O}
erx(1 - eX )2 dJ-l(x)
dr(x),
where r = 2c£o + (1 - eX)2 J-l. By the uniqueness assertion in Proposition 5.6 it follows that r and hence c and J-l are uniquely determined. Finally, b is also uniquely determined. D In the above result t/J is continuous if and only if d = 0, and if d = 0 the representation shows that t/J can be extended to a continuous negative definite function on ~+. The term -r(1 - eX) is a correction term analogous to a Levy function, cf. 4.3.17, and it can be replaced by any term of the form rqJ(x), where qJ is a locally bounded Borel function such that qJ(x) "" x for x---+-O and which is integrable at ± 00 with respect to all the measures J-l appearing in the representation. As examples we have qJ(x) = xe X and qJ(x) = x(1 + X 2)-1. This leads to the following result only formulated for continuous negative definite functions:
5.14. Theorem. A ./ilnction t/J: ~+ ---+- ~ is continuous and negative definite if' and only if'it has a representation 0./' the form ljJ(y) where a, b E ~,
=
a
C ~
+ by
- cy2
r (1 -
+
J~\{O}
eXY
+
1+xyx 2) dJ-l(X),
0 and J-l E M +(~\ {O}) satisfies
f
1
The representation is unique.
eXY dJ-l(x) <
00
for all
yE
~+
216
6. Positive Definite Functions and Moment Functions
Any function C+ given by l/J(z)
l/J
as above has an extension to the closed half-plane
i
= a + bz - cz 2 +
~\{O}
(1 - e
Xz
+ ~) 1+x
dJ.1(x).
(8)
This extension is continuous in C+ and holomorphic in the open half-plane and uniquely determined by these two properties. In particular, we find
l/J( -iy) = a -
iby
+ cy2 +
i
~\{O}
(1 - e-
ix
)' -
ixy 1+ x
2) dJ.1(x)
for
YE~, (9)
which is easily seen to be a continuous negative definite function on ~ in the group sense. In fact, formula (9) is the classical Levy-Khinchin representation, cf. Loeve (1963), with the exception that it is normally assumed that a = O. Note also that (9) does not give all continuous negative definite functions because of the integrability conditions on J.1 given in Theorem 5.14. Formula (9) gives all continuous negative definite functions if the only restriction on J.1 E M +(~\ {O}) is
i
X2
~\{O}
-1- - 2 dJ.1(X) <
+x
00.
With each continuous negative definite function l/J: ~ + sociated a convolution semigroup (J.1,)t ~o on ~ such that e-lojJ(y)
=
f
eX}'
dJ1,(X)
for
y
~
0,
t ~
-+> ~
o.
can be as-
(10)
In fact, by Theorem 3.2.2 exp( - tl/J) is continuous and positive definite, so the results in 5.6 and 5.8 ensure the existence of a unique measure J.1, E M~(~) such that (10) holds, and J.1, is concentrated on ~. By the uniqueness assertion in 5.8 it also follows that J.10 = £0 and J.1, * J.1s = J.1t+s. The function x ~ exp(xy) is continuous on Q! for y ~ 0, and we claim that the subspace of C(~) spanned by these functions is adapted. This follows by the fact that
belongs to CO(~) for y ~ O. Using the adaptedness we see by Remark 5.3 that t ~ J.1, is weakly continuous. The measure J.1, decreases exponentially on ~ + in the sense that for all and it follows that
y
~
0,
217
§5. Perfect Semigroups
is continuous on the closed half-plane {z Eel Re z ~ O} and holomorphic in its interior. By the uniqueness of holomorphic extensions we have by (8) that e-to/J(Z)
f
=
e
XZ
dpix)
for
Re z
~
0,
t
~
0,
so Y H I/I( - iy) is the continuous negative definite function in the group sense associated with (fJ.,),~o, cf. Berg and Forst (1975).
5.15. Example. For 0 < rt
~
1/I(X(y)
2 we define I/I(X:
=
~+ --+ ~
by
o<
y(X,
rt ~ 1, 1 < rt ~ 2.
(X { -y,
Then I/I(X is continuous and negative definite on ~+. We first remark that 1/11' -1/11 and 1/12 are all negative definite because if Yl' ... , YnE~+ and cl"",cnElRwith"LJ=1 cj = Owe have n
L
cjck(Yj
+ Yk) =
0,
j, k= 1
j.~
1 CjCk(Yj
+ Yk)2 =
2et CjYjY ~ O.
That I/I(X is negative definite follows now by Corollary 3.2.10 for 0 < rt < 1 and by Corollary 3.2.11 for 1 < rt < 2. Let (fJ.~(X»), ~ 0 be the corresponding convolution semigroup on ~, determined by the-formula e-to/J.(y)
=
f:oo
e
XY
dll:~)(X),
Y
~ 0,
t
~
O.
Since I/I(X(O) = 0 each fJ.~(X) is a probability measure, and I/I(X being nonnegative for 0 < rt ~ 1 we see by Corollary 4.4.5 that fJ.~(X) is concentrated on ] - 00, 0] for 0 < rt ~ 1 but not for 1 < rt ~ 2. For 0 < rt < 1 the measure fJ.~(X) (or its reflection in the origin) is a so-called one-sided stable measure of index rt, cf. Hall (1981). Clearly fJ.~l) = G_,. For rt E ]0, 2]\ {I}, we have Re z
~
0,
where z(X is the holomorphic extension of y(X to the right half-plane, i.e. z(X
= I z I(X exp(irt arg z)
with arg z E [ - n12, nI2]. In particular
(-iy)~ = IYI~ exp ( -ia ~ sgn(y») =
cos(a~) IYI~( 1 -
i sgn(y)
tan(a~)).
218
6. Positive Definite Functions and Moment Functions
hence
Iexp(t( -
iy)a sgn(1X - 1» I = exp(t sgn(1X - 1) COS(IX
~) Iy la)'
(11)
which is Lebesgue integrable since sgn(ex - 1) cos(ex(nj2)) < 0. By the Fourier inversion theorem Il~ex) has a density g~ex) E CO(~) given by
g:a)(x) = ~ eixYel(-iy)" 2n - 00
foo
sgn(a-
1)
dy.
Since the function in (11) remains integrable after multiplication with any power of y we see that g~ex) E Coo(~). The measure Il~ex) is a stable measure of index ex, cf. Hall (1981 ). For ex = 2 we have
g:2)(X)
= ~ foo 2n
dt = _1_
eiXYe-ly2
j4m
-00
e-x2/4l,
which is the density of a normal distribution and (1l~2»)l ~ 0 is the so-called Gaussian semigroup. For an explicit formula for gp /2)- see, for example, Berg and Forst (1975, p. 71). The restriction of t/J Cl to r\J 0 is negative definite on (r\J 0' +) so
°<< 1
is a Hamburger moment sequence. For ex sex(n) =
foo -00
exng\a)(x) dx
=
=f
ex ~ 1, ex ~ 2,
1 we have
foo tng\a)(log t)t0
dt,
1
n
~
0,
°
showing that Sex is a Stieltjes moment sequence. For < ex < 1 we have that g\ex)(log t) = for t ~ 1 so that Sex is completely monotone, and nCl is completely alternating. By comparison with Example 4.6 it can be seen that S2 is indeterminate.
°
5.16. Example. The important function from information theory t/J(y)
=
- y log y is continuous and negative definite on ~+ • In fact, from Example 5.15 we get that (- yex + y)(ex - 1)-1 is negative definite for ex E JO, 2J\{1}, and letting ex ~ 1 we get t/J(y) in the limit. Let (Ill)l ~ 0 be the corresponding convolution semigroup of probabilities on ~ determined by y
~
0,
In particular yY is continuous and positive definite on extension of t/J to the right half-plane is
t/J(z) = -zlogz = -z(loglzl
+ iargz),
t ~ 0.
~+ • The
Re z
~
holomorphic 0,
219
§5. Perfect Semigroups
where arg
n12, nI2]. In particular
Z E [ -
ljJ( -iy)
=
iY(IOgIYI - i
~ sgn(y») = ~ IYI + iy loglyl
so that
lyl + iy IOgIYI)} = f"ooe- iXY djlt(x).
exp{ -tG
Since y n exp{ - t(nI2) IY I} is Lebesgue integrable for all n E N, we see that J1l has a Coo-density gl given by
I eiXY p glx) = 2n f:oo ex {
-t(~ IYI + iy IOg/YI)} dy.
The measure J1l is a stable measure of index 1, cf. Hall (1981). The sequence (nn)n~o is positive definite on No, and we have
nn
=
foo -
enXg 1(x) dx = (00 tng 1(log t)t - 1 dt,
Jo
00
showing that nn is a Stieltjes moment sequence. Using Carleman's criterion, cf. Akhiezer (1965), it follows that nn is determinate. The result of 5.14 can be extended from IR + to IR\ and reads as follows:
5.17. Theorem. A jiinction t/J: IR\ ~ IR is continuous and negative definite if' and only if'it has a representation oj' the jorm ljJ(y)
=
a
+ (b, y)
- q(y)
+ {kIlO}
(1 - e<x,y) + 1~'lr~12) djl(x), YEIR\,
(12)
where a E IR, bE IRk, q(y) = L~,m= 1 anmYnYm is a nonnegative quadratic form on IRk and J1 E M + (IR k \ {O}) is such that
f
JO
f Jllxll >
e(x, Y)
Ilx 1 2 dJ1(X) <
dJ1(x) <
00,
jar all Y E IR\ .
00
1
The quadruple (a, b, q, J1) is uniquely determined. The proof is a modification of the proof of Theorem 5.18 below and will be left as an exercise for the reader. It should be noted that the continuous negative definite functions on IR\ are in one-to-one correspondence with the convolution semigroups (J1l)l ~ 0 on IRk satisfying
f.
~k
e(x, Y)
dJ1,(x) <
00
for
t ~
0,
Y E IR\ ,
220
6. Positive Definite Functions and Moment Functions
the correspondence being established by
=
e-'.p(y)
r
d/1,(x)
e(x,y)
J~k
t ~ 0,
for
Y E IR\ .
Formula (12) shows that t/J has a continuous extension to C\ which is holomorphic in its interior, and exp( - tt/J( - iy)) is the Fourier transform of J1,. Just as functions ({J E 9(Ok) automatically are continuous, so are functions which are negative definite in the semigroup sense on Ok. The following result can therefore be regarded as an integral representation of the cone %(Qk) or as an integral representation of the continuous functions t/J: IRk --+ IR which are negative definite on (IRk, +) in the semigroup sense. The result is due to Bickel and van Zwet (1980).
5.18. Theorem. A function t/J: IRk --+ IR is continuous and negative definite in the semigroup sense if' and only if'it has a representation oj' the jorm t/J(y) = a
+
+
f
(1
JlR/k\{O}
-
e
(x,y)
<x,y»)d ()
+ 1 + IIxl12
/1
x,
Y E IRk,
where a E IR, bE IRk, q(y) = L~,m= 1 anmYnYm is a nonnegative quadratic jorm on IRk and J1 E M +(lR k \ {O}) is such that
f
IIxl12 dJ1(x) <
0<
Ilxll
e(x, y) dJ1(x) <
J
1<
00,
~ 1
jor Y E IRk.
00
Ilxll
The quadruple (a, b, q, J1) is uniquely determined by t/J. PROOF. It is easy to see that any function of the above form is continuous and negative definite. Conversely, it suffices to prove that any continuous negative definite function t/J: IRk --+ IR with t/J(O) = 0 has a representation as above with a = O. By Theorem 5.11 there is a unique J1t E M ~ (IRk) such that
exp( - tl/J(y))
=
r
J~k
e(x, y)
d/1,(x)
From this equation together with the unicity of J1t we get J1, * J1s = J1t + s for t, s > 0, and using the fact that the subspace of C(lR k ) spanned by the functions x"-' exp( <x, y»), Y E IRk is adapted, we see that lim t _ o J1t = GO weakly, cf. Remark 5.3, i.e. (J1t)t~O is a convolution semigroup of probability measures on IRk where we put ~o = Go. Defining F(t, z)
=
r
J~k
e(x,Z)
d/1,(x)
for
t ~
0,
Z E
Ck,
~5.
221
Perfect Semigroups
then F: [0, 00 [ X C k ~ C is continuous and F(t, .) is holomorphic on C k for fixed t ~ O. Since F(t + s, z) = F(t, z)F(s, z) and F(O, z) = 1 we see that F(t, z) 4= 0, so there exists a unique holomorphic function til: C k ~ C such that F(t, z) = exp( - ttil(z)), and t'iJ is clearly an extension of t/J. In particular
exp( -tti!( -iy» =
r
J~k
e-i<x,y)
d/l,(x),
which shows that Y ~ tiI( - iy) is continuous and negative definite on IRk in the group sense. By the classical Levy-Khinchin formula in IRk, cf. Courrege (1964), we have !/J(-iy) = -i(b,y)
+ q(y) + {kl(O}
(1 -
/~'11~~2) d/l(x),
e-i<x,}') -
where bE IRk, q(y) = L~,m= 1 anmYnYm is a nonnegative quadratic form and J1 E M + (lR k \ {O}) satisfies
f
/Ix /I 2 dJ1(X) <
00
f
and
dJ1(x) <
00.
l
O
The Levy measure J1 is the vague limit as t ~ 0 of (ljt)J1,1 (lR k \ {O}). For each Y E IRk the family (ell/!(Y)e<X, y) dJ1,(X)), ~ 0
is a convolution semigroup of probabilities on IRk and its Levy measure is exp(x, y) dj1(x), and therefore
f
e<x, Y) dJ1(x)
IlxlI>
<
00
1
for each Y E IRk. It follows easily that
+
r
J~k\{O}
(1 -
e<x,z)
+
(x, z) 1
+ Ilx I
2) dJ1(X)
is well defined and holomorphic for z E C k , and since <1>( - iy) = t'iJ( - iy) for all y E IRk we get <1>(z) = t'iJ(z) for z E C k, in particular, <1>(y) = t/J(y) for y E IRk, which establishes the desired integral representation. The uniqueness assertion follows by the uniqueness statement in the classical Levy-Khinchin formula. 0
5.19. Exercise. For J1 E M +(lR k ) let x
~
All be the set of points y exp(x, y) is J1-integrable and define
.fiY) =
f
e<x,y)
E
IRk for which
d/l(x)
Show that All is a convex set and that f ll is a lower semicontinuous logarithmically convex function. Show that (u, v) ~ fll(u + v) is a positive definite kernel on (fAil) x (fAil)'
222
6. Positive Definite Functions and Moment Functions
Show finally thatfJl is COO in the interior of AJl and that for
C(
E
N~
if 0 is an interior point of AJl. The function f Jl is called the moment generating function of J1.
5.20. Exercise. Let t/J: IR + --+ IR be a continuous negative definite function and (fJ.t)t~O the convolution semigroup such that (10) holds. Show that lim t-+ o(l/t)fJ.t I(IR\ {O}) = fJ. vaguely, where fJ. is the measure in the representation of t/J in Theorem 5.14.
5.21. Exercise. Let S be a perfect semigroup and add to S an absorbing element
W,
cf. 4.1.3. Then S u {w} is again perfect.
5.22. Exercise. Let S be the subsemigroup of (IR+, +) generated by Q+ and an irrational number C( > O. Show that S is not perfect. 5.23. Exercise. Let qJ: IR --+ IR be a continuous function which is positive definite on the multiplicative semigroup (IR, .). Show that there are measures fJ. E M +(IR+), v E M +(]O, oo[) such that cp(s) =
r
J~+
1slY d,u(y)
+
r
1s I)' , sgn(s) dv(y),
s E IR.
J]o.oo[
5.24. Exercise. Show that (Q + , .) and (Q, .) are not perfect semigroups. Hint: Use that (Q+ \ {O}, .) is isomorphic with Z(oo).
Notes and Remarks In his fundamental paper" Recherches sur les fractions continues", Stieltjes formulated and solved the moment problem which bears his name. Later Hamburger generalized Stieltjes' result to moment sequences of measures on the whole real line. Moments of a measure have been studied, before Stieltjes, by Tchebycheff and others, but concerning the history of the moment problem we refer to Shohat and Tamarkin (1943). Our choice of the name "moment function" on a semigroup S is motivated by this classical theory, which corresponds to the semigroup (No, +). Likewise the symbol Yf(S) reflects the name of Hamburger. The results in 1.6-1.11 seem to be new but are, of course, known for some concrete semigroups. For a detailed ~tudy of Hamburger's moment problem, in particular of the set E + (IR, s), we refer the reader to the classical monographs by Akhiezer (1965) and Shohat and Tamarkin (1943). Results about denseness of the set of polynomials in 2 P(IR, fJ.) can be found in Berg and Christensen (1981,
223
Notes and Remarks
1983a). The F-moment problem in the case F = {x E IRlp(x) ~ O}, where p is a fixed polynomial, is studied in Berg and Maserick (1982). It contains a characterization of the polynomials p for which the set of {p ~ O}-moment sequences is equal to {sIs, p(E)s E &>(r\J o)}. Here N
(p(E)s)n =
L aksn+ k k=O
N
if p(x) =
L akxk. k=O
The F-moment problem where F = IR\U7= I ]a i , bi [ and a l < b l < a2 < b 2 < ... < an < bn is solved in Svecov (1939) for n = 1 and in Fil'stinskii (1964) for arbitrary n. Horn (1969b) showed that a nonzero positive definite sequence qJ: r\J o ~ IR is infinitely divisible if and only if t/J := -log qJ has the representation stated in Theorem 2.6. Already in 1888 Hilbert remarked that there exist nonnegative homogeneous polynomials in three variables which are not a sum of squares of homogeneous polynomials, cf. Hilbert (1888). His method is, however, rather involved and does not lead to a simple explicit example, cf. Gelfand and Vilenkin (1964, pp. 233-234) where Hilbert's construction is indicated. An example along these lines was found by Robinson (1969, 1973) and by Schmlidgen (1979). The polynomial xix~(xf + x~ - x~) + x~ obtained by making the polynomial of Lemma 3.1 homogeneous is probably as simple as possible since the degree must be at least 6, as shown by Hilbert. The paper by Motzkin (1967) contains a similar explicit example. A nonnegative homogeneous polynomial in two variables is a sum of two squares of homogeneous polynomials by Lemma 2.1. Theorem 3.4 is stated in Zarhina (1959) and also referred to in Gelfand and Vilenkin (1964, p. 235), but the proof seems incomplete. Zarhina states that it is unknown whether L(k) n A~k) is closed in A~k) and claims without proof that its closure is different from (A~k») + . Independently and almost simultaneously Theorems 3.2 and 3.4 were published by Berg et al. (1979) and Schmlidgen (1979). An explicit example of a function qJ E &>(r\J6)\~(r\J6) has been given by Friedrich in a paper to appear in 1984 (or later). Only few papers contain sufficient conditions for a function qJ E &>(Nt), k ~ 2, to be a moment function, cf. Devinatz (1957), Eskin (1960) and Nussbaum (1966). All these papers are based on results about the delicate problem of finding commuting self-adjoint extensions of symmetric operators which commute on a common dense domain in a Hilbert space. A survey of the operator theory approach to the moment problem is given by Fuglede (1983), cf. Exercise 3.15. The notion of Fmoment sequences and the .F-moment problem can be extended to k dimensions, and the first solution of the problem is due to Haviland (1935, 1936). His result is given in Exercise 3.16. For results in case of concrete sets F £; IRk, see Exercise 3.17 and McGregor (1980). If F is a compact convex set with nonvoid interior the problem is solved by Maserick (1977) and rediscovered by Cassier (1983). Semigroups of moments are considered in Berg (1984a), Buchwalter (1984) and Buchwalter and Cassier (1983a, b). In
224
6. Positive Definite Functions and Moment Functions
Petersen (1982) it is proved that qJ E Yf( N~) is determinate provided each of the one-dimensional moment sequences qJ(n 1 , 0, ... , 0), ... , qJ(O, ... , 0, nk ) is determinate. The results in 3.6-3.13 were summarized in Berg and Christensen (1983b). The two-sided moment problem has been studied by Jones et ale (1980, 1984). The first paper deals with the Stieltjes case where the representing measure is concentrated on JO, 00 [, and the authors use continued fractions as in Stieltjes' original work. In the second paper, where no restriction is required on the support of the measure, the authors study orthogonal Laurent polynomials and closely related Gaussian-type quadrature formulas to obtain the solution, thus following the method of Hamburger. Our approach can be called the Hahn-Banach method, first used by M. Riesz in connection with his solution of the Hamburger moment problem. This method was generalized to higher dimensions by Haviland in the papers cited above and later formulated in terms of adapted spaces by Choquet (1962). Theorem 4.3 is essentially in Wall (1931) although only extensions of s: No ~ [R to {k E 7L I k ~ - 2n} are discussed. That the log-normal distribution is indeterminate for the classical Stieltjes moment problem was noticed by Stieltjes (1894) and Heyde (1963). It seems difficult, but would be very interesting, to determine the structure of perfect semigroups. Theorems 5.4 and 5.5 show two possibilities of building perfect semigroups out of simpler ones. As another possibility let (Sn) be a sequence of semigroups. Then the direct sum 00
S
=
EB Sn =
n=1
{s = (S1' S2" ··)ISnESn' Sn =
o eventually}
is perfect if and only if each Sn is perfect. We sketch the proof and remark 1 This implies in particular that S* first that S* is isomorphic to need not be locally compact for a perfect semigroup S. For ({J E &J(S) and n E N there is a measure iln E E + = 1 S:) such that
flr=
S:.
(flZ
for
Sk E
il E E +
Sk' k
(flr=
1
=
1, ... , n. Itsuffices to prove the existence of a measure
S:) such that il Pn = iln for all n, where
Pn:
00
n
k= 1
k= 1
fl S: ~ fl S:
is the projection, and this is a consequence of a general result about projective systems, cf. Bourbaki (1965-1969, Chap. IX, p. 53). The semigroup of dyadic numbers is perfect, and this may be used to prove that every countable 2-divisible semigroup with the identical involution is perfect, see Berg (1984b). Continuous functions j': Ja, b[ ~ [R such that (x, y)~.l(x + y) is a positive definite kernel on J!a, !b[ x ]!a, !b[ were introduced by Bernstein
Notes and Remarks
225
who called them exponentially convexjilnctions. Both Bernstein and Widder proved Theorem 5.12 in the special case of H being one-dimensional, cf. Akhiezer (1965, p. 210). Akhiezer indicates how the result can be extended to k-dimensional intervals. See also Reeds (1979). Theorem 5.12 is also true if H is an open convex set, cf. Devinatz (1955). His proof uses the spectral theorem for a family of commuting self-adjoint operators. A complete proof of Theorem 5.17 is given in Berg (1984b).
CHAPTER 7
Hoeffding's Inequality and Multivariate Majorization
§1. The Discrete Case Many years ago Hoeffding (1956, Theorem 3) proved the following result: if Xl' ... , X n are independent Bernoulli random variables and if
1/1: {O, 1, ... , n}
--+
IR
is (strictly) concave, then (1) where on the right-hand side it is assumed that IFD(X i = 1) = Pi while on the 1 Pi for all i = 1, ... , n. left-hand side IFD(X i = 1) = P = (lin) Recently, Bickel and van Zwet (1980, Theorem 1.1) found a considerable extension of this result: let 1/1: IR m --+ IR and k E r\J be given and let Xl' , Xn be independent m-dimensional random vectors with distributions ill' , iln and the extra condition that there is a finite subset A ~ IR m (depending on ilb ... , iln) with cardinality k such that ilj(A) = 1 for allj = 1, ... , n. Then the following three conditions are equivalent:
2:7=
(i) the inequality
(2) holds for all n and all such ill' ... , iln; (ii) the inequality (2) holds for n = 2 and all ill' il2 of the type described above; (iii) the k x k matrix (I/I(Xi + Xj))i,j= 1, ... ,k is negative definite for every choice of X b ••• , X k E IR m •
227
§1. The Discrete Case
The interpretation of (2) is analogous to that of (1): on the right-hand side X i has the distribution J.1i while on the left-hand side the random vectors Xl' ... , X n are identically distributed with the average distribution fi = (lin) 1 JJi· Note that (2) reduces to (1) in case of {a, l}-valued random variables. The third condition in Bickel and van Zwet's result makes likely a certain kinship between negative definite functions and functions fulfilling inequalities of type (2) on more general semigroups than Just euclidean spaces. This, in fact, is true and will lead us to interesting characterizations of negative definite functions as well as of completely negative definite functions on arbitrary abelian semigroups. This last-mentioned class of functions has not yet been defined, but appears naturally in this connection, and the lower bounded members of this class turn out to be nothing more than the completely alternating functions. In this chapter we only consider abelian semigroups with the identical involution. First let S be an arbitrary nonempty set. If (Q, .91, P) is a probability space then X: Q ~ S is called an elementary random variable if {X = s} Ed for all S E S and IX(Q) I < 00, i.e. X assumes only finitely many value~. The distribution of an elementary random variable is, of course, a finite convex combination of one-point measures, or, as we have called it earlier, a molecular probability measure. If S is an abelian semigroup and X b . . . , X n are elementary S-valued random variables, then the sum 1 Xi is of this type, too. We shall make repeated use of the following straightforward identity:
L7=
L7=
1.1. Lemma. Let (ajk) be a symmetric real n x n matrix. Thenfor
Cb ... ,
Cn'
d 1, ... , dn E lR we have
~
i..J
j,k= 1 PROOF.
For j
. (C j + d j . Ck + dk _ .d) = a)k 2 2 c) k
~
i..J
j,k= 1
.
a)k
Cj -
2
dj . Ck - dk 2·
(3)
* k the two factors of ajk = akj on the left-hand side sum up to
2 . Cj
+ d j . Ck + dk _ C .dk _ Ck d . = 2 . Cj - d j . Ck - dk 2 2 ) ) 2 2
and the factor of a jj is given by
(
Cj
+ d j)2 2
_ c).d.) = (Cj -2 dj)2 '
thereby proving the equality (3). The next result is now nearly obvious:
D
228
7. Hoeffding's Inequality and Multivariate Majorization
1.2. Proposition. Let S be a nonempty set and let l/!: S x S ~ lR be symmetric. Then jor independent elementary S-valued random variables X and Y the inequality (4) [E;1[l/!(X, Y)J ~ [E1l1,1l2[l/!(X, Y)J holds for all J.11 and J.12 (In (4) we define and (2).)
if and only if l/! is negative definite.
fi = !(J.11 + J.12) and the interpretation is analogous to (1)
Let the two probability distributions J.11 and J.12 be concentrated on {S1' ... , sn} ~ S and denote (Xi = J.11({sil), Pi = J.12({sil). Then by Lemma 1.1
PROOF.
[E;1[l/!(X, Y)] - [E1l1,1l2[l/!(X, Y)]
((Xi
+ Pi (Xj + Pj
i.~ 1 l/J(Si , s) - 2 - - 2 - - (XiP j n
=
(X.-f3.l (X.-f3, l/!(Si' Sj) -l-2- . ~2),
n
.?: l,j=
)
(5)
1
and this is clearly nonpositive if l/! is negative definite. Suppose on the other hand that (4) holds for all J.11' J.12. Let S1' ... , Sn E S and C1 , ••. , Cn E lR with Ci = 0 be given. If not all Ci are zero then 0 < = Ci- =: C, and putting
L ct
L
L
the equality (5) shows n
L
i,j = 1
CiCjl/!(Si, Sj) ~ 0,
D
which had to be shown.
1.3. Definition. Let S be an abelian semigroup. Then l/!: S ~ lR is said to fulfil Hoeffding's inequality oforder n ~ 2 if for every sequence X l ' ... , X n of n independent elementary S-valued random variables the inequality
[E;1[l/!(X 1
+ ... + X n)]
~ [E1l1, ...,lln[l/!(X 1
+ ... + X n)]
holds, where on the right-hand side Xi has the distribution J.1i' 1 ~ i ~ n, while on the left-hand side all the Xi have the same average distribution fi:= (lin) 1 J.1i· The above inequality can also be expressed as
L7=
f l/J dji*" ~ f l/J d(/11 * ·.·* /1,,). Specializing in 1.2 the set S to a semigroup gives 1.4. Corollary. Ij'S is an abelian semigroup and l/!: S ~ lR a givenjunction on S, then l/! fulfils Hoeffding's inequality of order 2 if and only if l/! is negative
definite.
229
§1. The Discrete Case
A negative definite function on a semigroup need not however fulfil Hoeffding's inequalities of a higher order. As an example take S = No with usual addition. The function l/!:= 1{1, 3,5, ...} is negative definite on No because l/!(k) = t[l - (-l)kJ. If X 1, X 2, X 3 are independent Bernoulli random variables with parameters PI = 0, P2 = P3 = i, then p = and
t
[p[l/!(X 1 + X 2
+ X 3 )J = 3p(1
- p)2
+ p3 =!,
but
[Pl' P2, P3[l/!(X 1 + X 2 + X 3)J =
i·
The next result will show that Hoeffding's inequality of order 3 actually implies a sharpened form of negative definiteness.
1.5. Proposition. Let t/J: S -+ [R fulfil Hoeffding's inequality of order n ~ 3. Then for all a E S the translate E
PROOF.
+ (1 /12 := pea + (1 f.11
:=
pea
- p)f.1, - p)v,
/13:=··· = f.1n:= pea
+ (1 -
/1+v
p) -2-·
Then ji = f.1n and by easy calculation
ji*n - /11 * /12 * ... * f.1n = ji*
+ (1
f.1 + vJ*
* [ (1
/1 - VJ*2 - P)-2,
hence, dividing out (1 - p)2 and letting then p tend to 1, we get
and therefore
f
E(n-Z)atfJ
d[(f.1 ;
V)*ZJ
~f
By Corollary 1.4 E
E(n-Z)atfJ
d(f.1 * v). D
The new property showing up in the above result will be given a name: 1.6. Definition. Let S be an abelian semigroup. A function l/!: S -+ [R will be called completely positive (resp. negative) definite if E al/! is positive (resp. negative) definite for all a E S. For S = No the completely positive definite functions are precisely the Stieltjes moment sequences, cf. Theorem 6.2.5. For S = N~, k ~ 2, the completely positive definite functions are discussed in 6.3.7.
230
7. Hoeffding's Inequality and Multivariate Majorization
Our main result reads as follows:
1.7. Theorem. Let S be an abelian semigroup and let t/J be a real-valuedfunction defined on S. Then the following three conditions are equivalent: (i) (ii) (iii)
t/J fulfils Hoeffding's inequality of all orders; t/J fulfils Hoeffding's inequality of order 3; t/J is completely negative definite.
By Proposition 1.5 for n = 3 we know that (ii) implies (iii). Next let us assume that t/J: S ~ [R is completely negative definite. To see (i) it suffices by Corollary 1.4 to prove Hoeffding's inequality of order n for n ~ 3. We fix n probability measures of finite support J.11' ••• , J.1n on S and may assume that they all are concentrated on {s 1, ... , Sk} ~ S. The function PROOF.
is a mixture of negative definite functions, hence also of this type. Consequently, by Corollary 1.4, we have
ft/J
d(f.11
f ~f = ft/J
* f.12 * ... * f.1n) =
l{J d(f.11
* f.12)
l{J d[(f.11; f.12)*2]
d(f.11 + 2 f.12
* f.11 + 2 f.12 * J.13 * ... * J.1n ) •
The proof will be finished by showing that suitable successive pairwise averaging of a given vector x E [Rn converges to the vector of averages (x, X, ... , x) E [Rn where x = (lin) LI= 1 Xi. To see this we define the doubly stochastic n x n matrix T by 1
T=
1
2"
2"
0 0
0 0
0 0 1 0 0 1
0
0
0
0
1
0
2"
2"
1
0
1
2"
1
2"
0
0
0
1
0 0 0
0 0 0 0 1 0 0 0
1 0
2"
2"
1
1
0 0
1
0 0
0 0 0 0
0 0 0
0 0
1 0 0 1 0 0 1
0
0
0 0 0
0
1
1 0 0 0
0
0
231
§1. The Discrete Case
and observe that T is the transition matrix of a finite irreducible and aperiodic Markov chain. A well-known limit theorem for Markov chains of this kind (cf., for example, Billingsley (1979, Theorem 8.7» says that
as
m
~
00,
where Tei ~ 0 for all i and Ii= 1 Tei = 1. The limit matrix, however, must be doubly stochastic, too, and this is only possible if Te 1 = Te2 = ... = Ten = lin. Therefore Tm(x) ~ (x, x, ... ,x) for all x E [Rn as asserted. Identifying a probability measure v concentrated on {Sl' ... ,Sk} with the row vector (v( {S1}), ... , v( {Sk}» we now let T operate on n-tuples J.1 = (J.1 b ... ,jJn)' of such probability measures by defining T jJ as the n x k matrix
The sequence {TmJllm = 1,2, ...} converges to
(!)
as m -+
00,
and as we
have seen above
so that the obvious continuity of IE Jl [t/J(Li= 1 Xi)] as a function of jJ finally shows
thereby finishing the proof.
D
1.8. Corollary. On a 2-divisible abelian semigroup negative definiteness is }or each n ~ 2 equivalent with Hoeffding's inequality of order n. PROOF. It is sufficient to observe that a negative definite funct~on t/J on S is automatically completely negative definite. To see this, let a E S be given, i.e. a = b + b for some b E S by assumption. Then n
n
L CjckEat/J(Sj j,k=l if C1
+ ... + cn =
+ Sk) = I
j,k=l
CjCkt/J«b
+ Sj) + (b + Sk»
~ 0
0, so that all translates of t/J are negative definite, too. D
232
7. Hoeffding's Inequality and Multivariate Majorization
1.9. Remark. Looking once more to the results proved so far, it will become obvious that, in fact, something more has been shown. We could introduce a degree r by calling a kernel t/J: S x S ~ [R negative definite of degree r if all the matrices (t/J(Sj, Sk))j,k~m are negative definite for m = 1, 2, ... , r. If S is a semigroup then t/J: S ~ [R is called negative definite oj' degree r if the kernel t/J(s + t) has this property. Similarly, a degree could be added to Hoeffding's inequality of some given order n by requiring all the measures JJ l ' ... , JJn to be concentrated on one set of cardinality r. The proof of Proposition 1.2 then shows that negative definiteness of degree r is equivalent with the corresponding Hoeffding type inequality (4) of degree r. Proposition 1.5 tells us that Hoeffding's inequality of order 3 and degree r + 1 implies complete negative definiteness of degree r. Theorem 1.7 shows that a function which is completely negative definite of degree r still fulfils Hoeffding's inequality of degree r of all orders. Applying these remarks for r = 2 to the additive semigroup S = No we conclude that a function t/J: No ~ IR, such that all the 2 x 2 matrices
+ 2s) + S + t)
t/J(a ( with a, s, t
E
t/J(a
+ S + t)) t/J(a + 2t)
t/J(a
No are negative definite, fulfils all Hoeffding inequalities of
degree 2. But a 2 x 2 matrix
(~ ~) is negative definite if and only if y = f3
and lJ. + b ~ 2{3, hence complete negative definiteness of degree 2 is equivalent to t/J being concave. We therefore get back Hoeffding's original result mentioned at the beginning of this chapter, even slightly sharpened, the measures JJt, ... , JJn being allowed to be concentrated on any two-point set in No, not just {a, I}. The proof of Corollary 1.8 makes obvious that on a 2-divisible semigroup negative definiteness of degree r is equivalent to complete negative definiteness of degree r. Together with the preceding remarks this shows that the result of Bickel and van Zwet mentioned in the introduction generalizes from [Rm to any 2-divisible abelian semigroup. In Chapter 4 we have already met an interesting subclass of the negative definite functions, namely, the completely alternating functions. Their relation to completely negative definite functions will be clarified in the next theorem.
1.10. Theorem. Let S denote an abelian semigroup. Then a bounded jilnction if and only (l
PROOF.
If
~ [R
is completely monotone then it has by Theorem 4.6.4 the
233
§l. The Discrete Case
integral representation
f.
Js
pes) dJ1.(p), +
J-l being a Radon measure on the nonnegative bounded semicharacters. Then for any fixed a E Sand Sl' ... , Sn E S, Cl' ... , Cn E ~ we get n
L
n
CjckEaCP(Sj
+ Sk) = L
j,k= 1
CjCk
cp(a
+ Sj + Sk)
j,k= 1
showing that Ea cP is positive definite. Suppose on the other hand that cP is bounded and completely positive definite. By Theorem 4.2.8 there are Radon measures J-la on S, a E S, such that
Ea
s E S,
a E S.
We put J-l = J-lo and have to show that J-l is concentrated on S+. The open set S\S + = (9 a is the union of the open subsets (!J a := {p E SIp(a) < OJ, and J-l being a Radon measure it is enough to show that J-l«(9a) = 0 for all aES. We have
UaeS
for all
SE
S,
and the unicity of the integral representation for positive definite functions (cf. Theorem 4.2.5) implies J-la
=
p(a)J-l.
Therefore
o ~ J1.a(eJ a) = f
Jt9 a
pea) dJ1.(p)
~0
and we see J.1( (9 a) = O. The second statement concerning negative definite functions now follows easily: If t/J is lower bounded then t/J is completely negative definite if and only if exp( - tt/J) is completely positive definite and bounded for all t > 0, and by the first part of the proof this is true if and only if exp( - tt/J) is completely monotone which is equivalent with t/J being completely alternating; see 3.2.2 and 4.6.10. D A typical example of a completely negative definite function which is not bounded below is given on the additive semigroup (~, +) by t/J(x):= _x 2 • If Cl' ..• , Cn E ~ and L Cj = 0 then
L j,k
CjCk[ -(Xj
+ Xk)2]
=
-2(L
CjXj)2
~ 0,
234
7. Hoeffding's Inequality and Multivariate Majorization
and lR being 2-divisible l/J is, in fact, completely negative definite. But then, of course, the restriction l/J III to the non-2-divisible semigroup (1L, +) is also completely negative definite. This however is no coincidence, as we shall now see. We recall (cf. Definition 4.3.8) that a quadraticjorm q on an abelian group G, considered as a semigroup with the involution x* = -x, is a real-valued function q on G such that
q(x
+ y) + q(x
- y) = 2q(x)
+ 2q(y)
(6)
holds for all x, y E G. By Theorem 4.3.9 a nonnegative quadratic form is negative definite in the group sense on G, i.e. LCjCkq(X j j,k
xk) ~ 0
so that in view of (6) the function - q is negative definite in the semigroup sense on G. But we even have the following result: 1.11. Proposition. Ij'q is a nonnegative quadraticjorm on the abelian group G, then - q is completely negative definite in the semigroup sense, and hence fulfils all Hoeffding inequalities. PROOF.
In the proof of the already-mentioned Theorem 4.3.9 it was shown that
B(x, y) := q(x)
+ q(y) - q(x - y) = q(x + y) - q(x) - q(y) + X2' y) = B(x l , y) + B(X2' y) and B(x, YI + Y2)
is biadditive, i.e. B(x l B(x, Yl) + B(x, Y2) hold for all x, Xl' ... E G. We fix some a E G and consider the translate Eaq(x) = q(a h(x) := q(a + x) - q(x) - q(a) then
Eaq = q
=
+ x). Ifwe put
+ h + q(a),
so - Eaq is the sum of the negative definite function - q and a term which is obviously negative definite, being a constant plus an additive function. 0 1.12. Exercise. Let S denote an abelian semigroup. Denote by C n the closed convex cone generated by molecular signed measures of the form fi*n - III * 112 * ... * Iln' where Ill' ... , Iln are molecular probability measures on Sand where fi = (lin) 1 Ili· The closure is to be taken in Mol~(S) with respect to the finest locally convex topology on Mol~(S) which is just the topology induced by all real-valued functions on S via the canonical duality. Let (J E Mol~(S). Show that:
Li=
(i) if Jt/J d(J ~ 0 for all negative definite functions t/J: S ~ lR then (J E C 2; (ii) if f l/J d(J ~ 0 for all completely negative definite l/J: S -+ lR then (J E C 3. 1.13. Exercise. Show directly, by using the integral representation, that each completely alternating function fulfils all Hoeffding inequalities.
235
§2. Extension to Nondiscrete Semigroups
1.14. Exercise. If h: S definite.
~ [R
is additive then -exp(h) is completely negative
1.15. Exercise. The function l/J: S ~ [R is completely negative definite if and only if for all t > 0 the function - exp( - tl/J) has this property. 1.16. Exercise. Let (S, +) denote an abelian semigroup and let
+ ... + X n )]
~ E Il1 ,. •• ,lln[
~ [R
+ ... + X n )]
holds for all n if and only if it holds for n = 3, and if and only if
1.17. Exercise. Let PI' P2' ... E [0,1] be given such that P = L Pi < 00. Then the infinite convolution *~ 1 B(1, Pi) of the Bernoulli distributions B(l, Pi) = (1 - Pi)GO + Pi Gl exists, and for any concave function l/J: ~o ~ [R + we have
where 5.3.16.
1t p
is the Poisson distribution with parameter p. Hint: Use Exercise
§2. Extension to Nondiscrete Semigroups We know from the preceding section that negative definite and completely negative definite functions can be characterized by Hoeffding's inequality involving only probability measures of finite support. For applications, however, one would certainly also like to have Hoeffding's inequality in a nondiscrete situation. We begin with a counterpart to Proposition 1.2. 2.1. Theorem. Let S be a Hausdorffspace and let l/J: S x S ~ [R be a continuous negative definite kernel. Thenjor all Radon probability measures Ill' 112 on S we have
as soon as
l/J is integrable with respect to ji ® ji or l/J
~
o.
In a first step we assume that both III and 112 have compact support, i.e. Ill(K) = 1l2(K) = 1 for some compact subset K ~ S. By Proposition 2.3.5 there are nets (Ill,cx) and (1l2,cx) of molecular probability measures on K converging weakly to III (resp. 1l2) (it is no restriction to assume that both PROOF.
236
7. Hoeffding's Inequality and Multivariate Majorization
nets have the same index set). Then,ua:= !(Ill,a 2.3.3 we also have Ill,a ® 1l2,a
-+
III ® 1l2'
+ !J2,a) -+ ,u and by Theorem
,ua ®,ua
-+
,u ®,u
in the weak topology. On K x K the kernel l/J is a bounded continuous function, so that the portmanteau theorem (2.3.1) and Proposition 1.2 lead to the conclusion
f/J
d(ji ® ji)
=
li~ f l/J d(jia. ® jia.)
f
f
~ li~ l/J d(1l1,a. ® 1l2,a.) = l/J d(1l1 ® 112)·
(1)
In the second step we make use of the conditional probabilities 1l1,K(B):=
Ill(BnK) 1l1(K) ,
B E ~(S),
which certainly are defined for large enough K E %(S) and which immediately are seen to be again Radon measures. Putting IlK:= ¥!Jl,K + 1l2,K) we have by (1) the inequality
fl/J
for the above compact subsets K to ,u ® ,u gives, by Exercise 2.3.9,
fl/J
d(IlK ® 11K)
=
~
d(IlK ® 11K)
irf l/J
~
f
(2)
l/J(1l1,K ® 1l2,K)
S, and the integrability of l/J with respect
d(lll,K ® 1l1,K)
f
+ 2 l/J d(1l1,K ® 1l2,K)
+ f l/J d(1l2,K ® 1l2,K)J =
i [(11 1(~»2
Lx l/J ® + 1l1(K~Ilz{K) Lx/ K d(1l1
Ill)
d(1l1 ® 112)
+ (Ilz{~) )2 It =
irf l/J fl/J
L x
Kl/J d(1l2 ® 112)
d(1l1 ® Ill)
d(ji ® ji)
f
J
+ 2 l/J d(1l1 ® 112) +
fl/J
d(P2 ®
1l2)J
237
§2. Extension to Nondiscrete Semigroups
as well as
J
ljJ d(J11.K ® J12.K)
= J11(K;J12(K)
Lx/
d(J11 ® J12) -...
The same conclusion holds by Proposition 2.1.7 if ljJ (2) extends to the limit and we get
J
ljJ d(j1 ® ji)
~
~
J
ljJ d(J11 ® J12)'
0. Therefore inequality
J
ljJ d(J11 ® J12)
o
as asserted.
Let now (S, +) be a Hausdorff topological abelian semigroup with a neutral element. For two Radon probability measures Il, v on S the convolution Il * v is then a well-defined new Radon measure and furthermore (M ~ (S), *) is again a Hausdorff abelian topological semigroup, see Corollary 2.3.4. Theorem 2.1 has the following obvious consequence:
2.2. Corollary. Let ljJ be a continuous negative definite function on the abelian Hausdorff topological semigroup S. Thenfor all Ill' 112 E M~(S) we have
as soon as ljJ is integrable with respect to ji * ji or ljJ
~
0.
2.3. Example. Consider the semigroup S = (No, +). We have already mentioned that the indicator function ljJ = 1{1, 3, 5, ...} is negative definite on No. If X, Yare independent No-valued random variables with varying distributions Il (resp. v), but with a fixed given average distribution K = ¥Il + v), then
lPJL,v(X is minimized for Il
=
v
=
+
Y is odd)
= IE JL ,v[ljJ(X + Y)J
K.
2.4. Example. For (X E JO, 2J the function x 1-+ I X 1(% is negative definite in the group sense on IR (this follows, for example, from 3.1.10 and 3.2.10). If X, Y are independent real-valued random variables with distributions Il (resp. v), then under the restriction of a given average distribution K = !-(Jl + v) the expectation is minimized for Il = v =
K.
Theorem 1.7, the main result of§l, has the following nondiscrete extension:
2.5. Theorem. Let ljJ be a completely negative definite continuous function defined on the abelian Hausdorff topological semigroup S. Then for all Radon
238
7. Hoetfding's Inequality and Multivariate Majorization
probability measures Ill' ... , Iln on S we have
f
IjI d(ji*n)
~
f
IjI d(1l1
1 ji:= -
* ... * Iln),
n
'L Ili
(3)
ni=l
as soon as ljJ is integrable with respect to ji*n, and without this restriction if'ljJ is nonnegative. Of course, ljJ is integrable with respect to III * ... * Iln' too, in case 00. As in the proof of Theorem 2.1 we first assume that IlI(K) = 1l2(K) = ... = Iln(K) = 1 for some compact set K s;; S. Then both SUPP(1l1 * ... * Iln) and supp(ji*n) are contained in the compact set K n:= K + ... + K (n summands). Approximating Ill' ... ' Iln by molecular measures, the validity of (3) for these measures (Theorem 1.7) extends to the limit. Now we use again the conditional probability measures PROOF.
JIljJ I d(ji*n) <
Ili. K(B)
:=
lli(B n K) Il;(K) ,
~(S),
BE
1 ~ i ~ n,
K
E
f(S)
being well defined for large enough K, and have to go to the limit in the inequality
f IjI d«IlK)*n) ~ f IjI d(lll. K * ... * Iln. K),
(4)
'L7=1
where IlK:= (lin) lli,K· Evaluating the left-hand side we get
fiji d«IlK)*n) = ~n.
t _Il
11, ... , In -
n
I
fiji d(lli, * ... * lliJ
1 llij(K) Kn
j= I
and this converges, if either JIljJ I d(ji*n) < the right-hand side of (4) we obtain
fiji d(1l1.K * ... * Iln.K) =
n
1
00
Il lli(K)
or ljJ ~ 0, to
JljJ d(ji*n).
For
fiji d(1l1 * ... * Iln) K~l
i= I
thereby finishing our proof.
o
2.6. Example. On the additive semigroup (IR, +) the function ljJ(x) = -x 2 is completely negative definite, as we have seen above. Theorem 2.5 now implies that for independent real-valued random variables Xl' ... ' X n with distributions J1b ... , Iln (resp. the same average distribution ji = (lin) 'L7= I Ili)
239
§2. Extension to Nondiscrete Semigroups
If the first moments of J.ll' .•. , J.ln exist, then subtracting [lE p(Sn)]2 [IE JL1 , ... ,JLn(Sn)]2 in the above inequality we get
=
varii(Sn) ~ varJL1, ... ,JLn(Sn), i.e. the variance of the sum gets maximal, if all the Xi have the same distribution. On the other hand from "i.J
S;
- 2 = "i.J Xi2 - - , (Xi - X) n
-
Sn n
X=- ,
and IEti(L Xl) = IEJL1, ... ,JLn(L Xl) we see that (in case the second moments are finite)
-) IE-JL [ X· - X n _1-1 L (1
2] =< IEJL 1, ..., JLn [- )2], X· - X n _1-1 L (1
i.e. the mean empirical variance gets minimal in the i.i.d.-case. 2.7. Exercise. Let X, Y be independent Z-valued random variables with distribution J.l (resp. v) and a fixed average distribution K = t<J.l + v). Show that IP JL, veX - Y is odd) gets minimal for J.l
= v.
2.8. Exercise. Let Xl' , X n be independent real-valued random variables with distributions J.ll' , J.ln. Show that for any (continuous) increasing function l/J: [R -+ [R the inequality
IEti[l/J(
m~x Xi)] ~ IEJL1, ...,JLn[l/J( m~x Xi)]
l~J~n
l~J~n
holds as soon as the left-hand side is defined. Hint: Assume first that l/J is bounded below and adjoin the neutral element - 00 to the semigroup (IR, max). 2.9. Exercise. Let l/J be a Borel measurable completely alternating function defined on a Hausdorff topological abelian semigroup S, and assume that (p, s) H p(s) is Borel measurable on S x S. Then
holds for all Radon probabilities J.ll' ... , J.ln on S. Hint: Use Fubini's theorem.
240
7. Hoetfding's Inequality and Multivariate Majorization
§3. Completely Negative Definite Functions and Schur-Monotonicity An important and widely used ordering on ~n is given by the so-called majorization. A vector x E ~n is majorized by Y E [Rn if-intuitively-its components are" less spread out" than those of Y; this means precisely k
k
L X[i] ~ LY[i]'
i= 1
k = 1, ... , n - 1
for
i= 1
and n
n
L X[i] = LY[i]' i= 1
i= 1
where X[l]
~
X[2]
~
•••
~
X[n]
are the components of x in decreasing order; we write X~Y
in this case. An elementary example is given by
(!, ...,!) ~ (_1_, ... , _1-1' 0) ~ ... ~ (t, t, 0, ... , 0) n
n
n-l
~
n-
(1, 0, ... , 0).
Many interesting characterizations of majorization have been found. Of fundamental importance are the following four equivalent statements (where denotes the set of all n x n doubly stochastic matrices, cf. Exercise 2.5.15, and where Y1t for Y E [Rn and a permutation n of {I, ... , n} denotes the vector
an
(Y1t(1)' ... , Y1t(n»)):
(i) x~y; (ii) x' = Py' for some P E (iii) x E conv{Y1t In a permutation of {I, ... , n} }; (iv) Li= 1 j'(Xi) ~ Li= 1 j'(Yi) for each convex functionj': [R
an;
-+ [R.
For a proof of these results we refer to the monograph of Marshall and Olkin (1979), which furthermore contains a wealth of material concerning majorization and a bibliography on this subject of approximately 450 items. The real-valued functions preserving the order of majorization are called Schur-convex, i.e. f: A -+ [R (where A s;; [Rn) is Schur-convex if x, yEA and x ~ Y implies f(x) ~ j'(y). Iff instead reverses the order it is called Schurconcave. For example, f(x) = Li= 1 (Xi - X)2 is Schur-convex on IR n (x:= xJn), and the well-known entropy function H(Pl'···' Pn) = Pi log Pi is Schur-concave on [0, IJn. Another example is given in the following
-I
I
§3. Completely Negative Definite Functions and Schur-Monotonicity
3.1. Lemma. On [R~ thejilnctionj'(x) =
n7=1
Xi
241
is Schur-concave.
PROOF. Let X, Y E [R~ be given with X ~ y. Since the log-function is concave on [R+ (zero included in the obvious sense), the equivalence of (i) and (iv) above implies
n n
n
Xi
=
erlogxi
~ erlogYi
i= I
= nYi'
D
i= 1
Among the four equivalent statements (i)-(iv) property (ii) immediately leads to a multivariate generalization of the majorization order: 3.2. Definition. Let A be a convex subset of some real linear space E. For two vectors X = (Xl' ... , x n) and Y = (YI' ... , Yn) whose components Xi and Yi belong to A, we say that X is majorized by Y and write X -< Y if X' = Py' for some P = (Pij) E On in the sense that n
Xi
=
L PijYj
j= 1
holds for i = 1, ... , n. A real-valued function j' on An is Schur-convex if yEAn and X ~ Y impliesj'(x) ~f(y);j'is called Schur-concave if -fis Schur-convex.
X,
In the following A will be a convex subset of the family of all probability measures on a given measurable space, for example, the set of all molecular probabilities on an abstract set S. If J.l1' ... , J.ln are probability measures and ji = (lin) 1 J.l;, then
L7=
is the doubly stochastic matrix with equal entries lin, hence ,J.ln)' It seems natural to examine if the Hoeffding-type inequalities proved so far are only special cases of more general results stating that where I
n
(ji, ... ,ji) is majorized by (J.lI, ...
is under appropriate assumptions on l/! a Schur-convex function of (J.l1' ... ,J.ln)· For n = 2 this turns out to be true in full generality.
3.3. Proposition. Let t/J: S x S ~ [R be a negative definite kernel and let VI' Vz, J.lI' J.lz be molecular probability measures on S. Ij'(v1' vz) ~ (J.l1' J.lz) then [VI, V2[l/!(X,
Y)] ~ [1l1,1l2[l/!(X, Y)].
(The interpretation is analogous to that of (4) in §1.)
242
7. Hoetfding's Inequality and Multivariate Majorization
PROOF. For some P E Q 2 we have v = PIl where v = (VI' V2)' and Il = (Ilb 1l2)" and P has the form
p
=
(1 : a 1: a)
for some a E [0, 1]. Let III and 112 both be concentrated on {Sl' ... ,sn} Then VI and V2 are also concentrated there and
~
S.
n
= For j
L
j,k= 1
ljJ(Sj' Sk)[V l ({Sj})V2({Sk}) - Ill({Sj})1l2({Sk})]'
=f k we find [Vl({Sj})V2({Sk}) - Ill({Sj})1l2({Sk})] + [{V l ({Sk})V2({Sj}) - Ill({Sk})1l2({Sj})] = 2a( 1 - a) [Ill ({Sj}) - 112( { Sj})] . [Ill ({Sk}) - 112( { Sk} )],
furthermore Vl({Sj})V2({Sj}) - Ill({Sj})1l2({Sj})
= a(1 - a)[lll({Sj}) - 1l2({Sj})]2,
so that n
lEv [ljJ(X, Y)] - IEJL[ljJ(X, Y)] = a(1 - a)
L CjCk ljJ(Sj' Sk)' j,k = 1
with Cj = Ill({Sj}) - 1l2({Sj}), j = 1, ... , n. Since above is indeed ~ O.
L Cj = 0
the expression []
3.4. Example. For independent integer-valued random variables X, Y with distributions Ill' 112 the probability IP JLI, JL2(X - Y is odd) is Schur-convex in Ill' 1l2; in case X
~
0, Y
~
IP JLI, JL2(X
+
Y is odd).
0 so is
3.5. Example. Let X, Y be independent with values in a separable Hilbert space. Then
is Schur-concave in Ill' 1l2'
3.6. Definition. Let S be an abelian semigroup. A function ljJ: S called Schur-increasing oj'order n, n = 1, 2, ... , if
~ [R
is
243
§3. Completely Negative Definite Functions and Schur-Monotonicity
is a Schur-convex function of (J.lI, ... , J.ln) E Mol~(s)n, i.e. if v and J.l = (J.lI'· .. , J.ln) in Mol~(s)n are such that V ~ J.l then
f
r/J d( v1 * ... * v")
~
f
=
(VI" .. , vn)
r/J d(fll * ... * fl")·
Moreover, l/J is called Schur-increasing, if it is Schur-increasing of order n for all n = 1, 2, .... Finally, l/J is called Schur-decreasing (resp. oforder n), if -l/J is Schur-increasing (resp. of order n). Any constant function is both Schur-increasing and Schur-decreasing, and it is not difficult to see that the functions which are both Schur-increasing and Schur-decreasing are precisely the functions l/J(s) = c + h(s) where c E IR and h: S -+ IR is additive, cf. Exercise 3.16. The set of Schur-increasing functions of order n is a closed convex cone in IRs which decreases with n. If l/J is Schur-increasing of order n then l/J satisfies, of course, Hoeffding's inequality of order n, and for n = 2 each of these properties is equivalent with l/J E %(S) by Proposition 3.3 and Corollary 1.4. By Theorem 1.7 it is natural to examine if a completely negative definite function is Schur-increasing. Closely related is the question if every completely positive definite function is Schur-decreasing. These questions will be answered in the negative below, cf. 3.13, but the answer is positive under suitable boundedness conditions on the function as the following result shows.
3.7. Theorem. Let S be an abelian semigroup. Then: (i) every bounded completely positive definite function is Schur-decreasing; (ii) every lower bounded completely negative definite function is Schur-
increasing. PROOF. A bounded completely positive definite function qJ is completely monotone by Theorem 1.10 and has therefore by Theorem 4.6.4 a representation
=
L
p(s) dK(p),
s+
where K E M + (S +). It therefore suffices to prove that every pES + is Schurdecreasing. We prove that every pESt has this property. Let v = (VI" .. , vn ) and J.l = (J.lI' ... , J.ln) E Mol ~ (s)n satisfy n
L PijJ.lj, i = 1, ... , n, where (Pij) E Qn' Putting bi = J P dVb aj = J p dJ.lj' b = (b Vi =
j= I
(aI' ... , an) we have n
bi =
L Pijaj j= I
l , "',
bn) and a =
244
7. Hoeffding's Inequality and Multivariate Majorization
hence b
~
a, so by Lemma 3.1
f
p d(v 1
* ... * vn)
=
II bi ~ II ai
=
f
* ... * P,n)'
p d(P,l
If I/J is lower bounded and completely negative definite then exp( - tt/J) is bounded and completely positive definite hence Schur-decreasing by (i) for t > O. Finally, '" =
lim! (1 - e-"") t
1-+0
is Schur-increasing as pointwise limit of Schur-increasing functions.
0
The above theorem leaves open the question if, for example, t/J(x) = -x 2 is Schur-increasing on (lR, +). This in fact is true and could even be proved directly, but we will see below that a much more general result can be shown. First of all we consider the case where the underlying semigroup is perfect. We need the following: 3.8. Lemma. Let S be a per.fect semigroup and let lfJ: S -+ IR be completely positive definite. Then the Radon measure representing lfJ is concentrated on
st· PROOF.
Let J.,l E M +(S*) be the Radon measure such that cp(s)
=
f p(s) dp,(p),
s E S.
By assumption there are uniquely determined measures J.,la E M +(S*), a E S, such that
E"cp(s) = cp(a
f p(s) dp,,,(p),
+ s) =
sES
implying J.,la
The sets
Ta:= {p E S* Ip(a) <
= p(a)J.,l.
O} are open in S* and
p,,,(T,,) =
f
p(a) dp,(p)
Til
shows that J.,l( 7;,) = 0; therefore J.t(S*\St) =
J.,l(U 7;,) =
0
aeS
since J.,l is a Radon measure.
o
245
§3. Completely Negative Definite Functions and Schur-Monotonicity
3.9. Theorem. For an abelian semigroup S thejollowing conditions are equivalent: (i) Every completely positive definite junction is Schur-decreasing. (ii) Every completely negative definite junction is Schur-increasing.
Ij'S is perject or if'S = No then (i) and (ii) are verified. "(i) => (ii)" If l/J is completely negative definite then exp( - tl/J) is completely positive definite, hence Schur-decreasing for t > O. Therefore
PROOF.
t/J = lim ~ (l - e- f"') l-+O
t
is Schur-increasing. "(ii) => (i)" If qJ is completely positive definite then - qJ is completely negative definite and hence Schur-increasing. Every pESt is Schur-decreasing as shown in the proof of Theorem 3.7, and therefore every moment function qJ E .1f(S) with representing measure Il supported by st is Schur-decreasing. This shows that (i) holds for S = No because a completely positive definite function on No is a Stieltjes moment sequence, cf. 6.2.5. Furthermore (i) holds for a perfect semigroup by Lemma
3.8.
[J
3.10. Corollary. Let S be a Hausdorfftopological abelian semigroup containing a dense perject semigroup. Then every continuous completely positive (resp. negative) definite function qJ: S -+ lR is Schur-decreasing (resp. increasing). PROOF. Let two vectors Il = (Ilb . .. , Iln) and v = (VI, ... , vn) of molecular probability measures on S be given such that V ~ Il. There is a finite set A s;; S on which all Ili and hence all Vi are concentrated. Approximating each sEA by a suitable net of points from the given dense perfect subsemigroup P we can find nets Il(i) = (IlY), ... ,Il~i») and v(i) = (vy), ... , V~i») from Mol~(p)n such that v(i) ~ Il(i) and
J qJ
d(Vl
* ... * vn) =
li:n
~ li:n
J J qJ
d(vY)
*
* v~i)
qJ
d(J.LY)
*
* J.L~» =
J qJ
d(J.Ll
* ···* J.Ln),
where we have used that qJ IP is completely positive definite and hence Schur-decreasing. The case of qJ being completely negative definite is treated similarly. D This corollary applies, in particular, to the important semigroups (IR~, +) and (lR\ +). Since they are 2-divisible, we conclude that continuous negative definite functions are the same as continuous Schur-increasing functions on these semigroups.
246
7. Hoeffding's Inequality and Multivariate Majorization
In Proposition 1.11 we saw that a nonnegative quadratic form q on an abelian group G fulfils the reverse Hoeffding inequality of each order. As we shall see now q is even Schur-decreasing. It should be noticed that G is considered as a semigroup with the identical involution, so there is no contradiction in the fact that q is negative definite in the group sense and -q is completely negative definite in the semigroup sense.
3.11. Proposition. A nonnegative quadratic form q on an abelian group G is Sc hur-decreasing.
= 7L k for some k
PROOF. We consider first the case where G
~ 1. In the proof
of Theorem 4.3.9 it was shown that B(s, t)
:=
q(s)
+ q(t)
- q(s - t),
is biadditive and, furthermore, that B is a positive definite kernel. Denoting e1 = (1, 0, ... ,0), ... ,ek = (0, ... ,0, 1) we see, in particular, that the matrix (B(ei' ej))i,j= 1, ... ,k is positive definite. Noting q(O) = this implies
°
k
q(s)
=
=
tB(s, s)
tL
i,j= 1
sisjB(ej, ej)
and q is therefore the restriction of a continuous nonnegative quadratic form on IRk. The result follows now from 3.10 and 1.11 since IRk contains the dense perfect semigroup Qk. If G is arbitrary, let J.11' .•• , J.1n be all concentrated on the finite subset A = {S1"." Sk} S;; S implying vi(A) = 1 for all i = 1, ... , n if v -< J.1. We define a canonical group homomorphism h: 7l. k ~ G by h(z 1, ... ,Zk) := k L~= 1 ZjSj' Then q 0 h is a nonnegative quadratic form on 7L • Let ii and v k on 7L be defined by k
iii:=
k
L J.1iC {Sj} )eej'
Vi:=
j= 1
j= 1
Then ji~
= J.1i'
v~
=
Vi
for i
L viC {Sj} )eej'
= 1, ... , n and
f q d(Vl * ·..
*V n )
= =
=
~
V -< ji, implying
f q d(v~ * ·.. * v~)
f
q d[ (v 1 * ...
fq
* Vn)h]
h d(Vl
* .·.* vn)
q 0 h d(ill
* ... * iln)
0
f f
= q d(J.ll * .. ·* J.ln), where we have used the result of Exercise 2.3.12.
o
§3. Completely Negative Definite Functions and Schur-Monotonicity
247
3.12. As in the case of Hoeffding's inequality we can extend the results proved so far in this section to nondiscrete distributions, assuming, of course, continuity of the positive and negative definite functions (kernels) to be integrated, and assuming furthermore that the probabilities involved are Radon measures. To prove these extensions one uses the same ideas as in §2; cf., in particular, the proofs of Theorems 2.1 and 2.5. We will now show the existence of completely negative definite functions on N6 which are not Schur-increasing. By Theorem 3.9 this is equivalent with the existence of completely positive definite functions on N6 which are not Schur-decreasing. We prove even more: 3.13. Proposition. There exist completely positive definite functions on N6 which are not Schur-decreasing oj'order 3. PROOF.
Let B denote the set of polynomials
p(x, y) =
,
~
. k
CjkX)y
L
in two variables where cjk ~ 0 and p(l, 1) = Cjk = 1. In accordance with Definition 3.2 we consider the ordering -< on B 3 and define the set 13 = {qlq2q3 - PIP2P31p = (Pb P2' P3), q = (ql' q2' q3) E B 3, q -< p}. For a function qJ: N6
-+ ~
and a polynomial P E
A(2)
given as above we define
L qJ((j, k))Cjk'
and the vector spaces 1R~5 and A(2) form a dual pair under <.,.). With J1 E Mol~(N6) is associated a polynomial pEB by defining the coefficient of x j yk as C jk = J1( {(j, k)}). The mapping J1 r-+ P is clearly an affine isomorphism of Mol~(N6) onto B and convolution of measures corresponds to products of polynomials. Therefore qJ: N6 -+ IR is Schur-decreasing of order 3 if and only if
~
0
for all
rEB.
In the proof of Theorem 6.3.7 we introduced the cone C(2) with the property that
for all r
E C(2)
if and only if qJ is completely positive definite. Since C(2) is a closed convex cone by Lemma 6.3.9 it suffices to prove that B\C(2) =t= 0. In fact, having found a polynomial Po EB\C(2), there exists a function qJ: N6 -+ ~ by the separation theorem (1.2.3) such that
and
for all
r E C(2).
This function qJ is completely positive definite but not Schur-decreasing of order 3. The proof will be finished by the following lemma:
248
7, Hoeffding's Inequality and Multivariate Majorization
3.14. Lemma. Let P = (x, y, 1) E B 3, let PIX E 0. 3 be the matrix
° ° ( rJ.
= 1-
PIX
1-
rJ.
1-
rJ.
rJ.
rJ.)
rJ.
rJ.E
,
[0,1],
°
and let q = (qb q2' q3) be defined by q' = PIXP'· Then rlX := q1q2q3 - P1P2P3 is given by rlX(x, y) = rJ.(1 - rJ.)[rJ.(x + y2 + x 2y) + (1 - rJ.)(y + x 2 + xy2) - 3xy], and rlX E B\C(2) if'and only it'rJ. E [0, 1]\{0,!, I}. PROOF.
The formula for rlX is easily verified and we see that ro = r 1 =
that
°
and
8r1/2 = x + Y + x 2 + y2 - 6xy + x 2y + xy2 = (x - y)2 + x(1 - y)2 + y(1 - X)2, which shows that r1/2 E C(2). We show next that 'IX:= {rJ.(1 - rJ.)} -1 rlX E C(2) implies rJ. = !. If 'IX = a + xb + yc + xyd with a, b, c, dE 'L(2) we see that a, band care of even degree ;£ 2 and d is a nonnegative constant because 'IX is of degree 3. By assumption " 2 a -_i..J ai'
'IX(O,O)
=
°= L
'IX(X,O)
=
rJ.x
'IX(O, y)
= (1 -
+
C
=
L cf,
ai(O, 0)2, (1 - rJ.)x 2 =
L alx, 0)2 + x L bi(x, 0)2, rJ.)y + rJ.y2 = L ai(O, y)2 + Y L Ci(O, y)2,
which shows that ai(O, O) = 0, bi(x,O) = Pi' Ci(O, y) = Yi for certain constants Pi' Yi E IR. This means that we can write
bi(x, y) = Pi Ci(X, y) = Yi
+ p~y, + y~x,
for certain constants bi , Ci' P~, y~ E IR. If we insert these expressions and compare coefficients we find
L Pf = L cf = L y~2, 1 - rJ. = L yf = L bf = L p~2, -3 = 2 L biCi + 2 L PiP~ + 2 L YiY~ + d. rJ. =
§3. Completely Negative Definite Functions and Schur-Monotonicity
249
We apply the Cauchy-Schwarz inequality to the sums in the last formula and get
!(d
+ 3) ~
3Je«1 - e<),
and since d ~ 0 this implies Je«1 - e<) ~ ~ 0, hence e< = !.
!
which is equivalent to (e< _ !)2
D
3.15. Remarks. (i) If rEB then r(x, y) ~ 0 for (x, y) E IR~ and r(l, 1) =
o.
The positivity assertion follows from Lemma 3.1. The existence of a polynomial Po E B\ C(2) also implies that there exists a completely positive definite function on N6 which is not a Stieltjes moment function. This assertion is weaker than the result of Theorem 6.3.7. (ii) The result of 3.13 is best possible in the sense that every positive definite function and hence afortiori every completely positive definite function on an abelian semigroup is Schur-decreasing of order 2.
3.16. Exercise. Show that the sum of a constant function and an additive function is both Schur-increasing and Schur-decreasing. Show conversely that if qJ: S ~ IR is both Schur-increasing and Schur-decreasing of order 2, then qJ is sum of a constant and an additive function.
3.17. Exercise. Show that if qJ: S ~ IR is Schur-increasing of order n + 1 then Ea qJ is Schur-increasing of order n for all a E S. Show that if qJ is Schurincreasing then so is Ea qJ for all a E S. 3.18. Exercise. Show that
where C(k) ~ A(k) is the cone introduced in the proof of Theorem 6.3.7. Hint: If the polynomial in question is not in C(k), then there exists a completely positive definite function on N~ which does not verify the reverse Hoeffding inequality of order k.
3.19. Exercise. Let X I'
... , X n be independent integrable real-valued random variables with distributions Ill' ... , Iln. Then
var I11 , •.•
,l1n(t Xi) 1=
I
is a Schur-concave function of Ill' ... , Iln' whereas [111, ...
,l1n[_I-1 I. (Xi n i=l
is Schur-convex in Ill' ... , Iln (if [(Xf) <
CfJ
xyJ
for all i).
250
7. Hoeffding's Inequality and Multivariate Majorization
3.20. Exercise. Show that the semigroup (Z, +) has the two equivalent properties of Theorem 3.9. Hint: See Exercise 6.4.7. 3.21. Exercise. A doubly stochastic matrix PEOn is called elementary if at most two off-diagonal entries of P are nonzero. Let r n £; On be the set of all products of finitely many elementary doubly stochastic n x n matrices. If t/J: S -+ IR is completely negative definite and v, ,u E (Mol~(s»n are such that v' = P,u' for some P E r n, the closure of r n, then
ft/J
d( VI
f
* ... * vn) ~ t/J d(Jil *... * Jin)·
3.22. Exercise. Let B(k) denote the set of polynomials p(x) = variables with C~ ~ 0 and p(l) further B~k)
= 2, 3,
(i) B~)
£;
L
C(X
k
= 1, where I = (1, ... , 1) E IRk. Let
qn - P1 ... Pn Ip = (Pb ... , Pn), q = (q1' ... , qn) E (B(k»n, q
= {q1
for n
=
L caxa in
-< p}
Show that
B~) ~
"', B~k) ~ {p E A~)([O, oo[k)lp(l) = O};
(ii)
C(B~k»
= {p E L(k) Ip(l) = O},
C(B~k»
;2
{p E C(k) Ip(l) = O},
where the smallest convex cone in A(k) containing a subset M ~ is denoted C(M), and C(k) is defined in 6.3.7; (iii) qJ: N~ -+ IR is Schur-decreasing of order n if and only if
~ 0
for all
A(k)
p E B~k).
Notes and Remarks Most of the results in §§1 and 2 were published in Christensen and Ressel (1981) and in Ressel (1982a). The nondiscrete extension of Hoeffding's inequality is shown in these two papers to hold even for r-smooth probability measures on arbitrary abelian topological semigroups (the Hausdorff property is not essential). Theorem 1.7 shows that Hoeffding's inequality of order 3 plays a particular role; it is unclear if the number 3 can be replaced by some larger kEN. Exercise 1.17 is taken from Karlin and Novikoff (1963). Our Definition 3.2 of multivariate majorization coincides for A ~ IR m with that given in Marshall and Olkin (1979, Ch. 15, A.2). On page 14 of this monograph it is stated that the classical Schur-convex functions should rather be called Schur-increasing, since they preserve the Schur majorization order x -< y. This is the reason for the terminology introduced in 3.6. Sharpening the majorization order on (Mol~(s»n by requiring the doubly stochastic matrix in the definition of that ordering to belong to r n(cf. Exercise 3.21), we
Notes and Remarks
251
get what Marshall and Olkin (1979, Ch. 15, 0.1) called chain majorization. Lemma 3.14 together with Exercise 3.21 imply, in particular, that f 3 is not dense in 0 3 . More precisely the matrices P ex defined in Lemma 3.14 belong to Q 3 \ f 3 for rJ. E ]0, t[ u ]1, 1[. But even P 1/2 ~ f 3 and this follows from the fact that the set of doubly stochastic n x n matrices P = (Pij) such that Idet P I ~ = 1 Pi, n(i) for at least one permutation 1t of {I, ... , n} is a closed subsemigroup of On containing f n, and that (for n = 3) the matrix P 1 / 2 does not have this property (Fuglede and Johansen, private communication). The existence of completely positive definite functions which are not Schurdecreasing of order 3 was announced in Berg and Christensen (1983b). The result of Exercise 3.18 has been shown by explicit formulas by Motzkin (1967), but for k ~ 3 such formulas are complicated. After the completion of this manuscript the following result has been obtained: On a 2-divisible semigroup every negative definite function is Schur-increasing. See Berg (1984b).
TIi
CHAPTER 8
Positive and Negative Definite Functions on Abelian Semigroups Without Zero
The integral representation theorems for posItIve and negative definite functions obtained so far were all proved under the assumption that the semigroup contained a neutral element, i.e. a "zero" with respect to the additively written semigroup operation. We also saw that some boundedness conditions were necessary for the functions under consideration in order to prove the main representation results. In this chapter we shall see that on abelian semigroups without zero, too, some interesting integral representation theorems for positive and negative definite functions can be proved, as long as unboundedness (resp. unboundedness below) is only allowed"" near the (nonexisting) zero element ", a condition to be made precise below. All semigroups considered in this chapter are assumed to carry the identical involution.
§1. Quasibounded Positive and Negative Definite Functions Let (H, +) denote an abelian semigroup. We now have in mind a semigroup without zero, but formally this will not be required. An element h E H is called maximal if it is not of the form hi + h 2 with hi' h 2 E H, cf. Choquet (1954, p. 262). Let H' := H + H be the subsemigroup of nonmaximal elements of H. Note that H' = H whenever H is 2-divisible, but, for example, H' = H also for the non-2-divisible semigroup H = (] -1,1[, .). Put S := H u {O} with some element 0 ~ H. Let 0 + h = h + 0 := h for any h E H, and 0 + 0:= 0; then S is an abelian semigroup with neutral element 0, and Scontains the absorbing element 1{O}' cf. 4.1.3.
253
§ 1. Quasibounded Positive and Negative Definite Functions
If we use the original definition of positive and negative definite functions also for semigroups without zero, then clearly the values at maximal elements of H do not enter into the defining property. A positive (resp. negative) definite function on H is therefore a function qJ: H' -+ ~ such that the kernel (h, k) ~ qJ(h + k) is positive (resp. negative) definite on H x H. The set of positive (resp. negative) definite functions on H is denoted fJJ(H) (resp. %(H». If H already contains a neutral element, then H' = H and the notion agrees with that already given in Chapter 4. Let us agree in calling a function qJ: H' -+ ~ quasibounded (resp. quasibounded below) if
+ s) I <
CIJ
for all
h'
E
H'
+ s) > -
CIJ
for all
h'
E
H').
sup IqJ(h' seS (resp. inf qJ(h' seS
Furthermore we introduce r!J>'l(H) to be the set of all quasibounded positive definite functions on H. Some simple properties of r!J>'l(H) are stated below.
1.1. Lemma. (i) The set r!J>'l(H) is a closed convex cone in ~H', stable under multiplication. (ii) For qJ E r!J>'l(H) and s E S the two functions h' ....... qJ(h') ± qJ(s + h') belong also to fJJq(H). (iii) If qJ E fJJq(H) and h, k E H, then s ....... qJ(s
+ 2h) + qJ(s + 2k)
- 2qJ(s
+ h + k)
belongs to fJJb(S). (i) If qJ E r!J>'l(H) then s ....... qJ(s + 2h) is bounded and positive definite on S, hence by Proposition 4.1.12 sUPseS IqJ(s + 2h) I = qJ(2h) and furthermore PROOF.
IqJ(h + k + s) I ~ [qJ(2h)qJ(2k + 2s)] 1/2, showing immediately the closedness of r!J>'l(H). Stability under multiplication is obvious. (ii) If h b . . . , hn E Hand Cl' ... , Cn E ~ then <1>: S -+ ~ defined by n
<1>(s):=
L
i,j= 1
CiCjqJ(S
+ hi + hj)
is easily seen to belong to fJJb(S), cf. the proof of Theorem 4.1.14, hence, again by 4.1.12, 1<1>(s) 1 ~ <1>(0) for all S E S, i.e.
L Cicj[qJ(h i + hj) ± qJ(s + hi + hj)] ~ o.
i,j= 1 C2
(iii) Take <1> as defined above for n = -1.
= 2, hI = h, h 2 = k,
Cl
= 1 and D
254
8. Positive and Negative Definite Functions on Abelian Semigroups Without Zero
Let qJ E fJJq(H) and put qJh, k(S) := qJ(s + h + k) for S E Sand h, k E H. We mentioned already that qJh := qJh,h E fJJb(S), hence by Theorem 4.2.8 there are uniquely determined Radon measures Ilh E M +(5) such that
for all
SE
S.
Furthermore, in view of the above lemma, there are measures Vh,k E M +(5) with
+ 2h)
- 2
+ h + k) +
for all S E Sand h, k E H. Put Ilh. k := !(Ilh Ilh,h = Ilh E M +(5) and
+ h + k) =
+ Ilk
Lp(s) dllh,k(P)
-
for
Lp(s) dVh,k(P) Vh.k),
SE
then Ilh. k E M(5),
S, h, k E H.
Our aim is to get an integral representation for quasibounded positive definite functions on H, but in order to avoid pathologies we have to restrict ourselves to the subcone q fJJ'b(H):= {qJEfJJ (H)lllh,k({l{o}}) =
o for all h, kEH}.
Notice that qJ E fJJq(H) belongs to fJJ'b(H) if and only if for all
h, k E H,
and from this remark it follows that fJJ'b(H) is an extreme subset of [J}Jq(H). The example H = ([1,00[, +) and qJ = 1f2l shows the possibility of fJJ'b(H) c fJJq(H). For semigroups without maximal elements, however, :f
examples of this kind are not possible, as the next lemma shows.
1.2. Lemma. Let qJ belong to fJJq(H). Then we have: (i) ~f 2H + S = H' and Ilh( {I {Ol}) = 0 for all h E H, it .follows qJ E r!J6(H). (ii) If H = H', then fJJ'b(H) = &q(H).
(i) Let h, k E H; we can find ho E H and a E S such that h + k = 2h o + a. For any S E S we have
PROOF.
Lp(S) dllh,k(P) =
+ h + k) =
+ 2h o + a)
=
255
§l. Quasibounded Positive and Negative Definite Functions
(ii) For given h, k E H we find hI' h2
Lp(S} dllh,k(P}
E
H such that h = hI
+ h2 • From
=
for all
SE
S we deduce J1h. k = P(k)J1h t • h2' and therefore J1h,k({I{o}})
D
= O.
1.3. Remarks. (1) If H contains already a neutral element, then it is easily seen that
is a topological semigroup isomorphism, showing, in particular, that l{o} in this case is an isolated point of S. (2) The set H":= 2H + S is always a subsemigroup of H'. For H = (N, +) we have H" = H' <:f H. That H" <:f H' is possible is shown by the semigroup H = (~\{1}, +). 1.4. Theorem. Let qJ E P/J1J(H) be given. Then there exists a uniquely determined Radon measure J1 on the locally compact space H := S\ {l{o}} such that
jor all h' E H'.
= J,P(h'} dll(P}
hz, k l , k 2 E H we have
Lp(S + hi + hz } dll k"k2(p} =
= Lp(S + k l + k z } dll ht ,h2(p} for all s E S, giving us
p(h l Let
(!)h
+ h2 )J1k l,k2 =
p(k l
+ k 2 )J1hl,h2·
= {p E SIp(h) =F O}; then (!)h is open and
UheH (!)h
= H. Setting
1 J1
= p2(h) J1h
we obtain by Theorem 2.1.18 a well-defined Radon measure J1 on H. We claim that In fact, on
(!)k
we have
p(h l
+ h z}1l =
p(h l p(k
+ h2 ) + k} 11k =
Ilh"h2'
256
8. Positive and Negative Definite Functions on Abelian Semigroups Without Zero
(This also shows that J1 h t, h2 is concentrated on for h' = hI + h2 E H' l
J,P(h dJi(p) )
(!)ht
n
(!)h2')
Therefore, we get
= Jih',h,(H) = Jihl,h,(S) =
where we used that by hypothesis qJ E fJJ6(H). The unicity of J1 is easily seen in the following way: if v is another Radon measure on H representing qJ then
+ 2h) = Lp(s)p 2 (h) dJi(p) = Lp(s)p 2 (h) dv(p),
implying J11 (!)h = v I(!)h for each h E H and therefore J1 = v by 2.1.18.
D
1.5. Example. Let us consider the open half-line H = ]0, oo[ with usual addition. We have S = [0,00[, S ~ [0,00] (cf. 4.4.1) where 00 corresponds to the semicharacter s 1--+ e- 00 • s, i.e. to 1{O}. Hence the locally compact space H can be identified with [0, 00 [. In view of Lemma 1.2 and Theorem 1.4 each quasibounded positive definite function qJ: ]0, 00 [ ~ [R has the representation
=
r
e-
AS
dJi(A)
)[O,OO[
for some uniquely determined Radon measure J1 on [R +. This shows that quasibounded positive definite functions on ]0, 00 [ are exactly the completely monotone functions in the classical sense, cf. Corollary 4.6.14. A well-known example is the function l/s, for which the measure J1 in the above representation is just Lebesgue measure on [R + .
1.6. Example. Let H be the additive semigroup of natural numbers, i.e. = (N, +). Then S = ~o, S = [ -1,1] and H can be identified with [ -1, u J 0, 1J. The function n 1--+ lin is positive definite and even bounded
H
°[
on N. Its integral representation is simply
-1 = n
II 0
tn -
1
dt =
1 ]O,I[
tn .dt -. t
The function n 1--+ ( -l)n . l/n is positive definite, too, and has the representing measure It 1- 1 . 1[-1, O[(t) dt. A natural question arises at this point: Under what conditions can a given quasibounded positive definite function on H be extended to a bounded positive definite function on S? An answer is given by
1.7. Proposition. Let qJ belong to fJJ6(H) and let J1 be its desintegrating measure given by Theorem 1.4. Then qJ can be extended to ajunction <1> E fJJb(S) if'and only if'J1(H) < 00.
257
§l. Quasibounded Positive and Negative Definite Functions
Iii
PROOF. If J.1(H) is finite then (s) := p(s) dJ.1(p) defines a positive definite extension. On the other hand, suppose that (s) = p(s) dv(p) with v E M +(S) is an extension of qJ. Then by the unicity statement in Theorem 1.4 we get J.1 = v on H, in particular, J.1(H) < 00. D
Is
From this result we see, for example, that the positive definite function n ~ lin on N, although being bounded, cannot be extended to a positive definite function on No. We introduce now %q(H), the set of all negative definite functions l/J: H' -+ ~ which are quasibounded below. If l/J E %q(H) and h E H then s ~ l/J(2h + s) is negative definite and bounded below, hence t/J(2h + s) ~ l/J(2h) by Corollary 4.3.2. Some simple properties of %q(H) are contained in the following lemma:
1.8. Lemma. (i) %q(H) is a closed convex cone in
~H'.
(ii) - f!JJ'l(H) £; %q(H). (iii) l/J E %q(H) if and only if exp( - tt/J) E f!JJ'l(H) for all t > O. (iv) Put ~al/J(h'):= l/J(a + h') - t/J(h')jor h' E H' and a E S. Then ifl/J we get ~at/J E gpq(H)jor all a E S.
E
%q(H)
PROOF. (ii) is a trivial property. (iii) follows from Theorem 3.2.2. (iii) together with Lemma 1.1 implies (i). Concerning (iv) we observe that if t/J E -f!JJ'l(H) then ~at/J E f!JJ'l(H) by Lemma 1.1. Using the closedness of f!JJ'l(H) and the fact that
IjJ = lim ~ [l - exp( - tljJ)], 1-0
we get
~a
t
l/J E f?j>q( H) for every l/J E JVq( H).
D
Let us consider for a moment the semigroup S with neutral element. If
l/J E %(S) is bounded below and ljJ(s)
= c + a(s) +
r
(l - p(s» dJi(p)
JS\{l}
is the uniquely determined representation given by Theorem 4.3.20, where CE~, rx:S-+ ~+ is additive and J.1EM+(S\{l}), and if ~at/J(s) = t/J(s + a) - l/J(s) = fja(s), then (1)
and (1 - p(a))J.1
=
(ja
on S\{l},
implying J.1({1{0}}) = (ja({l{O}}) for all a E S\{O} = H. Motivated by this result and taking into account the last lemma we define %fi(H) := {t/J
E
%q(H) I~at/J E gp'6(H) for aU a E H}.
258
80 Positive and Negative Definite Functions on Abelian Semigroups Without Zero
It is easily seen that JVfi(H) is an extreme subcone of J¥°q(H), and from Lemma 1.2 it follows immediately that the two cones coincide in case H is without maximal elements.
1.9. Theorem. Let t/J E JV'6(H) be given. Then there exist an additive function (1:S -+ ~+ and a Radon measure J1 on the locally compact space H\{I} such that for h', k' E H' ljJ(h') - ljJ(k')
= 1_
(p(k') - p(h'» dJ1(p)
+ rx(h')
- rx(k').
JH\{l}
The pair «(1, J1) is uniquely determined and given in the following way:
+ ns)
t/J(2h
(1() s = 11m 0
n
n-cc
independent oj' h E H, (!)a = {p E Hlp(a) < I},
on the open set where (Ja is the Radon measure on H satisfying AaljJ(h')
h'EH'.
= LiP(h') d(Ja(P),
PROOF. From ~al/JEg;q(H) we get, using Lemma 1.8, that and hence ~b( -~at/J) E 9;q(H). We have
-~at/JEJVq(H)
+ b) + ~at/J(h') + b + h') + t/J(h' + b) + t/J(h' + a)
[~b( -~at/J)](h') = -~at/J(h'
= -t/J(a
= L/(h') d(Ja(P) - L/(h'
= L,P(h')[l - p(b)] d(Ja(P)
+
- t/J(h')
b) d(Ja(P)
= L,P(h')[l - p(a)] d(Jb(P)
for all h' E H', and the unicity statement in Theorem 1.4 tells us that
[1 - p(b)](Ja = [1 - p(a)](Jb
on
Taking into account that the open sets (!) a' a E H, cover again Theorem 2.1.18 we conclude that
J1 := [1 - p(a)] - 1 (J a
on
H. H\ {I}, and invoking
(!) a
is a well-defined Radon measure on H\{l} satisfying [1 - p(a)]J1 = (Ja on H\{l}. Hence AaljJ(2h)
=
Iv JH\{l}
for all h E H and a E S.
p 2 (h)[l - p(a)] dJ1(p)
+ (Ja({l})
259
§l. Quasibounded Positive and Negative Definite Functions
Now we observe that ~st/J(2h) = t/J(2h + s) - t/J(2h) =: t/Jh(S) is nonnegative and negative definite as a function of s; hence t/J h has the representation ljIh(S)
= ah(s) + fA
[1 - p(s)] dJ1.h(P)
JS\{l}
with an additive function (Xh: S
-+
+ s)
f,
ljI(2h
- ljI(2h)
=
~+ and f.1h
E
M +(S\{l}). We get
p2(h)[1 - p(s)] dJ1.(p)
+ o-s({I})
JH\{l}
f
= ah(s) +
[l - p(s)] dJ1.h(P)
JS\{l}
for all hE H, S E S. From relation (1) we learn that (Xh(S) = !s,h({l}) where is,h = ~sljJh' Put h' = 2h + a, then ~st/J(h')
= t/J(2h + a + s) - t/J(2h + a) =
~st/Jh(a)
= L-/(h') do-s(p) = f/(a)p2(h) do-s(p), and therefore !s, h = p2(h)O's implying (Xh(S) = !s.h({I}) = O's({I}), independent of h and additive in S. Put (X(s) := O's( {I}), then indeed
SE
ljI(h'
+ s) - ljI(h')
=
=
t!sljl(h') = Lp(h') do-s(p)
f,
p(h') do-.(p)
+ o-s({I})
JH\{l}
=
f,
p(h')[1 - p(s)] dJ1.(p)
+ a(s)
JH\{1}
for all h' E H' and S E S. Computing in this way t/J(h' ljJ(h' + k') - t/J(k') for h', k' E H' gives us, finally, ljI(h') - ljI(k')
=
f,
[p(k') - p(h')] dJ1.(p)
JH\{l}
+ k')
+ a(h')
- t/J(h') and
- a(k')
as asserted. From Theorem 4.3.20 we know that a(s)
=
ah(s)
= lim! ljIh(ns) = lim! ljI(2h + ns) n-+oo
n
n-+oo
n
independent of h E H. Hence (X is uniquely determined and so then is f.1, because any representing measure must coincide on (!Ja with [1 - p(a)]-lO'a D by immediate calculation, using the unicity statement in Theorem 1.4.
260
8. Positive and Negative Definite Functions on Abelian Semigroups Without Zero
The integral representations of 1.4 and 1.9 lead naturally to the question about a converse statement, to be answered now. 1.10. Proposition. Put Xh'(P):= p(h')jor h' E H' and P E
H.
(i) If J.1 is a Radon measure on fl such that Xh' E !£1(J.1) for each h' E H', then qJ: H' -+ IR defined by qJ(h') := Xh' dJ.1 belongs to f!JJ1J(H). (ii) Ij' J.1 is a Radon measure on H\ {I} such that Xh' - Xk' E fLJI(J.1) jor all h', k' E H', and if'a: S -+ IR+ is additive, then t/J: H' -+ IR defined by
Iii
r.
t/J(h'):=
+ rx(h'),
(Xk' - Xh') dJi
JH\{l}
k' E H' arbitrary but fixed, belongs to %'6(H). PROOF. (i) Of course
suplqJ(h' seS
qJ is positive definite and for each h' E H'
+ s)1 ~
+ s)1 dJ.1(p) ~
sup fIP(h' seS
fIP(h')1 dJ.1(p) <
00
showing qJ to be quasibounded. The measures J.1h,k occurring in the definition of r!Pb(H) are given by
J.1h,k = p(h)p(k)J.1 so that J.1h,k({I{o}}) = 0 for all h, k E H and therefore qJ E f!JJb(H). (ii) Let hi, ... , hn E Hand Cb ... , Cn E IR be given such that Ci = O. Then
L
.t
1,J-
c;cjt/J(h;
+ h) =
I
f L. c;clp(k') -
p(h;
+ h)] dJi(p)
t, )
+ L cicj[a(h i) + a(h j)] i,j
Furthermore, for h' t/J(h'
+ s) ~
f
= hi + h2 E H'
[p(k') - p(h'
~
+ s)] dJi(p)
f [p(k') - J p(2h J p(2h + 2s)] dJi(p) ~ f [p(k') - tp(2h tp(2h + 2s)] dJi(p) 1) •
1) -
~t >
f
[p(k') - p(2h 1 )] dJi(p)
-00
2
2
+t
f
[p(k') - p(2h 2 )] dJi(p)
261
§l. Quasibounded Positive and Negative Definite Functions
for all
SE
S; hence
l/J is quasibounded below. Finally, for a E Sand h' E H'
~aljJ(h') = r~
[1 - p(a)]p(h') dJi(p)
+ o:(a)
JH\{l}
=
L/(h') daa(p),
where aa(P) = [1 - p(a)]Jl on H\{l} and aa({l}) = Cl(a), implying ~al/J E f7Jb(H) by (i), hence indeed l/J E %b(H) as asserted. 0 Looking once more at the proof of 1.9, we see that only the property Ia E H} £; &'b(H) has been used. Together with the above proposition this means, that any function l/J: H' --+ ~ such that ~a l/J E &'b(H) for all a E H belongs already to %b(H), a partial converse to Lemma 1.8(iv). {~al/J
1.11. Example. Let again H be the additive semigroup ]0, 00[. In 1.5 we saw that fI can be identified with [0, 00 [, where the point corresponds to the semicharacter 1, hence H\{l} ~ ]0,00[. We have JVb(H) = %q(H) because H is 2-divisible. By Theorem 1.9 any negative definite function l/J: ]0, 00 [ --+ ~, quasibounded below, has the representation
°
ljJ(s) - ljJ(t)
=
r J]O,
(e- A'
-
e- AS) dJi()')
+ o:(s -
t)
oo[
for all s, t E ]0, 00 [, where Cl E IR + and Jl E M + (]O, 00 E). From this and Proposition 1.10 it follows that the functions in JVq(]O, 00 [) can be characterized as
where
ljJ(s)
=
IR, Cl ~
°
dJi()') <
00
CE
f).
c
+ o:s + {Xl(e- A -
e- AS ) dJi().),
S
> 0,
(2)
and Jl E M +(]O, oo[) is such that
From (2) we see that
f"
and
e- SA dJi()') <
00,
for all
S
> 0.
l/J is Coo and that ljJ'(s)
=
0:
+
{Xl
).e- AS
dJi()')
is completely monotone. Conversely, if t/J: ]0, 00 [ -+ IR is Coo and t/J' is completely monotone, we have by Corollary 4.6.14 that l/J' = !fa for some a E M + ([0, 00 [) and therefore ljJ'(s)
=
0:
+
{Xlk -
AS
dJi().),
with Cl = a( {O}) and Jl = A-I (a 1]0, 00 C), hence longs to %q(]O, 00 E).
l/J has the form (2) and be-
262
8. Positive and Negative Definite Functions on Abelian Semigroups Without Zero
An important example is given by the log-function whose integral representation reads log s - log
=
t
[00
J
o
(e- A1
e- AS )
-
dA
T'
s, t > 0,
or, taking t = 1, s>
o.
1.12. Example. For the additive semigroup N we found in 1.6 that N ~ [-1, 1]\{0}, hence N\{I} ~ [-1, 1]\{0, I}. A measure J.1 on N\{I} for which Proposition 1.10(ii) can be applied, is, for example, given by dt
The corresponding negative definite function tjJ: N' the values (choosing k' = 2) tjJ(2)
= 0,
tjJ(3)
t/J(2n
= tjJ(4) = 1,
+ 1) =
t/J(2n
tjJ(5)
=
{2, 3,4, ... } -+
~
has
= tjJ(6) = 1 + t, ... , 1
+ 2) = 1 + t + t + ... + -. n
The log-function restricted to N' is of course also negative definite there. It has the integral representation log n - log m =
f
(t m
-
tn)
]O,l[
From the description of q>q(] 0, 00 [) and JV'l(]O, 1.5 and 1.11 we have the following result:
-t
dt I . ogt
00 [)
1.13. Theorem. Ij' qJ E &>'l(]0, oo[) then tjJ: ]0, oo[
given in the Examples
-+ ~
defined by tjJ(s):=
J1 qJ(t) dt belongs to JVq(]O, oo[). Conversely, the derivative oj' any jiinction in JVq(]O,
00 [)
belongs to g>q(]O,
00
E).
1.14. Exercise. Let H be N k with componentwise addition. Show that fI can be identified with ([ -1, O[ u ]0, 1])k. Show that both n = (n t , ••• , nk) ~ (nt+···+nk)-l and n~(-l)nl+···+nl'(nl+···+nk)-t are positive definite (and quasibounded) and try to find their integral representations. 1.15. Exercise. Let H be the real line with maximum operation. Then g>q(H) is the set of all nonnegative decreasing functions, and JVq(H) consists of all increasing functions on ~. The same is true if we replace the line by an arbitrary nonempty subset of ~.
~2.
263
Completely Monotone and Completely Alternating Functions
1.16. Exercise. If H is the additive semigroup JO, oo[n, then FJ == [R~ (by natural identification). Show that all first partial derivatives of a function in %q(]O, 00 en) belong to gPq(]O, 00 en). 1.17. Exercise. Show that IIxll- I is not positive definite on ]0, oo[n for n > 1, 11·11
denoting the euclidean norm.
1.18. Exercise. Let l/J be a strictly positive, negative definite function on the abelian semigroup H. Then if j'E gPq(]O, oo[) we have f l/J E gPq(H), and if g E%q(]O, oo[), then g ffE %q(H). 0
0
1.19. Exercise . Use 1.18 to show that x 1---+ log(2:7 = I xJ is negative definite 00 [" and that x 1---+ (2:7=1 xJ - t is positive definite for all t > 0.
on JO,
1.20. Exercise. Let K be a compact Hausdorff space and let H be the set of all strictly positive continuous functions on K with pointwise addition. Show that H can be identified with M +(K), the set of Radon measures on K. 1.21. Exercise. Let l/J E %6(H) have the representation given in Theorem 1.9. Then l/J can be extended to a negative definite function on S, bounded below, if and only if SH\{l} [1 - p(h)] dp,(p) < 00 for all hE H.
°
1.22. Exercise. Let S be a perfect semigroup with zero element and identical involution, and assume that H:== S\{O} fulfils H == H + H. Then H is a perfect semigroup without zero in the sense that for every qJ E f2P(H) there is a unique Radon measure J1 on S* \ r 1{a}} such that qJ(h)
=
fp(h)d~(p),
hEH.
§2. Completely Monotone and Completely Alternating Functions Again let (H, +) denote an arbitrary abelian semigroup and put, as in the preceding section, S :== H u {a}, H :== S\ {I fO}}. In the definition of completely monotone (resp. alternating) functions on H no change is necessary compared with the definition for functions on semigroups with neutral element (cf. 4.6.1): we again require (resp.
~
0)
for all n E N, (hI' . .. ,hn) E Hn and all hE H. If one of the h/s is 0, this expression is still well defined and equal to zero, so that the condition is not changed by allowing (hI' ... , hn ) E sn. Of course a completely monotone function is also required to be nonnegative.
264
8" Positive and Negative Definite Functions on Abelian Semigroups Without Zero
We denote by ---'I(H) (resp. d(H)) the set of all completely monotone (resp. alternating) functions on H. Note that in contrast with the positive and negative definite functions which in a natural way are only defined on the subsemigroup H' of nonmaximal elements of H, the completely monotone (resp. alternating) functions are defined on all of H. For f: H -+ [R and h E H let fh: S -+ [R be defined by fh(S) := f(h + s), and for fixed s E S define ~sf: H -+ [R by ~sf(h) := f(h + s) - f(h). Let us first state some simple properties of ---'I(H) and d(H). 2.1. Lemma. (i) Both ---'I(H) and d(H) are closed convex cones in [RH. (ii) - ---'I(H) ~ d(H). (iii) A function qJ: H -+ [R belongs to ---'I(H) (resp. d(H)) if and only
---'I(S) (resp. qJh E d(S))for all hE H. (iv) A function t/J: H -+ ~ belongs to d(H) if and only function ~a t/J belongs to ---'I(H). (v) qJb qJ2 E ---'I(H) implies qJl . qJ2 E ---'I(H). (vi) l/J E d(H) if"and only if e-tl/J E ---'I(H)jor all t > O. PROOF.
if qJh E
if jor every a E S the
(i) and (ii) need no proof. (iii) follows from the identity
Val··· VanqJh(S) = Val··· VanqJ(h valid for all hE H, S E S, n E N, (a 1 , We get (iv) from
••• ,
+ s)
an) E sn.
Val··· Van(~al/J) = - Val··· VanVal/J, which holds for all a E S, n E N and (at, ... , an) E sn. Using (iii) and the fact that ---'I(S) is closed under pointwise multiplication we get property (v) observing also that (qJl . qJ2)h = (qJt)h . (qJ2)h. In the same D way (vi) follows from Proposition 4.6.10. Let qJ be a completely monotone function on H. Then the above lemma tells us that for each h E H the translate qJh belongs to ---'I(S) implying by Theorem 4.6.5 the representation
+ s) =
I
Js
p(s) dJ1.h(P)
A
+
with uniquely determined Radon measures J.1h on S+. Again we have to require that J.1h( {I {Ol}) = 0 for all h E H in order to get an integral representation on H. We put
which is an extreme subcone of ---'I(H). 2.2. Lemma. If H = H', in particular, if H is 2-divisible or contains a neutral element, then ---'I(H) = ---'lo(H).
if H already
265
§2. Completely Monotone and Completely Alternating Functions
Let qJ E .A(H) and let hE H have the form h = hi h2 EH. From
PROOF.
+ h2
with h b
for all s E S we conclude
whence
D 2.3. Theorem. For every qJ E .A o(H) there exists a uniquely determined Radon measure J.1 E M +(FI +) such that
jor all hE H. PROOF.
It is easy to see that
for all h, k E H; hence the Radon measure J.1 defined by 1 J.1:= p(h) J.1h is well defined on
on
fJ + and satisfies
fJ + . Furthermore,
gives for s
=0
for all h E H. This proves existence of J.1 and the uniqueness follows in the usual way. D By Lemma 2.1 we know that t/J E d(H) if and only if each a E S. We therefore introduce. d o(H)
and note that d(H)
:=
~at/J E
{t/J E d(H) I~at/J E.A o(H) for all a E S}
= d o(H) if H = H' by Lemma 2.2.
.A(H) for
266
8. Positive and Negative Definite Functions on Abelian Semigroups Without Zero
2.4. Theorem. Let l/J Ed o(H) be given. Then there exist an additive junction \1: S -+ ~+ and a Radon measure J.l E M +(fI + \{1}) such thatjor all h, k E H t{J(h) - t{J(k)
=
r v
[p(k) - p(h)] dJl(p)
+ rx(h)
- rx(k).
JH+\{l}
Both
(X
and J.l are uniquely determined.
We do not give the proof because it is very similar to that of Theorem 1.9. Proposition 1.10 has also its obvious counterpart. From Chapter 4, §6 we know that for an abelian semigroup S with neutral element we always have Jt(S) £; &,b(S) and d(S) £; JV1(S) with equality in case S is 2-divisible. Similar relations are, of course, also to be expected on semigroups without zero. We state them in the following theorem.
2.5. Theorem. The restriction mappings A 0(H) -+ P/6(H) (resp. d 0(H) -+ JVb(H)) given by j'~ j'l H' are injective. If' H' = H then .A o(H) is an extreme subcone oj'&'b(H) and .S# o(H) is extreme in JVb(H). For 2-divisible H we have A o(H) = &'b(H) and d o(H) = %b(H). PROOF. Let qJ E Jt o(H) be given. Then qJ has an integral representation qJ(h) = p(h) dJ.l(p) as stated in Theorem 2.3, from which it is immediately seen that qJ IH' is positive definite and, in fact, belongs to &'b(H). If qJl' qJ2 E Jt o(H) have the same restrictions to H', then their representing measures coincide on each (!) 2h = {p E S+ Ip(2h) > O} and are therefore equal on fI +. If H = H', then A o(H) £; P/b(H) and .A o(H) consists exactly of those functions in &'b(H) whose representing measure is concentrated on fI + , showing Jt o(H) to be extreme in &'b(H). In the 2-divisible case, of course, fI + = fI implying Jt o(H) = &'b(H). The corresponding statements concerning negative definite and completely alternating functions are proved analogously. D
J
2.6. Example. If H = (N, +) then fI = [ -1, O[ u ]0,1], see 1.6. Hence the positive part fI + is given by the interval ]0, 1]. The function n ~ lin is completely monotone, its representing measure 1]0, 1](t) . dtlt being concentrated on fI + . Likewise on N k the function
+ ... + nk)- 1 character 1 E No corresponds
n = (nl' ... , nk) ~ (nl
is completely monotone. Since the to 1 E [ -1,1], we have N+ \{1} ~ ]0,1[, and we see that the log-function already treated in Example 1.12 is indeed completely alternating with representing measure 1]0, l[(t) . dtl( - t log t).
2.7. Let (H, +) be an idempotent abelian semigroup without neutral element, i.e. h + h = h for all h E H. Then H = H' and defining the ordering ~ on H by h ~ k if h + k = k as in 4.4.16 we see that every positive definite
§2. Completely Monotone and Completely Alternating Functions
267
function qJ: H ~ IR is nonnegative, decreasing and quasibounded, hence by Lemma 1.2, Lemma 2.2 and Theorem 2.5 that f!JJ(H) = f!JJq(H) = f!JJb(H) = vIt(H) = uIto(H). Similarly, every negative definite function l/J: H ~ IR is increasing and quasibounded below and .AI(H) = .Alq(H) = .Alb(H) = d(H) = d o(H). If p E fl then Jp := {h E HI p(h) = 1} is a subsemigroup of H which is hereditary on the left and p I H = 1J ,cf. 4.4.16, and it is easily seen that p t---+ J p is a bijection of fl onto the set !/ ;f (nonempty) subsemigroups of H which are hereditary on the left. The bijection becomes a homeomorphism if !/ is equipped with the coarsest topology in which the sets h:={JE!/lhEJ},
hEH
are clopen (i.e. open and closed). Each set h is compact in !/. With the above notation we can specialize the Theorems 2.3 and 2.4 to give the following extension of Proposition 4.4.17:
2.8. Proposition. Let (H, +) be an idempotent semigroup. For every qJ E f!JJ(H) there is a unique J.1 E M +(!/) such that qJ(h) = J.1(h), and for
hEH,
l/J E .AI(H) there is a unique J.1 E M + (!/\ {H}) l/J(h) - l/J(k) = J.1(k\h) - J.1(h\k),
such that
h, k E H.
2.9. Example. Let X be a locally compact space and let H = (%, n) be the semigroup of compact subsets of X, the semigroup operation being intersection n. The empty set is an absorbing element, and H has a neutral element if and only if X is compact. The semigroup H is idempotent and the ordering of 2.7 is given by K ~ L<=> K ;2 L. The space !/ consists of the nonempty families J ~ eX with the following properties K, L
K
~
E
L, K
J E
=> K n L E J,
J => L E J,
and the topology on !/ is the coarsest in which the sets
K = {JE!/IKEJ},
KE%
are clopen. In continuation of the results of 4.6.15-4.6.19 we will study the completely monotone capacities on %, i.e. the functions qJ E uIt(H) which are continuous on the right in the sense of 4.6.15. We show below that such functions admit an integral representation over the set of semicharacters on % which are continuous on the right. We first show that the set of semicharacters on % which are continuous on the right can be identified with %. Let!/,. denote the set of J E!/ for which 1J is continuous on the right.
268
8. Positive and Negative Definite Functions on Abelian Semigroups Without Zero
2.10. Lemma. For LE% let J L = {KE%IL ~ K}. Then Lt---+J L is a bijection oj'% onto !/,., and the inverse mapping is J t---+ nKeJ K. It is obvious that J L E 9;. and that we consider the set L E % defined by
PROOF.
nKeJ L
K = L. Conversely, if J
E
9;.
and have clearly J L ;2 J. Assume L ¢ J. Since 1J is continuous on the right there is an open neighbourhood G of L such that r(L, G) n J = 0 where r(L, G) = {K E %IL ~ K ~ G}. We have
nK\G = L\G = 0,
KeJ
in particular, for K o E J
n(K o n K)\G = 0·
KeJ
By the finite intersection property for the compact space K o\G we can find K b . . . , K n E J such that (K o n K 1 n··· n Kn)\G
= 0,
~ G for K := K o n K 1 n ... n K n E J, so we get the contradiction K E r(L, G) n J. We therefore have L E J, hence J L = J. D
hence K
We now introduce the so-called myope topology on x. Let ~ (resp. <§) denote the set of closed (resp. open) subsets of X and define for a subset B~X
%B
= {K E %IK
%B
=
n B
{KE%IK nB
= 0},
=f
0}
=
X\%B.
The myope topology on % is the coarsest topology in which the sets %F, FE ff and % G, G E <§ are open. The sets %F n % Gl n ... n % Gn with FE ff, G 1 , • •• , Gn E ~ form a base for the topology which is easily seen to be Hausdorff. It is well known that % is locally compact, cf. Matheron (1975), and a proof of this is included in Remark 2.12 below. 2.11. Proposition. The mapping b: !/ -+ % defined by b(J)
=
nK,
KeJ
is continuous and proper and maps 9;. bijectively onto $: PROOF. By Lemma 2.10 we already know that b is a bijection of !/,. onto $: For the continuity of b it suffices to prove that b - 1(%1") and b - 1(%G) are open in !/ for F E ff and G E~. We first prove the following biimplication for J E !/ and G E ~: (1) b(J) ~ G <=> 3L E J: L ~ G.
269
~2. Completely Monotone and Completely Alternating Functions
Here" <=" is clear, and if b(J)
£;
G then
nK\G
= 0,
KeJ
and as in the proof of Lemma 2.10 we see that there is L = 0. By (1) it follows that
b- 1(%F) = {JEg;lb(J)
£;
E
J such that L \G
UL
Fe} =
Le:K LSFc
which shows that b- l(%F) is open in g; when F E~. For G E ~ and JoE b- 1(% G) we choose open relatively compact sets WI' W2 £; X such that b(J 0) £; WI £; WI £; W2 and a point Xo E b(J 0) (\ G. We next choose an open neighbourhood V of X o such that V £; G and define @= {JEg;lwIEJ,W2\V~J},
which is an open subset of g;. We have JoE @ because b(J 0) £; WI implies by (1) that WI E J o. Furthermore, W2 \ V E J 0 is impossible since Xo E V (\ b(J 0). Finally, we claim that @ £; b - 1(%G) which shows that b - 1(%G) is open in g;. In fact, for J E @, we have b(J) £; WI £; W2 and b(J) $ W2 \ V, hence b(J) (\ V =f 0 and a fortiori J Eb- 1 (%G). To see that b(J) $ w 2\V we remark that the assumption b(J) £; W2 \ V leads by (1) to W2 \ V E J. We show finally that b is proper, i.e. that b- 1 (r) is compact for each compact set r £; %. Let ~ 0 denote the family of all open relatively compact subsets of X. Then (%GC)Ge r9 o is an open covering of % so if 1/ £; % is compact there are G 1 , ••• , Gn E ~o such that 1/ is covered by the sets %Gf, i = 1, ... ,n. It is therefore sufficient to show that b- l(%GC) is relatively compact for G E ~ 0' but this is clear since
b- 1 (%GC) = {J E g;lb(J) and
£;
G}
£;
K is compact for K E %, cf. 2.7.
G, D
2.12. Remark. The above proof shows that a compact subset 1/ of % is contained in a set of the form %KC
for a suitable K 2.11 since
E
=
{LE%IL
£;
K}
%. That %KC is compact in % follows from Proposition for
K E%.
This shows in particular that % is locally compact. In fact, if K o E %F where F E !F, then %KC is a compact neighbourhood of K o contained in %F provided K is a compact neighbourhood of K o disjoint with F. And if K o E % G where G E~, then %KC (\ % H is a compact neighbourhood of K o contained in % G provided K is a compact neighbourhood of K o and H is a closed subset of G such that II (\ K o =f 0.
270
8. Positive and Negative Definite Functions on Abelian Semigroups Without Zero
2.13. Theorem. For every qJ E uIt(.Yt, n) which is continuous on the right there exists a unique J.1 E M
+(~)
such that
jor
KE~,
(2)
and for every J.1 E M +(~) this formula defines a function qJ E A(.Yt, n) which is continuous on the right. PROOF. Suppose first that qJ E uIt(~, n) is continuous on the right. By Proposition 2.8 there exists v E M +(!/) such that qJ(K) = v(K) for K E~. As in the proof of Theorem 4.6.18 we consider the image measure J.1 = v'\ which is a Radon measure on ~ because b is continuous and proper. For fixed K E ~ we define a decreasing net (L)LeA where A = {LE.YtIK ~ L} and the ordering of A is defined by L 1 ~ L 2 <=> L 1 ;2 L 2 , and we have
nL = {JE!/lb(J)~K}
LeA
as an easy consequence of (1). Using that qJ is decreasing with respect to and continuous on the right we find by Lemma 2.1.3 that qJ(K)
~
= inf{qJ(L)ILEA} = inf{v(L)/LEA} =
v( nL) = v({J LeA
E
!/lb(J)
~ K}) = J.1(~KC).
Conversely, if J.1 E M + (.Yt) then (2) defines a nonnegative real-valued function since ~Kc is compact in ~. By induction it is easy to establish that
V' K1 V'K 2
•••
V'KnqJ(K)
=
J.1(~KC n ~K~ n··· n ~K~)'
which shows that qJ is completely monotone on (~, n). For fixed K E.Yt and A as above (~LC)LeA is a decreasing net of compact subsets of ~ with intersection ~Kc. By Lemma 2.1.3 follows that qJ(K)
= inf{qJ(L)ILEA},
which shows that qJ is continuous on the right. The proof of uniqueness of the representing measure is similar to the proof in Theorem 4.6.18 and is left as an exercise to the reader. D
2.14. Exercise. Give an example of a function qJ E g>6(~) whose representing measure is concentrated on ~ +, but such that qJ cannot be extended to a function in uIt o( ~). 2.15. Exercise. Let qJ: H ~ [R be a function such that for all hE H the translate s t---+ qJ(h + s) is positive definite and bounded on S (resp. negative definite and bounded below on S). Then qJ is completely monotone (resp. alternating) on H. Show that the converse statement holds also. 2.16. Exercise. Let t/J E d o(H) be strictly positive. Then j·o t/J E uIt o(H) for fE q>q(]O, oo[) and g t/J Edo(H) for g E JVq(]O, oo[). 0
271
Notes and Remarks
2.17. Exercise. Let X be a locally compact space and (f, n) the semigroup of compact subsets of X. Then the mapping (a, J1) ~ ljI, where ljI(K) = a - J1(K\{%})
is an affine bijection of (a, J1) ~ ljI, where ljI(K)
=
[R
for K
E
%,
x M +(g;\{f}) onto .>:1(%, n). Furthermore
a - J1(%K
C \
{0})
for KEf,
is an affine bijection of [R x M+(%\{0}) onto the set of ljIE.s;1(%, n) which are continuous on the right.
Notes and Remarks Many of the results in this chapter were published in Ressel (1979). Choquet (1954, §47) introduced completely monotone and alternating functions on arbitrary abelian semigroups, but his main interest was directed towards the special case where S = (%(X), n) is the semigroup of all compact subsets of some given locally compact space X with respect to intersection. Theorem 2.13 is due to Choquet (1954, §50) and is also formulated in Talagrand (1976). The results of2.9-2.13 are similar to the results of 4.6.15-4.6.19. Our method of proof seems to be new. Some of the results of §1 (for example Theorem 1.4) have obvious extensions to semigroups with involutions other than the identical one which we assumed throughout this chapter in order to make the exposition more readable. It is also possible to weaken the condition of quasi-boundedness and still have unique integral representation; see Stochel (1983) for some results in this direction.
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List of Symbols
1. Sets of Numbers N = {I, 2, 3, } No = {O, 1, 2, } 7L = {O, ± 1, ± 2, ...} Q IR C IK Q+ IR +
rational numbers real numbers complex numbers common symbol for IR or C nonnegative rational numbers nonnegative real numbers
~ = [ - 00, oo[ C+ = {zECIRez ~ O} D = {zECllzl ~ I} lr = {z E C z I = I} lrd lr considered as a discrete group
213
II
166
2. Operations on Sets, Functions and Measures
A
C
A\B
complement of A
=A
n Be
1 .4 iJA = 1\.4 ex(A) conv(A)
AO A.l fAg fvg f+ = f
V
0
closure of A interior of A (topological) boundary of A extreme points of A convex hull of A polar of A dual cone minimum of f and g maximum of f and g
55 12 13
282
f- = -(f /\
List of Symbols
0)
(f x g)(x, y) = (f(x), g(y)) (f ® g)(x, y) = f(x)g(y) tensorproduct of f and g . I if x EA. 1A(X) = { Indicator functIon of A o if x ~ A
sgn = l]O,oo[ - 1]_ 00, O[ signum function supp(f) = {f =*= O} support off
f
E o(g)
fiB J11 B sUPP(J1) II J111 J1f J1 ® v J1 * v
restriction of the function f to B restriction of the measure J1 to B support of a Radon measure J1 total variation of J1 image measure product measure of J1 and v convolution of J1 and v J1*n = J1 * ... * J1 (n times) f J1 = f dJ1 measure with density f with respect to J1 J1+, J1Jordan-Hahn decomposition Gx(B) = I B(x) one-point measure, Dirac measure .91 ® (JI product of two a-algebras ,r;1 and (JI yX set of all mappings from X to Y ,r;1 + := {f E ,r;11 f(x) ~ 0 for all x E X} if ,r;1 ~ ex
39 42 34 21,53 22 40 28 26 29 22 41,44,45 22
3. Families of Subsets of a Hausdorff Space X C§ ~
= l§(X) = ~(X)
open subsets lJl' X closed subsets of X
~B'~B
% = %(X)
compact subsets of X
%B'%B
(JI( X)
OIl
Borel subsets of X filter of neighbourhoods of zero weak topology Mackey topology
a(E, F) r(E, F) "Y(K,G) = {LE%IK ~ L ~ G}
137,268 136,268 137 17,136,267 268 17 1 11
12 136,268
4. Spaces of Functions and Measures Associated with a Hausdorff Space X C(X) C b( X) CC(X) CO(X) M +(X) M~(X)
MC+ (X) M~(X)
M(X) MC(X)
Mol+(X)
continuous (real-valued) functions on X continuous bounded functions continuous functions with compact support continuous functions tending to zero at infinity Radon measures on X finite (= bounded) Radon measures Radon measures with compact support Radon probability measures signed Radon measures signed Radon measures with compact support molecular measures
39 39 39 39 17 29,45 93 49 40 96 22
283
List of Symbols
5. Operators, Function Spaces and Spaces of Measures Associated with an Abelian Semigroup (S, +, *) I Ea
identity operator shift operator
~a = Ea - I fa = t(E a + Ea.) Va = -~a Vnf(s; a 1 ,· •• ,an) dual semigroup S* restricted dual semigroup S evaluation function on S* Xs V(~), V(C) 9(S) positive definite functions on S 9 1(S) = {qJ E 9(S)lqJ(O) = I} f??(S) ex-bounded positive definite functions &'i(S) = {qJ E 9 CX(S) IqJ(O) = I} g e(S) exponentially bounded positive definite functions 9 b(S) bounded positive definite functions 9~(S) = {qJ E &>"(S) IqJ(O) = I} &:.(B) %(S) negative definite functions %'(S) lower bounded negative definite functions %b(S) bounded negative definite functions A(S) completely monotone functions A 1(S) = {qJ E A(S) IqJ(O) = I} d(S) completely alternating functions Yr(S) moment functions E+(S*) = {,u E M +(S*)IJ IXsl d,u < 00 for all s E S} E+(S*, qJ) = {,u E E+(S*)IJ XS d,u = qJ(s) for all S E S} E(S*) = {,ul - ,u2 + i(,u3 - ,u4)I,ul,···,,u4 E E+(S*)} 2 set of Levy measures
129 91, 123 110, 129, 130 103 130 130 92 96
180 181 88 88 94 94 94
96 96
153 89 99
105 130 130 179 179
180 203 105
6. Notations Related to an Abelian Semigroup (H, +) which (Possibly) has no Zero (S:= H u {O}) H' = H + H
fJ = S\ {l{o}} 9(H)
the nonmaximal elements of H
&"l(H)
positive definite functions on H quasibounded positive definite functions
96(H) %(H) %q(H)
negative definite functions on H negative definite functions, quasibounded below
%~(H)
A(H) Ao(H) d(H) do(H)
completely monotone functions on H completely alternating functions on H
252 255 253 253 254 253 257 257 264 264 264
265
284
List of Symbols
7. Miscellaneous Symbols 3
PA
seminorm determined by A supremum norm E* algebraic dual space E' topological dual space alg span(r), alg span + ( r) r-positive linear functionals
II . 1100
A:
/P
= {x E IR
'pP(Jl) LP(Jl)
=
I I IJI Ix(n)IP < oo}, 0 < p <
,P
A~)
A ~k) A~)(F)
IP IE var i.i.d. B(n, p) TCb
=
{p E
123, 124 124 148, 153
00
J
= {f: X ~ elf measurable, If IP dJl < oo}, 0 < p < 00 the associated Banach space (if p ~ 1) Laplace transformation 75, 114, 115 96,203 generalized Laplace transform of J1 161 generating function of J1 E M ~ (r\J 0) polynomials in k variables with real coefficients 129, 190 polynomials in k variables with complex coefficients 197 polynomials in A (k) of degree at most d 191 A(k) Ip(x) ~ 0 for all x E F} 196 probability expectation variance independent and identically distributed 165 binomial distribution 164,165,235 Poisson distribution 240 order of majorization
'pP(X, J1)
{l {l A(k)
10 10
x -< y Jl-<JV 1 = (1, ... , 1) On doubly stochastic n x n matrices !7 set of subsemigroups, hereditary on the left !7,. = {J E Y II] is right continuous} RKHS reproducing kernel Hilbert space
58 60,240 119,136,267 136,267 81
Index
abelian semigroup 86, 92 with involution 86, 92 a-bounded 90 absolute polar set 13 value on a semigroup 89 absolutely convex subset 2 decreasing function 140 monotone function 140 absorbing element 87, 100, 107 subset 2 adapted cone 42 space 42, 52, 181 *-additive function 101 admissible subset 123 Alaoglu-Bourbaki's theorem 14,64 algebraic dual space 10 almost positive kernel 66 alternating of infinite order 130 analytic space 64 Appert-Varadarajan space 33 approximation lemma 145 balanced subset 2 barycentre 56 base of a convex cone 59, 88 Bauer simplex 59, 95, 96, 131, 158, 169 Bernoulli distribution 235 random variable 226, 229
Bernstein function 114, 141 Bernstein's theorem 135 Bickel and van Zwet's theorem 226 bimeasure 24, 33, 64, 206 binomial distribution 165 bipolar theorem 12 Birkhoff's theorem 60 bivariate generating function 165 Bochner's theorem 96, 157 Borel measure 17 subset 17 boundary measure 58 bounded with respect to an absolute value 90 Bretagnolle, Dacunha-Castelle and Krivine's theorem 149, 176 BY-functions 143
canonical topology of a finite dimensional space 13 capacity 139, 267 centre of gravity 55, 56 chain majorization 251 Choquet-Bishop-de Leeuw's theorem 59 Choquettheory 65 classical Levy-Khinchin formula 216, 221 completely alternating capacity 139 function 130, 263
286
Index
completely monotone capacity 267 function 130, 263 in the classical sense 136 completely negative definite function 229 completely positive definite function 193,229 complex moment 117 function 117 problem 117, 193 conditionally positive definite kernel 66 cone 13 continuity theorem for generating functions 162 for Laplace transforms 123 continuous on the right 136 convex combination 55 polyhedron 55 convolution of finite Radon measures 29 semigroup 100
Faraut and Harzallah's theorem 174,177 finest locally convex topology 4 finite signed measure 39 F-moment
problem 186 sequence 186 Fubini's theorem 26 function bounded with respect to an absolute value 90 of positive type 88 functions operating on positive definite matrices 157
Gaussian semigroup 218 generalized Laplace transform 96, 203 transformation 96, 203 generating function 156, 161, 165 generator set 127 geometric version of Hahn-Banach's theorem 6
determinate measure 180 moment function 180 positive definite sequence 200 Dirac measure 22 doubly stochastic matrix 60, 240, 251 Douglas'theorem 61 dual map (homomorphism) 97, 207 pair 11 semigroup 93 Dvoretzky's theorem 152, 176 Dynkin class 28
Einhorn's theorem 153, 154 elementary dou bly stochastic matrix 250 random variable 227 entropy function 218, 240 Euler's constant 140 exponential of J.l 50 exponentially bounded function 90 positive definite function 90, 92 exponentially convex function 225 extension theorem for Radon contents 18 version of Hahn-Banach's theorem
extreme boundary 55 point 55 ray 83 subset 55
8
Hahn-Banach theorems 5 Hamburger's moment problem 186 Hamburger's theorem 186 Hankel matrix 122 Hausdorff's moment problem 134,187 Hausdorff topological semigroup 29 Haviland's moment problem 187 Helly's selection theorem 64 hereditary on the left 119 hermitian function on a semigroup with involution 88 kernel 67 matrix 66 Hilbert matrix 122 Hilbert space representation of negative definite kernels 82 of positive definite kernels 81, 82 Hildebrandt-Schoenberg's theorem 134
287
Index
Hoeffding's inequality of all orders 230 of order n 228 and degree r 232 homomorphism of semigroups 87 *-homomorphism of *-semigroups 87 hyperplane 5
idempotent commutative semigroup 119 identical involution 87 image measure 28 indeterminate measure 180 moment function 180 positive definite sequence 200 two-sided moment sequence 201 infinitely divisible positive definite kernel 76 probability measure on No 163, 164 inner measure 18 integral representation of bounded positive definite functions 96 integral representations, theory of 55 integrale superieure essentielle 62 involution 86, 123 isomorphism of semigroups 87
Jordan-Hahn decomposition
39
Kisynski's theorem 18,63 Krein-Milman's theorem 57, 65
Laplace transform 75, 114, 115 Levy continuity theorem 97 function 108 measure 104 Levy-Khinchin formula 108, 141 localization principle 30 locally convex topological vector space 2 finite Radon measure 17, 33, 61, 63 log-normal distribution 201 lower semicontinuous 20
Mackey-Arens'theorem 15 Mackey topology 12, 15 majorization 240, 241
marginal distribution 50 maximal elements in a semigroup 252 mean empirical variance 239, 249 mesure exterieure 62 method of moments 185 molecular measure 22, 49, 55, 227 moment 116 function 116, 179, 222 generating function 222 monotone convergence theorem for Radon measures 20 of infinite order 130 multi-dimensional moment problem 190 multivariate majorization 241 myope topology 268 negative definite function 89, 253 kernel 67 matrix 66 negative definiteness in the group sense 88 in the semigroup sense 88 of degree r 232 Neumann's result 59, 60 Nevanlinna extremal measure 200 parametrization 200 nonmetrizable first countable compact space 121 norm 3 dependent positive definite function 151 normal distri bution 190 one-dimensional moment problem 185 one-point measure 22 one-sided stable measure of index lI. 217 ordering of an idempotent semigroup 119 parallelogram law 153 perfect semigroup 203 without zero 263 Poisson distribution 164, 165, 235 polar subset 12 Polish space 16 Pollard- Tops~e's theorem 35,36 portmanteau theorem 45, 46
288 positive definite function 87, 253 kernel 67 matrix 66 positive definiteness in the group sense 88 in the semigroup sense 88 positive type 88 potential kernel 142 product measure 26 semigroup 87, 97, 204 proper mapping 29 pseudo-concentrated 58 purely imaginary homomorphism
Index
101
quadratic form 101, 234, 246 quasibounded 253 below 253 positive definite function 253
Radon bimeasure 24 content 17 measure 17 which is not locally finite 33 real infinite dimensional hyperbolic space 173 relatively compact subset of M +(X) 51,52 representation of a semigroup S 90 representing measures, set of 52, 180, 181 reproducing kernel Hilbert space (RKHS) 81 restricted dual semigroup 96 restriction of a Radon measure 21,53 resultant 56 reverse Hoeffding inequality 235 Riesz representation theorem 39, 64
Schoenberg triple 148 Schoenberg's theorem 74, 85 for the complex Hilbert sphere 171 for the real Hilbert sphere 159 Schur concave function 240, 241 convex function 240,241 decreasing function (of order n) 243 increasing function 243 of order n 242
Schur's theorem 69 semicharacter 92 on an idempotent semigroup 119 semigroup 86 with involution 86 *-semigroup 86 seminorm 3 determined by a subset 3 separation version of Hahn-Banach's theorem 7 shift operator 91, 123 signed measure 17 Radon measure 39 simplex 59 Slutsky's theorem 64 Sorgenfrey topology 31 stable measure of index lI. 218,219 standard p-function 142 Stieltjes function 142 Stieltjes moment function 193 problem 187 sequence 187 Stieltjes' theorem 187 stochastic matrix 60 strictly negative definite kernel 67 positive definite function 87 kernel 67 strong Hamburger moment problem 198 sublinear function 8 subsemigroup 87 *-subsemigroup 87 support of a function 39 of a Radon measure 22, 32 symmetric kernel 68
tensor product of positive definite kernels 70 topological dual space 10 group 1 semigroup 29, 87 vector space topology compatible with a1duality 11 induced by a family of seminorms total variation norm 40
4
289
Index
r-positive function 123, 127 linear functional 124 r-smooth measure 21, 250 2-divisible semigroup 98, 113, 133, 224, 231,251 two-sided moment problem 198 sequence 198 two-sided Stieltjes moment sequence 202
vague convergence of Radon measures topology 50
50
weak convergence of finite Radon measures 45 topology on M~(X) 45 u(£, F) 11 Wells and William's theorem 153
