Harmonic analysis on semigroups, christian berg, jens peter reus christensen, paul ressel

Graduate Texts in Mathematics 100 Editorial Board F W Gehring P R Halmos (Managing Editor) c C Moore Graduate Texts in Mathematics I 10 II 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 TAKEUTl/ZARING Introduction to Axiomatic Set Theory 2nd ed OXTOBY Measure and Category 2nd ed SCHAEFFER Topological Vector Spaces HILTON/STAMMBACH A Course in Homological Algebra MACLANE Categories for the Working Mathematician HUGHEs/PIPER Projective Planes SERRE A Course in Arithmetic TAKEUTl/ZARING Axiometic Set Theory HUMPHREYS Introduction to Lie Algebras and Representation Theory COHEN A Course in Simple Homotopy Theory CONWAY Functions of One Complex Variable 2nd ed BEALS Advanced Mathematical Analysis ANDE~SON/FuLLER Rings and Categories of Modules GOLUBITSKy/GUlLLEMIN Stable Mappings and Their Singularities BERBERIAN Lectures in Functional Analysis and Operator Theory WINTER The Structure of Fields ROSENBLATT Random Processes 2nd ed HALMos Measure Theory HALMos A Hilbert Space Problem Book 2nd ed., revised HUSEMOLLER Fibre Bundles 2nd ed HUMPHREYS Linear Algebraic Groups BARNEs/MACK An Algebraic Introduction to Mathematical Logic GREUB Linear Algebra 4th ed HOLMES Geometric Functional Analysis and its Applications HEWITT/STROMBERG Real and Abstract Analysis MANES Algebraic Theories KELLEY General Topology ZARISKIISAMUEL Commutative Algebra Vol I ZARISKIISAMUEL Commutative Algebra Vol II JACOBSON Lectures in Abstract Algebra I: Basic Concepts JACOBSON Lectures in Abstract Algebra II: Linear Algebra JACOBSON Lectures in Abstract Algebra III: Theory of Fields and Galois Theory HIRSCH Differential Topology SPITZER Principles of Random Walk 2nd ed WERMER Banach Algebras and Several Complex Variables 2nd ed KELLEy/NAMIOKA et al Linear Topological Spaces MONK Mathematical Logic GRAUERT/FRITZSCHE Several Complex Variables ARVESON An Invitation to C*-Algebras KEMENy/SNELL/KNAPP Denumerable Markov Chains 2nd ed APOSTOL Modular Functions and Dirichlet Series in Number Theory SERRE Linear Representations of Finite Groups GILLMAN/JERISON Rings of Continuous Functions KENDIG Elementary Algebraic Geometry LOEvE Probability Theory I 4th ed LOEVE Probability Theory II 4th ed MOISE Geometric Topology in Dimensions and continued after Index Christian Berg J ens Peter Reus Christensen Paul Ressel Harmonic Analysis on Semigroups Theory of Positive Definite and Related Functions Springer Science+Business Media, LLC Christian Berg Jens Peter Reus Christensen Mathematisch-Geographische Matematisk Institut Kl1Jbenhavns Universitet Universitetsparken DK-2100 K~benhavn ~ Denmark Katholische Universităt Eichstătt Residenzplatz 12 D-8078 Eichstătt Federal Republic of Germany Paul Ressel Fakultăt Editorial Board P R Halmos F W Gehring c C Moore Managing Editor Department of Mathematics Indiana University Bloomington, IN 47405 U.S.A 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, 44AlO, 44A60, 46A55, 52A07,60E15 Library of Congress Cataloging in Publication Data Berg, Christian Harmonic analysis on semigroups (Graduate texts in mathematics; 100) Bibliography: p Includes index Harmonic analysis Semigroups Christensen, Jens Peter Reus II Ressel, Paul III Title IV Series QA403.B39 1984 515'.2433 83-20122 With Illustrations © 1984 by Springer Science+Business Media New York Originally published by Springer-Verlag Berlin Heidelberg New York Tokyo in 1984 Softcover reprint of the hardcover Ist edition 1984 All rights reserved No part of this book may be translated or reproduced in any form without written permission from Springer Science+Business Media, LLC Typeset by Composition House Ltd., Salisbury, England 321 ISBN 978-1-4612-7017-1 DOI 10.1007/978-1-4612-1128-0 ISBN 978-1-4612-1128-0 (eBook) 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 St, , Sn from S The three basic results mentioned above correspond to (~, +, 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 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 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 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 Preface vii 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 Undervisningsministerium, as well as our home universities Thanks are due to Flemming Topsq,e for his advice on Chapter We had the good fortune to have Bettina Mann type the manuscript and thank her for the superb typing March 1984 CHRISTIAN BERG JENS PETER REus CHRISTENSEN PAUL REsSEL Contents CHAPTER 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 11 15 CHAPTER 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 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 x Contents CHAPTER 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 t-Positive Functions §6 Completely Monotone and Alternating Functions Notes and Remarks 86 86 92 98 113 123 129 141 CHAPTER 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 Positive Definite Functions and Moment Functions §1 §2 §3 §4 Moment Functions The One-Dimensional Moment Problem The Multi-Dimensional Moment Problem The Two-Sided Moment Problem §S Perfect Semigroups Notes and Remarks 178 178 185 190 198 203 222 CHAPTER 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 Positive and Negative Definite Functions on Abelian Semigroups Without Zero §l 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 285 276 References HARZALLAH, K (1969) Fonctions operant sur les fonctions definies-negatives Ann Inst Fourier (Grenoble) 19 (2),527-532 HAVILAND, E K (1935) On the momentum problem for distributions in more than one dimension Amer J Math 57, 562-568 HAVILAND, E K (1936) On the momentum problem for distribution functions in more than one dimension, II Amer J Math 58, 164-168 HELMS, L L (1969) Introduction to Potential Theory New York:Wiley HERZ, C S (1963a) Fonctions operant sur les fonctions definies-positives Ann Inst Fourier (Grenoble) 13, 161-180 HERZ, C S (1963b) Une ebauche d'une theorie generale des fonctions definies-negatives Seminaire Brelot-Choquet-Deny (Theorie du Potentiel), 7· annee Inst Henri Poincare, Paris HEWITT, E and H S ZUCKERMANN (1956) The It-algebra of a commutative semigroup Trans Amer Math Soc 83, 70-97 HEYDE, C C (1963) On a property of the lognormal distribution J Roy Statist Soc., Ser B, 25, 392-393 HEYER, H (1977) 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Greifswald: Ernst-Moritz-Arndt Universitiit VARADARAJAN, V S (1965) Measures on topological spaces Amer Math Soc Transl., Ser II, 48, 161-228 WALL, H S (1931) On the Pade approximants associated with a positive definite power series Trans Amer Math Soc 33, 511-532 WELLS, J H and L R WILLIAMS (1975) Embeddings and Extensions in Analysis Berlin-Heidelberg-New York: Springer-Verlag WIDDER, D V (1941) The Laplace Transform Princeton: Princeton University Press WILLIAMSON, J H (1967) Harmonic analysis on semi groups J London Math Soc 42, 1-41 ZARHINA, R B (1959) On the two-dimensional problem of moments Dokl Akad Nauk SSSR 124, 743-746 (in Russian) a List of Symbols Sets of Numbers N = {1, 2, 3, } No = {O, 1, 2, } 7! = {O, ±1, ±2, } iQ rational numbers IR real numbers C complex numbers II{ common symbol for IR or C iQ + nonnegative rational numbers IR+ nonnegative real numbers ~ = [-00, oo[ C+ = {z E CiRe Z ~ O} D = {zECllzl ~ 1} If = {zECllzl = 1} lfd If considered as a discrete group 213 166 Operations on Sets, Functions and Measures AC A\B A = A (\ B" A aA = A\A eX(A) conv(A) AO A.L fl\g fvg f+ = f v complement of A 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 maximum of f and ss 12 13 282 List of Symbols r = -(f /\ 0) (f x g)(x, y) = (f(x), g(y» (f ® g)(x, y) = f(x)g(y) tensorproduct of f and g if x eA.10d·lcator functIOn 0fA ifxjA sgn = 1]0 co! - I j - co • o! signum function supp(f) = {f =F O} support off f e o(g) fiB restriction of the function f to B JlI B restriction of the measure Jl to B supp(Jl) support of a Radon measure Jl "Jl" total variation of Jl JlI image measure Jl ® v product measure of Jl and v Jl * v convolution of Jl and v Jl*" = Jl * * Jl (n times) fJl = f dJl measure with density f with respect to Jl Jl+, JlJordan-Hahn decomposition eiB) = la(x) one-point measure, Dirac measure d ® f1I product of two a-algebras d and f1I yX set of all mappings from X to Y d + := {f e d I f(x) ~ for all x e X} if d ~ eX lA(x) = { I o 39 42 34 21,53 22 40 28 26 29 22 41,44,45 22 Families of Subsets of a Hausdorff Space X ,,§(X) open subsets of X closed subsets of X fJ' = fJ'(X) fJ'B, fJ'B Y{ = 'y{(X) compact subsets of X Y{B, fa f1I(X) Borel subsets of X 0/1 filter of neighbourhoods of zero aCE, F) weak topology 1:(E, F) Mackey topology '"f/"(K, G) = {Le.Y{IK ~ L ~ G} "§ = 137,268 136,268 137 17,136,267 268 17 11 12 136,268 Spaces of Functions and Measures Associated with a Hausdorff Space X C(X) Cb(X) CC(X) CO(X) M + (X) M"+ (X) M"+ (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 Operators, Function Spaces and Spaces of Measures Associated with an Abelian Semigroup (S, +, *) I E identity operator shift operator d = E - I r = !(E + E•• ) V = -d VnJ(s; a 1, ·, an) S* dual semigroup restricted dual semigroup S evaluation function on S* Xs V(IR), V(C) &(S) positive definite functions on S &1 (S) = {ep E &(S) Iep(O) = I} &"'(S) IX-bounded positive definite functions &Ii (S) = {ep E &"'(S) Iep(O) = I} &'(S) exponentially bounded positive definite functions g>b(S) bounded positive definite functions ~(S) = {ep E g>b(S) Iep(O) = I} ~(B) ';v(S) negative definite functions ;V1(S) lower bounded negative definite functions ;Vb(S) bounded negative definite functions A(S) completely monotone functions A1(S) = {ep E A(S) Iep(O) = I} d(S) completely alternating functions if(S) moment functions E+(S*) = {Jl E M +(S*)IJ IXsl dJl < OCJ for all s E S} E+(S*, ep) = {Jl E E+(S*)IJ XS dJl = ep(s) for all s E S} E(S*) = {Jl1 - Jl2 + i(Jl3 - Jl4)IJl1"'" Jl4 E E+(S*)} set of Levy measures Notations Related to an Abelian Semi group (H, +) which (Possibly) has no Zero (S:= H u {O}) the nonmaximal elements of H H' = H +H Ii = S\{l{o}} &(H) [Jl'I(H) r?b(H) ;V(H) ;Vq(H) ;VMH) A(H) Ao(H) d(H) do(H) positive definite functions on H quasi bounded positive definite functions negative definite functions on H negative definite functions, quasi bounded below completely monotone functions on H completely alternating functions on H 129 91, 123 110, 129, 130 103 130 130 92 96 180 181 88 88 94 94 94 96 96 153 89 99 105 130 130 179 179 180 203 105 252 255 253 253 254 253 257 257 264 264 264 265 284 List of Symbols Miscellaneous Symbols PA 11·1100 E* seminorm determined by A supremum norm algebraic dual space E' topological dual space alg span(r), alg span+(,) A~ '.positive linear functionals [P = {x E [RN IJ1lx(nw < oo}, 0< p < 10 10 123, 124 124 148, 153 00 f It'P(Jl) = It'P(X, Jl) = {f: X elf measurable, If IP dJl < OO}, < p < 00 U(Jl) the associated Banach space (if p ~ 1) It' Laplace transformation 75,114,115 96,203 {1 generalized Laplace transform of Jl (1 generating function of Jl E M~(I~Jo) 161 A(k) polynomials in k variables with real coefficients 129, 190 A~) polynomials in k variables with complex coefficients 197 A~k) polynomials in A(k) of degree at most d 191 A~)(F) = {p E A(k)lP(x) ~ for all x E F} 196 ltD probability IE expectation var variance i.i.d independent and identically distributed binomial distribution B(n, p) 165 Poisson distribution 164, 165, 235 7tb order of majorization 240 x-