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Harmonic Analysis on Semigroups

Theory of Positive Definite and Related Functions

  • Christian Berg
  • Jens Peter Reus Christensen
  • Paul Ressel

Part of the Graduate Texts in Mathematics book series (GTM, volume 100)

Table of contents

  1. Front Matter
    Pages i-x
  2. Christian Berg, Jens Peter Reus Christensen, Paul Ressel
    Pages 1-15
  3. Christian Berg, Jens Peter Reus Christensen, Paul Ressel
    Pages 16-65
  4. Christian Berg, Jens Peter Reus Christensen, Paul Ressel
    Pages 66-85
  5. Christian Berg, Jens Peter Reus Christensen, Paul Ressel
    Pages 86-143
  6. Christian Berg, Jens Peter Reus Christensen, Paul Ressel
    Pages 144-177
  7. Christian Berg, Jens Peter Reus Christensen, Paul Ressel
    Pages 178-225
  8. Christian Berg, Jens Peter Reus Christensen, Paul Ressel
    Pages 226-251
  9. Christian Berg, Jens Peter Reus Christensen, Paul Ressel
    Pages 252-271
  10. Back Matter
    Pages 273-291

About this book

Introduction

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 mono­ tone 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.

Keywords

Fourier transform Functions Halbgruppe Harmonische Analyse Hilbert space Positiv definite Funktion Vector space harmonic analysis

Authors and affiliations

  • Christian Berg
    • 1
  • Jens Peter Reus Christensen
    • 1
  • Paul Ressel
    • 2
  1. 1.Matematisk InstitutKøbenhavns UniversitetKøbenhavn ∅Denmark
  2. 2.Mathematisch-Geographische FakultätKatholische Universität EichstättEichstättGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-1128-0
  • Copyright Information Springer-Verlag New York Inc. 1984
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4612-7017-1
  • Online ISBN 978-1-4612-1128-0
  • Series Print ISSN 0072-5285
  • Buy this book on publisher's site