Overview
- Authors:
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Christian Berg
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Matematisk Institut, Københavns Universitet, København ∅, Denmark
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Jens Peter Reus Christensen
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Matematisk Institut, Københavns Universitet, København ∅, Denmark
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Paul Ressel
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Mathematisch-Geographische Fakultät, Katholische Universität Eichstätt, Eichstätt, Germany
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Table of contents (8 chapters)
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- Christian Berg, Jens Peter Reus Christensen, Paul Ressel
Pages 1-15
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- Christian Berg, Jens Peter Reus Christensen, Paul Ressel
Pages 16-65
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- Christian Berg, Jens Peter Reus Christensen, Paul Ressel
Pages 66-85
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- Christian Berg, Jens Peter Reus Christensen, Paul Ressel
Pages 86-143
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- Christian Berg, Jens Peter Reus Christensen, Paul Ressel
Pages 144-177
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- Christian Berg, Jens Peter Reus Christensen, Paul Ressel
Pages 178-225
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- Christian Berg, Jens Peter Reus Christensen, Paul Ressel
Pages 226-251
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- Christian Berg, Jens Peter Reus Christensen, Paul Ressel
Pages 252-271
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Back Matter
Pages 273-291
About this book
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.
