Ebook: Harmonic Analysis on Semigroups: Theory of Positive Definite and Related Functions
- Tags: Topological Groups Lie Groups
- Series: Graduate Texts in Mathematics 100
- Year: 1984
- Publisher: Springer-Verlag New York
- Edition: 1
- Language: English
- pdf
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.
Content:
Front Matter....Pages i-x
Introduction to Locally Convex Topological Vector Spaces and Dual Pairs....Pages 1-15
Radon Measures and Integral Representations....Pages 16-65
General Results on Positive and Negative Definite Matrices and Kernels....Pages 66-85
Main Results on Positive and Negative Definite Functions on Semigroups....Pages 86-143
Schoenberg-Type Results for Positive and Negative Definite Functions....Pages 144-177
Positive Definite Functions and Moment Functions....Pages 178-225
Hoeffding’s Inequality and Multivariate Majorization....Pages 226-251
Positive and Negative Definite Functions on Abelian Semigroups Without Zero....Pages 252-271
Back Matter....Pages 273-291
Content:
Front Matter....Pages i-x
Introduction to Locally Convex Topological Vector Spaces and Dual Pairs....Pages 1-15
Radon Measures and Integral Representations....Pages 16-65
General Results on Positive and Negative Definite Matrices and Kernels....Pages 66-85
Main Results on Positive and Negative Definite Functions on Semigroups....Pages 86-143
Schoenberg-Type Results for Positive and Negative Definite Functions....Pages 144-177
Positive Definite Functions and Moment Functions....Pages 178-225
Hoeffding’s Inequality and Multivariate Majorization....Pages 226-251
Positive and Negative Definite Functions on Abelian Semigroups Without Zero....Pages 252-271
Back Matter....Pages 273-291
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