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AuthorBerg, Christian. author
TitleHarmonic Analysis on Semigroups [electronic resource] : Theory of Positive Definite and Related Functions / by Christian Berg, Jens Peter Reus Christensen, Paul Ressel
ImprintNew York, NY : Springer New York : Imprint: Springer, 1984
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Descript X, 292 p. online resource


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 + Skยป 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 monoยญ 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


1 Introduction to Locally Convex Topological Vector Spaces and Dual Pairs -- ยง1. Locally Convex Vector Spaces -- ยง2. Hahn-Banach Theorems -- ยง3. Dual Pairs -- Notes and Remarks -- 2 Radon Measures and Integral Representations -- ยง1. Introduction to Radon Measures on Hausdorff Spaces -- ยง2. The Riesz Representation Theorem -- ยง3. Weak Convergence of Finite Radon Measures -- ยง4. Vague Convergence of Radon Measures on Locally Compact Spaces -- ยง5. Introduction to the Theory of Integral Representations -- Notes and Remarks -- 3 General Results on Positive and Negative Definite Matrices and Kernels -- ยง1. Definitions and Some Simple Properties of Positive and Negative Definite Kernels -- ยง2. Relations Between Positive and Negative Definite Kernels -- ยง3. Hubert Space Representation of Positive and Negative Definite Kernels -- Notes and Remarks -- 4 Main Results on Positive and Negative Definite Functions on Semigroups -- ยง1. Definitions and Simple Properties 86 ยง2. Exponentially Bounded Positive Definite Functions on Abelian Semigroups -- ยง3. Negative Definite Functions on Abelian Semigroups -- ยง4. Examples of Positive and Negative Definite Functions -- ยง5. T-Positive Functions -- ยง6. Completely Monotone and Alternating Functions -- Notes and Remarks -- 5 Schoenberg-Type Results for Positive and Negative Definite Functions -- ยง1. Schoenberg Triples -- ยง2. Norm Dependent Positive Definite Functions on Banach Spaces -- ยง3. Functions Operating on Positive Definite Matrices -- ยง4. Schoenbergโ{128}{153}s Theorem for the Complex Hilbert Sphere -- ยง5. The Real Infinite Dimensional Hyperbolic Space -- Notes and Remarks -- 6 Positive Definite Functions and Moment Functions -- ยง1. Moment Functions -- ยง2. The One-Dimensional Moment Problem -- ยง3. The Multi-Dimensional Moment Problem -- ยง4. The Two-Sided Moment Problem -- ยง5. Perfect Semigroups -- Notes and Remarks -- 7 Hoeffdingโ{128}{153}s Inequality and Multivariate Majorization -- ยง1. The Discrete Case -- ยง2. Extension to Nondiscrete Semigroups -- ยง3. Completely Negative Definite Functions and Schur-Monotonicity -- Notes and Remarks -- 8 Positive and Negative Definite Functions on Abelian Semigroups Without Zero -- ยง1. Quasibounded Positive and Negative Definite Functions -- ยง2. Completely Monotone and Completely Alternating Functions -- Notes and Remarks -- References -- List of Symbols

Mathematics Topological groups Lie groups Mathematics Topological Groups Lie Groups


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