半群上的调和分析

CHAPTER 1 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

CHAPTER 2 Radon Measures and Integral Representations

1.Introduction to Radon Measures on Hausdorff Spaces

2.The Riesz Representation Theorem

3.Weak Convergence of Finite Radon Measures

4.Vague Convergence of Radon Measures on Locally Compact Spaces

5.Introduction to the Theory of lntegral Representations

Notes and Remarks

CHAPTER 3 General Results on Positive and Negative Definite Matrices and Kernels

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

CHAPTER 4 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 Defnite Functions

5.Positive Functions

6.Completely Monotone and Alternating Functions

Notes and Remarks

CHAPTER 5 Schoenberg-Type Results for Positive and Negative Definite Functions

1.Schoenberg Triples

2.Norm Dependent Positive Definite Functions on Banach Spaces

3.Functions Operating on Positive Definite Matrices

4.Schoenberg's Theorem for the Complex Hilbert Sphere

5.The Real Infinite Dimensional Hypcrbolic Space

Notes and Remarks

CHAPTER 6 Positive Definite Functions and Moment Functions

1.Moment Functions

2.The One-Dimensional Moment Problem

3.The Multi Dimensional Moment Problem

4.The Two-Sided Moment Problem

5.Perfect Semigroups

Notes and Remarks

CHAPTER 7 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

CHAPTER 8 Positive and Negative Definite Functions on Abelian Semigroups Without Zero

1.Quasibounded Positive and Negative Definite Functions

2.Completely Monotone and Completely Alternating Functions

Notes and Remarks

References

List of Symbols

Index

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 ourier transformations， as well as the generating functions for integer valued 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.