Harmonic analysis on semigroups: theory of positive definite and related functions

Description

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.- 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 Integral Representations.- Notes and Remarks.- 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. Hubert Space Representation of Positive and Negative Definite Kernels.- Notes and Remarks.- 4 Main Results on Positive and Negative Definite Functions on Semigroups.- 1. Definitions and Simple Properties 86 2. Exponentially Bounded Positive Definite Functions on Abelian Semigroups.- 3. Negative Definite Functions on Abelian Semigroups.- 4. Examples of Positive and Negative Definite Functions.- 5. T-Positive Functions.- 6. Completely Monotone and Alternating Functions.- Notes and Remarks.- 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 Hyperbolic Space.- Notes and Remarks.- 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.- 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.- 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.