Harmonic Analysis on Semigroups

1 Introduction to Locally Convex Topological Vector Spaces and Dual Pairs.- x1. Locally Convex Vector Spaces.- x2. Hahn-Banach Theorems.- x3. Dual Pairs.- Notes and Remarks.- 2 Radon Measures and Integral Representations.- x1. Introduction to Radon Measures on Hausdorff Spaces.- x2. The Riesz Representation Theorem.- x3. Weak Convergence of Finite Radon Measures.- x4. Vague Convergence of Radon Measures on Locally Compact Spaces.- x5. Introduction to the Theory of Integral Representations.- Notes and Remarks.- 3 General Results on Positive and Negative Definite Matrices and Kernels.- x1. Definitions and Some Simple Properties of Positive and Negative Definite Kernels.- x2. Relations Between Positive and Negative Definite Kernels.- x3. Hubert Space Representation of Positive and Negative Definite Kernels.- Notes and Remarks.- 4 Main Results on Positive and Negative Definite Functions on Semigroups.- x1. Definitions and Simple Properties 86 x2. Exponentially Bounded Positive Definite Functions on Abelian Semigroups.- x3. Negative Definite Functions on Abelian Semigroups.- x4. Examples of Positive and Negative Definite Functions.- x5. T-Positive Functions.- x6. Completely Monotone and Alternating Functions.- Notes and Remarks.- 5 Schoenberg-Type Results for Positive and Negative Definite Functions.- x1. Schoenberg Triples.- x2. Norm Dependent Positive Definite Functions on Banach Spaces.- x3. Functions Operating on Positive Definite Matrices.- x4. Schoenberg's Theorem for the Complex Hilbert Sphere.- x5. The Real Infinite Dimensional Hyperbolic Space.- Notes and Remarks.- 6 Positive Definite Functions and Moment Functions.- x1. Moment Functions.- x2. The One-Dimensional Moment Problem.- x3. The Multi-Dimensional Moment Problem.- x4. The Two-Sided Moment Problem.- x5. Perfect Semigroups.- Notes and Remarks.- 7 Hoeffding's Inequality and Multivariate Majorization.- x1. The Discrete Case.- x2. Extension to Nondiscrete Semigroups.- x3. Completely Negative Definite Functions and Schur-Monotonicity.- Notes and Remarks.- 8 Positive and Negative Definite Functions on Abelian Semigroups Without Zero.- x1. Quasibounded Positive and Negative Definite Functions.- x2. Completely Monotone and Completely Alternating Functions.- Notes and Remarks.- References.- List of Symbols.