<br><br>Harmonic analysis on semigroups<br><br><br><br>Theory of positive definite and related functions<br><br><br>Christian berg<br>Jenspeter reus christensen<br>Paul ressel<br><br><br><br><br>Contents<br>CHAPTER 1 <br>Introduction to Locally Convex Topological Vector Spaces and<br>Dual Pairs 1<br>§1. Locally Convex Vector Spaces 1<br>§2. Hahn-Banach Theore 5<br>§3. Dual Pairs 11<br>Notes and Remarks 15<br>CHAPTER 2 <br>preasures and Integral Representations 16<br>§1. Introduction to Radon Measures on Hausdorff Spaces 16<br>§2. The Riesz Representation Theorem 33<br>§3. Weak Convergence of Finite Radon Measures 45<br>§4. Vague Convergence of Radon Measures on Locally Compact Spaces 50<br>§5. Introduction to the Theory of Integral Representations 55<br>Notes and Remarks 61<br>CHAPTER 3 <br>General Results on Positive and Negative Definite Matrices and<br>Kernels 66<br>§1. Definitions and Some Simple Properties of Positive and Negative<br>Definite Kernels66<br>§2. Relations Between Positive and Negative Definite Kernels 73<br>§3. Hubert Space Representation of Positive and Negative Definite Kernels 81<br>Notes and Remarks 84<br>CHAPTER 4<br>Main Results on Positive and Negative Definite Functions on<br>Semigroups 86<br>§1. Definitions and Simple Properties 86<br>§2. Exponentially Bounded Positive Definite Functions on<br>Abelian Semigroups 92<br>§3. Negative Definite Functions on Abelian Semigroups 98<br>§4. Examples of Positive and Negative Definite Functions 113<br>§5. -[-Positive Functions 123<br>§6. Completely Monotone and Alternating Functions 129<br>Notes and Remarks 141<br>CHAPTER 5<br>Schoenberg-Type Results for Positive and Negative Definite<br>Functions 144<br>§1. Schoenberg Triples 144<br>§2. Norm Dependent Positive Definite Functions on Banach Spaces 151<br>§3. Functions Operating on Positive Definite Matrices 155<br>§4. Schoenberg’s Theorem for the Complex Hubert Sphere 166<br>§5. The Real Infinite Dimensional Hyperbolic Space 173<br>Notes and Remarks 176<br>CHAPTER 6<br>Positive Definite Functions and Moment Functions 178<br>§1. Moment Functions 178<br>§2. The One-Dimensional Moment Problem 185<br>§3. The Multi-Dimensional Moment Problem 190<br>§4. The Two-Sided Moment Problem 198<br>§5. Perfect Semigroups , 203<br>Notes and Remarks 222<br>CHAPTER 7<br>Hoeffding’s Inequality and Multivariate Majorization 226<br>§1. The Discrete Case 226<br>§2. Extension to Nondiscrete Semigroups 235<br>§3. Completely Negative Definite Functions and Schur-Monotonicity 240<br>Notes and Remarks 250<br>CHAPTER 8 M<br>Positive and Negative Definite Functions on Abelian Semigroups <br>Without Zero 252<br>§1. Quasibounded Positive and Negative Definite Functions 252<br>§2. Completely Monotone and Completely Alternating Functions 263<br>Notes and Remarks 271<br>References 273<br>List of Symbols 281<br>Index 285<br>I<br>