An Introduction to the Theory of Reproducing Kernel Hilbert Spaces (Cambridge Studies in Advanced Mathematics, Series Number 152)

Reproducing kernel Hilbert spaces have developed into an important tool in many areas, especially statistics and machine learning, and they play a valuable role in complex analysis, probability, group representation theory, and the theory of integral operators. This unique text offers a unified overview of the topic, providing detailed examples of applications, as well as covering the fundamental underlying theory, including chapters on interpolation and approximation, Cholesky and Schur operations on kernels, and vector-valued spaces. Self-contained and accessibly written, with exercises at the end of each chapter, this unrivalled treatment of the topic serves as an ideal introduction for graduate students across mathematics, computer science, and engineering, as well as a useful reference for researchers working in functional analysis or its applications. Front Matter 1 Contents 5 Preface 9 Part I General theory 11 1 Introduction 13 1.1 Definition 13 1.2 Basic examples 14 1.2.1 Cn as an RKHS 14 1.2.2 A non-example 15 1.3 Examples from analysis 16 1.3.1 Sobolev spaces on [0,1] 16 1.3.2 The Paley-Wiener spaces: an example from signal processing 19 1.4 Function theoretic examples 21 1.4.1 The Hardy space of the unit disk H2(D) 21 1.4.2 Bergman spaces on complex domains 23 1.4.3 Two multivariable examples 25 1.5 Exercises 25 2 Fundamental results 27 2.1 Hilbert space structure 27 2.2 Characterization of reproducing kernels 33 2.3 The Reconstruction Problem 37 2.3.1 The RKHS induced by a function 40 2.3.2 The RKHS of the Min function 40 2.3.3 The RKHS induced by a positive matrix 42 2.3.4 The RKHS induced by the inner product 43 2.4 Exercises 45 3 Interpolation and approximation 47 3.1 Interpolation in an RKHS 47 3.2 Strictly positive kernels 50 3.3 Best least squares approximants 51 3.4 The elements of H(K) 52 3.5 Exercises 55 4 Cholesky and Schur 57 4.1 Cholesky factorization 57 4.1.1 Cholesky’s algorithm 60 4.2 Schur products and the Schur decomposition 62 4.3 Tensor products of Hilbert spaces 64 4.4 Kernels arising from polynomials and power series 65 4.4.1 Polynomial kernels 65 4.4.2 Power series on disks 66 4.4.3 Power series on balls 68 4.4.4 The Drury–Arveson space 70 4.4.5 The Segal–Bargmann space 71 4.4.6 Other power series 72 4.5 Exercises 72 5 Operations on kernels 75 5.1 Complexification 75 5.2 Differences and sums 76 5.3 Finite-dimensional RKHSs 78 5.4 Pull-backs, restrictions and composition operators 79 5.4.1 Composition operators 81 5.5 Products of kernels and tensor products of spaces 82 5.6 Push-outs of RKHSs 85 5.7 Multipliers of a RKHS 87 5.8 Exercises 90 6 Vector-valued spaces 92 6.1 Basic theory 92 6.1.1 Matrices of operators 96 6.2 A Vector-valued version of Moore’s theorem 98 6.3 Interpolation 100 6.4 Operations on kernels 107 6.5 Multiplier algebras 108 6.6 Exercises 109 Part II Applications and examples 111 7 Power series on balls and pull-backs 113 7.1 Infinite direct sums of Hilbert spaces 113 7.2 The pull-back construction 114 7.3 Polynomials and interpolation 116 7.3.1 The Drury-Arveson space as a pull-back 119 7.3.2 The Segal–Bargmann space 120 7.4 Exercises 120 8 Statistics and machine learning 122 8.1 Regression problems and the method of least squares 122 8.1.1 Linear functions 123 8.1.2 Affine functions 124 8.1.3 Polynomials 125 8.2 The kernel method 126 8.3 Classification problems and geometric separation 128 8.4 The maximal margin classifier 130 8.5 Summary: a functional perspective 134 8.6 The representer theorem 135 8.7 Exercises 138 9 Negative definite functions 139 9.1 Schoenberg’s theory 139 9.2 Infinitely divisible kernels 144 9.3 Exercises 145 10 Positive definite functions on groups 147 10.1 Naimark’s theorem 147 10.2 Bochner’s theorem 149 10.3 Negative definite functions and cocycles 150 10.4 Exercises 153 11 Applications of RKHS to integral operators 155 11.1 Integral operators 155 11.2 The range space of an integral operator 156 11.2.1 The Volterra integral operator and the min kernel 159 11.3 Mercer kernels and integral operators 160 11.3.1 An application of Mercer’s theory 169 11.4 Square integrable symbols 170 11.5 Exercises 173 12 Stochastic processes 175 12.1 Covariance 175 12.2 Stochastic processes 177 12.3 Gaussian random variables 179 12.4 Karhunen–Loève theorem 186 12.5 Exercises 188 Bibliography 190 Index 191 Part 1 -- General Theory: 1. Introduction -- 2. Fundamental Results -- 3. Interpolation And Approximation -- 4. Cholesky And Schur -- 5. Operations On Kernals -- 6. Vector-valued Spaces -- Part 2 -- Applications And Examples: 7. Power Series On Balls And Pull-backs -- 8. Statistics And Machine Learning -- 9. Negative Definite Functions -- 10. Positive Definite Functions On Groups -- 11. Applications Of Rkhs To Integral Operators -- 12. Stochastic Processes. Vern I. Paulsen, Mrinal Raghupathi. Includes Bibliographical References And Index. Covering the fundamental underlying theory as well as a range of applications, this unique text provides a unified overview of reproducing kernel Hilbert spaces. It offers an unrivalled and accessible introduction to the field, ideal for graduate students and researchers working in functional analysis or its applications.