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......Page 1 Contents......Page 5 Preface......Page 9 Part I General theory......Page 11 1.1 Definition......Page 13 1.2.1 Cn as an RKHS......Page 14 1.2.2 A non-example......Page 15 1.3.1 Sobolev spaces on [0,1]......Page 16 1.3.2 The Paley-Wiener spaces: an example from signal processing......Page 19 1.4.1 The Hardy space of the unit disk H2(D)......Page 21 1.4.2 Bergman spaces on complex domains......Page 23 1.5 Exercises......Page 25 2.1 Hilbert space structure......Page 27 2.2 Characterization of reproducing kernels......Page 33 2.3 The Reconstruction Problem......Page 37 2.3.2 The RKHS of the Min function......Page 40 2.3.3 The RKHS induced by a positive matrix......Page 42 2.3.4 The RKHS induced by the inner product......Page 43 2.4 Exercises......Page 45 3.1 Interpolation in an RKHS......Page 47 3.2 Strictly positive kernels......Page 50 3.3 Best least squares approximants......Page 51 3.4 The elements of H(K)......Page 52 3.5 Exercises......Page 55 4.1 Cholesky factorization......Page 57 4.1.1 Cholesky’s algorithm......Page 60 4.2 Schur products and the Schur decomposition......Page 62 4.3 Tensor products of Hilbert spaces......Page 64 4.4.1 Polynomial kernels......Page 65 4.4.2 Power series on disks......Page 66 4.4.3 Power series on balls......Page 68 4.4.4 The Drury–Arveson space......Page 70 4.4.5 The Segal–Bargmann space......Page 71 4.5 Exercises......Page 72 5.1 Complexification......Page 75 5.2 Differences and sums......Page 76 5.3 Finite-dimensional RKHSs......Page 78 5.4 Pull-backs, restrictions and composition operators......Page 79 5.4.1 Composition operators......Page 81 5.5 Products of kernels and tensor products of spaces......Page 82 5.6 Push-outs of RKHSs......Page 85 5.7 Multipliers of a RKHS......Page 87 5.8 Exercises......Page 90 6.1 Basic theory......Page 92 6.1.1 Matrices of operators......Page 96 6.2 A Vector-valued version of Moore’s theorem......Page 98 6.3 Interpolation......Page 100 6.4 Operations on kernels......Page 107 6.5 Multiplier algebras......Page 108 6.6 Exercises......Page 109 Part II Applications and examples......Page 111 7.1 Infinite direct sums of Hilbert spaces......Page 113 7.2 The pull-back construction......Page 114 7.3 Polynomials and interpolation......Page 116 7.3.1 The Drury-Arveson space as a pull-back......Page 119 7.4 Exercises......Page 120 8.1 Regression problems and the method of least squares......Page 122 8.1.1 Linear functions......Page 123 8.1.2 Affine functions......Page 124 8.1.3 Polynomials......Page 125 8.2 The kernel method......Page 126 8.3 Classification problems and geometric separation......Page 128 8.4 The maximal margin classifier......Page 130 8.5 Summary: a functional perspective......Page 134 8.6 The representer theorem......Page 135 8.7 Exercises......Page 138 9.1 Schoenberg’s theory......Page 139 9.2 Infinitely divisible kernels......Page 144 9.3 Exercises......Page 145 10.1 Naimark’s theorem......Page 147 10.2 Bochner’s theorem......Page 149 10.3 Negative definite functions and cocycles......Page 150 10.4 Exercises......Page 153 11.1 Integral operators......Page 155 11.2 The range space of an integral operator......Page 156 11.2.1 The Volterra integral operator and the min kernel......Page 159 11.3 Mercer kernels and integral operators......Page 160 11.3.1 An application of Mercer’s theory......Page 169 11.4 Square integrable symbols......Page 170 11.5 Exercises......Page 173 12.1 Covariance......Page 175 12.2 Stochastic processes......Page 177 12.3 Gaussian random variables......Page 179 12.4 Karhunen–Loève theorem......Page 186 12.5 Exercises......Page 188 Bibliography......Page 190 Index......Page 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.