Gaussian Processes, Function Theory, and the Inverse Spectral Problem

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ISBN: 9780486462790 | 048646279X
Cover: Paperback
Copyright: 2/29/2008

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This text offers background in function theory, Hardy functions, and probability as preparation for surveys of Gaussian processes, strings and spectral functions, and strings and spaces of integral functions. It addresses the relationship between the past and the future of a real, one-dimensional, stationary Gaussian process. 1976 edition.

Authors' Note	p. ix
Preface	p. xi
Introduction	p. 1
Background: Function Theory
Harmonic Functions	p. 10
Positive Harmonic Functions on the Upper Half-Plane	p. 13
Analytic Functions of Exponential Type	p. 16
Phragmen-Lindelof Principle	p. 17
Poisson-Jensen Formula	p. 18
Quotients of Exponential Type	p. 20
Carlson's Theorem	p. 21
Hadamard Products	p. 21
Nevanlinna's Formula	p. 22
Background: Hardy Functions
Fourier Integrals	p. 26
Paley-Wiener Theorem	p. 28
Hardy Functions of Class H[superscript 2]	p. 30
Projection	p. 33
A Basis	p. 34
An Inequality	p. 35
Inner and Outer Functions	p. 36
Tests for Outer Functions	p. 39
Hardy Functions of Class H[superscript 1]	p. 42
Hardy Functions of Class H[superscript infinity]	p. 46
Hilbert Transforms and Conjugate Functions	p. 48
Phases	p. 52
Inner Functions Again	p. 53
Quasi-Analytic Classes	p. 55
Background: Probability
Gaussian Families	p. 61
Independence and Perpendicularity	p. 63
Angles	p. 65
Degree of Independence	p. 67
Mutual Information	p. 68
Kolmogorov's Modulus	p. 69
Stationary Gaussian Processes	p. 71
Spectral Functions	p. 74
Metric Transitivity	p. 76
Ivankov's Alternative	p. 78
Past and Future
Kolmogorov-Wiener Prediction Problem	p. 82
Szego's Alternative	p. 83
Doing the Prediction	p. 87
Wiener's Formula	p. 90
Filters and White Noise	p. 92
Past and Future	p. 96
Rational Spectral Densities	p. 100
Type Spaces	p. 108
Splitting	p. 117
Germ and Gap	p. 121
Okabe's Formulas	p. 126
Mixing	p. 130
Interpolation	p. 136
Strings and Spectral Functions
Strings	p. 147
Self-Adjoint Operators	p. 153
Feller's Theorem	p. 159
Green Functions	p. 162
Principal Spectral Functions	p. 176
Traces	p. 181
Transforms	p. 185
The Backwards Problem	p. 194
Stieltjes' Investigations	p. 203
Nonprincipal Spectral Functions	p. 208
Strings and Spaces of Integral Functions
De Branges Spaces	p. 221
Short de Branges Spaces	p. 228
Short Spaces and Short Strings	p. 233
Type Spaces	p. 241
Nesting of Short Spaces	p. 247
Recovering the String	p. 252
Hardy Densities	p. 255
Dual Strings	p. 262
Change of Scale and Mass	p. 264
Prediction Using Part of the Past	p. 279
Short Processes	p. 291
A New Class of Strings	p. 293
Strings for Interpolation	p. 306
Interpolation: Examples	p. 318
References	p. 323
Index	p. 329
Table of Contents provided by Ingram. All Rights Reserved.

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Gaussian Processes, Function Theory, and the Inverse Spectral Problem

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Gaussian Processes, Function Theory, and the Inverse Spectral Problem

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Author Biography

Table of Contents

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