Gaussian Processes, Function Theory, and the Inverse Spectral Problem

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  • Format: Paperback
  • Copyright: 2008-02-29
  • Publisher: Dover Publications

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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.

Author Biography

H. Dym is Professor of Mathematics at the Weizmann Institute of Science, Rehovot, Israel H. P. McKean is Professor of Mathematics at New York University's Courant Institute

Table of Contents

Authors' Notep. ix
Prefacep. xi
Introductionp. 1
Background: Function Theory
Harmonic Functionsp. 10
Positive Harmonic Functions on the Upper Half-Planep. 13
Analytic Functions of Exponential Typep. 16
Phragmen-Lindelof Principlep. 17
Poisson-Jensen Formulap. 18
Quotients of Exponential Typep. 20
Carlson's Theoremp. 21
Hadamard Productsp. 21
Nevanlinna's Formulap. 22
Background: Hardy Functions
Fourier Integralsp. 26
Paley-Wiener Theoremp. 28
Hardy Functions of Class H[superscript 2]p. 30
Projectionp. 33
A Basisp. 34
An Inequalityp. 35
Inner and Outer Functionsp. 36
Tests for Outer Functionsp. 39
Hardy Functions of Class H[superscript 1]p. 42
Hardy Functions of Class H[superscript infinity]p. 46
Hilbert Transforms and Conjugate Functionsp. 48
Phasesp. 52
Inner Functions Againp. 53
Quasi-Analytic Classesp. 55
Background: Probability
Gaussian Familiesp. 61
Independence and Perpendicularityp. 63
Anglesp. 65
Degree of Independencep. 67
Mutual Informationp. 68
Kolmogorov's Modulusp. 69
Stationary Gaussian Processesp. 71
Spectral Functionsp. 74
Metric Transitivityp. 76
Ivankov's Alternativep. 78
Past and Future
Kolmogorov-Wiener Prediction Problemp. 82
Szego's Alternativep. 83
Doing the Predictionp. 87
Wiener's Formulap. 90
Filters and White Noisep. 92
Past and Futurep. 96
Rational Spectral Densitiesp. 100
Type Spacesp. 108
Splittingp. 117
Germ and Gapp. 121
Okabe's Formulasp. 126
Mixingp. 130
Interpolationp. 136
Strings and Spectral Functions
Stringsp. 147
Self-Adjoint Operatorsp. 153
Feller's Theoremp. 159
Green Functionsp. 162
Principal Spectral Functionsp. 176
Tracesp. 181
Transformsp. 185
The Backwards Problemp. 194
Stieltjes' Investigationsp. 203
Nonprincipal Spectral Functionsp. 208
Strings and Spaces of Integral Functions
De Branges Spacesp. 221
Short de Branges Spacesp. 228
Short Spaces and Short Stringsp. 233
Type Spacesp. 241
Nesting of Short Spacesp. 247
Recovering the Stringp. 252
Hardy Densitiesp. 255
Dual Stringsp. 262
Change of Scale and Massp. 264
Prediction Using Part of the Pastp. 279
Short Processesp. 291
A New Class of Stringsp. 293
Strings for Interpolationp. 306
Interpolation: Examplesp. 318
Referencesp. 323
Indexp. 329
Table of Contents provided by Ingram. All Rights Reserved.

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