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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 |
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