xii CONTENTS
Chapter 5. Harmonic Analysis in £2(T, //) 93
5.1. Generalized Fourier Series 93
5.2. Bases of Exponentials in
L2(T,/i)
96
5.3. Harmonic Conjugates 98
5.4. The Helson-Szego Theorem 99
5.5. An Example 102
5.6. Comments 103
5.7. Exercises and Further Results 104
5.8. Notes and Remarks 129
Chapter 6. Transfer to the Half-Plane 143
6.1. A Unitary Mapping from
LP(T)
to
LP(R)
143
6.2. Cauchy Kernels and Fourier Transforms 144
6.3. The Hardy Spaces Hp+ = HP(C+) 144
6.4. Canonical Factorization and Other Properties 147
6.5. Invariant Subspaces 148
6.6. Exercises and Further Results 150
6.7. Notes and Remarks 151
Chapter 7.' Time-Invariant Filtering 153
7.1. The Language of Signal Processing 153
7.2. Frequency Characteristics of Causal Filters 154
7.3. Design Problems (Filter Synthesis) 155
7.4. Inverse Analysis Problems, or How to Tackle a Filter 157
7.5. Exercises and Further Results 159
7.6. Notes and Remarks 160
Chapter 8. Distance Formulae and Zeros of the Riemann ^-Function 163
8.1. Distance Functions 163
8.2. Zeros and Singular Measures via Distance Functions 165
8.3. Localization of Zeros of the Riemann ^-Function 166
8.4. Invariant Subspaces Related to the ^-Function 169
8.5. Exercises and Further Results 170
8.6. Notes and Remarks 171
Part B. Hankel and Toeplitz Operators 173
Chapter 1. Hankel Operators and Their Symbols 179
1.1. Hankel Matrices and Hankel Operators 179
1.2. The Hardy Space Representation 180
1.3. Symbols of Hankel Operators and the Nehari Theorem 181
1.4. Two Proofs of the Nehari Theorem 182
1.5. An appendix on Hilbert space operators 186
1.6. Exercises and Further Results 188
1.7. What is a Hankel operator? A brief survey 195
1.8. Notes and Remarks 205
Chapter 2. Compact Hankel Operators 211
2.1. Essential Norm and the Calkin Algebra 211
2.2. The Adamyan-Arov-Krein Version of Hartman's Theorem 212
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