Contents
Thanks vii
Chapter 1. Mathematical Shapes of Uncertainty 1
1. Basic notations 3
2. Variety of mathematical forms of UP 4
3. Function theoretic background 10
Chapter 2. Gap Theorems 17
1. Classical Gap Theorems 18
2. Spectral gap as a property of the support 22
3. Toeplitz kernels and uniform approximation 23
4. A formula for the gap characteristic of a set 24
5. Examples and applications 27
6. Appendix: Proof of the gap formula 29
7. Appendix: Technical lemmas 51
8. Appendix: De Branges’ theorem 66 in Toeplitz form 61
Chapter 3. A Problem by olya and Levinson 67
1. olya sequences 67
2. A theorem on existence of a de Branges’ space in
L2
68
3. Beurling–Malliavin densities 69
4. Two theorems on Toeplitz kernels 70
5. A description of olya sequences 71
6. Technical lemmas 72
7. Proofs of theorems 74
Chapter 4. Determinacy of Measures and Oscillations of High-pass Signals 77
1. Sign changes of a measure with a spectral gap 77
2. Entire functions and densities 79
3. A lemma on d-uniform sequences 80
4. M. Riesz-type criterion and its consequences 81
5. Extreme measures in the indeterminate case 85
6. Measures annihilating Paley-Wiener spaces 90
7. Sign changes of measures with spectral gap 93
Chapter 5. Beurling–Malliavin and Bernstein’s Problems 95
1. A problem on completeness of exponentials 95
2. Structure of proof of BM Theorem: BM theory 99
3. Bernstein’s problem 101
4. Semi-continuous weights 102
5. Characteristic sequences 103
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