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 P´ olya and Levinson 67

1. P´ 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 P´ 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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