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 v

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