vi CONTENTS
6. Equivalence between weighted uniform and
Lp-approximation
104
7. A criterion for completeness of polynomials in CW 105
8. Lemmas and proofs 106
9. Examples and corollaries 114
Chapter 6. The Type Problem 119
1. General case p = 2 120
2. Known examples 121
3. Polynomial decay 122
4. Completeness of exponentials in Lp and CW 123
5. Main results 124
6. Classical results and further corollaries 126
7. Auxiliary statements 130
8. Proofs of main results 139
Chapter 7. Toeplitz Approach to UP 143
1. Spaces of entire functions and their zero sets 143
2. Model spaces and Toeplitz operators 144
3. Spectral theory 145
4. Outer, inner and Herglotz functions 146
5. Model spaces 148
6. Weyl inner functions 148
7. Modified Fourier transform 149
8. Entire functions 151
9. Square root transformation 153
10. Dimension and triviality of Toeplitz kernels 153
11. General form of Levinson’s completeness theorem 161
12. Applications to UP 164
Chapter 8. Toeplitz Version of the Beurling–Malliavin Theory 183
1. The language of Toeplitz kernels 184
2. Super-exponential case 186
3. Sub-exponential case 187
4. The structure of BM theory 189
5. One-sided Lipschitz condition for the Hilbert transform 189
6. Triviality of Toeplitz kernels 193
7. Non-triviality of Toeplitz kernels in Smirnov–Nevanlinna class 197
8. Multiplier theorem 200
9. Non-triviality of Toeplitz kernels in Hardy spaces 204
Bibliography 211
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