1. Representation Theorems |
1.1 Theorems on representation at a point |
1.2 Integral operators. Convergence in Lp-norm and almost everywhere |
1.3 Multidimensional case |
1.4 Further problems and theorems |
1.5 Comments to Chapter 1 |
2. Fourier Series |
2.1 Convergence and divergence |
2.2 Two classical summability methods |
2.3 Harmonic functions and functions analytic in the disk |
2.4 Multidimensional case |
2.5 Further problems and theorems |
2.6 Comments to Chapter 2 |
3. Fourier Integral |
3.1 L-Theory |
3.2 L2-Theory |
3.3 Multidimensional case |
3.4 Entire functions of exponential type. The Paley-Wiener theorem |
3.5 Further problems and theorems |
3.6 Comments to Chapter 3 |
4. Discretization. Direct and Inverse Theorems |
4.1 Summation formulas of Poisson and Euler-Maclaurin |
4.2 Entire functions of exponential type and polynomials |
4.3 Network norms. Inequalities of different metrics |
4.4 Direct theorems of Approximation Theory |
4.5 Inverse theorems. Constructive characteristics. Embedding theorems |
4.6 Moduli of smoothness |
4.7 Approximation on an interval |
4.8 Further problems and theorems |
4.9 Comments to Chapter 4 |
5. Extremal Problems of Approximation Theory |
5.1 Best approximation |
5.2 The space Lp. Best approximation |
5.3 Space C. The Chebyshev alternation |
5.4 Extremal properties for algebraic polynomials and splines |
5.5 Best approximation of a set by another set |
5.6 Further problems and theorems |
5.7 Comments to Chapter 5 |
6. A Function as the Fourier Transform of A Measure |
6.1 Algebras A and B. The Wiener Tauberian theorem |
6.2 Positive definite and completely monotone functions |
6.3 Positive definite functions depending only on a norm |
6.4 Sufficient conditions for belonging to Ap and A* |
6.5 Further problems and theorems |
6.6 Comments to Chapter 6 |
7. Fourier Multipliers |
7.1 General properties |
7.2 Sufficient conditions |
7.3 Multipliers of power series in the Hardy spaces |
7.4 Multipliers and comparison of summability methods of orthogonal series |
7.5 Further problems and theorems |
7.6 Comments to Chapter 7 |
8. Summability Methods. Moduli of Smoothness |
8.1 Regularity |
8.2 Applications of comparison. Two-sided estimates |
8.3 Moduli of smoothness and K-functionals |
8.4 Moduli of smoothness and strong summability in Hp(D), 0erences |
Author Index |
Topic Index. |