Author | Trigub, Roald M. author |
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Title | Fourier Analysis and Approximation of Functions [electronic resource] / by Roald M. Trigub, Eduard S. Bellinsky |
Imprint | Dordrecht : Springer Netherlands : Imprint: Springer, 2004 |
Connect to | http://dx.doi.org/10.1007/978-1-4020-2876-2 |
Descript | XIV, 586 p. online resource |
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