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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
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Searching... | 30000004598193 | QA403.5 T74 2004 | Open Access Book | Book | Searching... |
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Summary
Summary
In Fourier Analysis and Approximation of Functions basics of classical Fourier Analysis are given as well as those of approximation by polynomials, splines and entire functions of exponential type.
In Chapter 1 which has an introductory nature, theorems on convergence, in that or another sense, of integral operators are given. In Chapter 2 basic properties of simple and multiple Fourier series are discussed, while in Chapter 3 those of Fourier integrals are studied.
The first three chapters as well as partially Chapter 4 and classical Wiener, Bochner, Bernstein, Khintchin, and Beurling theorems in Chapter 6 might be interesting and available to all familiar with fundamentals of integration theory and elements of Complex Analysis and Operator Theory. Applied mathematicians interested in harmonic analysis and/or numerical methods based on ideas of Approximation Theory are among them.
In Chapters 6-11 very recent results are sometimes given in certain directions. Many of these results have never appeared as a book or certain consistent part of a book and can be found only in periodics; looking for them in numerous journals might be quite onerous, thus this book may work as a reference source.
The methods used in the book are those of classical analysis, Fourier Analysis in finite-dimensional Euclidean space Diophantine Analysis, and random choice.
Table of Contents
Dedication |
Preface1: 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 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 úIiú |
The Wiener Tauberian theorem |
6.2 Positive definate and completely monotone functions |
6.3 Positive definate functions depending only on a norm |
6.4 Sufficient conditions for belonging to Ap 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), 0 |
Further problems and theorems |
8.6 Comments to |
Chapter 8 9: Lebesgue Constants and Approximation |
9.1 Upper and lower estimates |
9.2 Examples of Lebesgue constants in the multiple case |
9.3 Asymptotics of Lebesgue constants and approximation |
9.4 Further problems and theorems |
9.5 Comments to |
Chapter 9 10: Widths, Polynomial Approximation |
10.1 Entrophy numbers |
10.2 Polynomials with free spectrum |
10.3 Kolmogorov widths |
10.4 Further problems and theorems |
10.5 Comments to |
Chapter 10 11: Functions with Bounded Mixed Derivative |
11.1 Hyperbolic cross polynomials |
11.2 Estimates of entrophy numbers |
11.3 Widths |
11.4 Further problems and theorems |
11.5 Comments to |
Chapter 11 Appendices |
Prerequisites |
Weierstrass approximation theorems |
The modulus of continuity of a function |
The Riemann-Stieltjes integral |
Summability of series |
Analytic functions |
Measure and integral. Complex-valued measures |
Hilbert spaces. Classical orthonormal systems |
Banach spaces and linear operators |
Fourier multipliers |
Several classical theorems |
Polynomials with integral coefficients |
Some inequalities |
&mathR;m and its subsets |
Functional notations |
References |
Author Index |
Topic Index |