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140801 ||| eng |
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|a 9781402028762
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|a Trigub, Roald M.
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|a Fourier Analysis and Approximation of Functions
|h Elektronische Ressource
|c by Roald M. Trigub, Eduard S. Belinsky
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| 250 |
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|a 1st ed. 2004
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| 260 |
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|a Dordrecht
|b Springer Netherlands
|c 2004, 2004
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| 300 |
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|a XIV, 586 p
|b online resource
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|a 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 --
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|a 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
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|a 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 --
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| 653 |
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|a Functional analysis
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| 653 |
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|a Measure theory
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| 653 |
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|a Functional Analysis
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| 653 |
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|a Fourier Analysis
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| 653 |
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|a Sequences, Series, Summability
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| 653 |
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|a Approximations and Expansions
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| 653 |
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|a Measure and Integration
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| 653 |
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|a Sequences (Mathematics)
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| 653 |
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|a Approximation theory
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| 653 |
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|a Fourier analysis
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| 700 |
1 |
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|a Belinsky, Eduard S.
|e [author]
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| 041 |
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7 |
|a eng
|2 ISO 639-2
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| 989 |
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|b SBA
|a Springer Book Archives -2004
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| 028 |
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|a 10.1007/978-1-4020-2876-2
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| 856 |
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|u https://doi.org/10.1007/978-1-4020-2876-2?nosfx=y
|x Verlag
|3 Volltext
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|a 511.4
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| 520 |
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|a 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
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