## ＜電子ブック＞Fourier Analysis and Approximation of Functions

責任表示 by Roald M. Trigub, Eduard S. Bellinsky Trigub, Roald M Bellinsky, Eduard S SpringerLink (Online service) English (英語) Springer Netherlands Imprint: Springer 2004- Dordrecht, Netherlands 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.続きを見る 1. Representation Theorems1.1 Theorems on representation at a point1.2 Integral operators. Convergence in Lp-norm and almost everywhere1.3 Multidimensional case1.4 Further problems and theorems1.5 Comments to Chapter 12. Fourier Series2.1 Convergence and divergence2.2 Two classical summability methods2.3 Harmonic functions and functions analytic in the disk2.4 Multidimensional case2.5 Further problems and theorems2.6 Comments to Chapter 23. Fourier Integral3.1 L-Theory3.2 L2-Theory3.3 Multidimensional case3.4 Entire functions of exponential type. The Paley-Wiener theorem3.5 Further problems and theorems3.6 Comments to Chapter 34. Discretization. Direct and Inverse Theorems4.1 Summation formulas of Poisson and Euler-Maclaurin4.2 Entire functions of exponential type and polynomials4.3 Network norms. Inequalities of different metrics4.4 Direct theorems of Approximation Theory4.5 Inverse theorems. Constructive characteristics. Embedding theorems4.6 Moduli of smoothness4.7 Approximation on an interval4.8 Further problems and theorems4.9 Comments to Chapter 45. Extremal Problems of Approximation Theory5.1 Best approximation5.2 The space Lp. Best approximation5.3 Space C. The Chebyshev alternation5.4 Extremal properties for algebraic polynomials and splines5.5 Best approximation of a set by another set5.6 Further problems and theorems5.7 Comments to Chapter 56. A Function as the Fourier Transform of A Measure6.1 Algebras A and B. The Wiener Tauberian theorem6.2 Positive definite and completely monotone functions6.3 Positive definite functions depending only on a norm6.4 Sufficient conditions for belonging to Ap and A*6.5 Further problems and theorems6.6 Comments to Chapter 67. Fourier Multipliers7.1 General properties7.2 Sufficient conditions7.3 Multipliers of power series in the Hardy spaces7.4 Multipliers and comparison of summability methods of orthogonal series7.5 Further problems and theorems7.6 Comments to Chapter 78. Summability Methods. Moduli of Smoothness8.1 Regularity8.2 Applications of comparison. Two-sided estimates8.3 Moduli of smoothness and K-functionals8.4 Moduli of smoothness and strong summability in Hp(D), 0erencesAuthor IndexTopic Index.続きを見る Full text available from SpringerLink ebooks - Mathematics and Statistics (Archive)

### 詳細

レコードID 1464375 QA401-425 511.4 Mathematics. Fourier analysis. Functional analysis. Sequences (Mathematics). Mathematics. Approximations and Expansions. Fourier Analysis. Sequences, Series, Summability. Measure and Integration. Functional Analysis. ssj0000937074 9789048166411[9048166411](print) 9781402028762[1402028768] 2014.09.18 2017.11.26