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Fourier analysis and approximation of functions /

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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 i...

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Bibliographic Details
Main Authors: Trigub, Roald M. (Author), Belinsky, Eduard S. (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Dordrecht ; London : Springer Science+Business Media, 2004.
Subjects:
Fourier analysis.
Approximation theory.
Functions.
Electronic books.
Online Access:Connect to the full text of this electronic book
Table of Contents:
  • 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.