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Fourier Analysis and Approximation of Functions

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., Belinsky, Eduard S. (Author)
Format: eBook
Language:English
Published: Dordrecht Springer Netherlands 2004, 2004
Edition:1st ed. 2004
Subjects:
Functional Analysis
Measure Theory
Fourier Analysis
Sequences, Series, Summability
Approximations And Expansions
Measure And Integration
Sequences (mathematics)
Approximation Theory
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
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
  • 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
  • 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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