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2004 | Buch

Representation Theorems Kapitel lesen Erstes Kapitel lesen

Fourier Analysis and Approximation of Functions

verfasst von: Roald M. Trigub, Eduard S. Bellinsky

Verlag: Springer Netherlands

Enthalten in: Professional Book Archive

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SUCHEN

Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Representation Theorems
Abstract
One of the central objects of Fourier Analysis is operators of the form
$$ f\left( \right) \mapsto \int\limits_{{R^m}} {f\left( { + \lambda u} \right)d\mu \left( u \right)} $$
(the convolution of a function and a measure) . When λ = 1 these operators commute with the translation operator: \( f\left( \right) \mapsto f\left( { + t} \right)! \) , that is, are translation invariant, and this property characterizes such operators completely.
Roald M. Trigub, Eduard S. Bellinsky
Chapter 2. Fourier Series
Abstract
In this chapter trigonometric Fourier series are studied. Convergence and divergence problems, the Fejér and the Abel-Poisson summability methods, and the association of the Fourier series problems with those for the Hardy spaces in the circle are among the subjects. Onedimensional series are considered in Sections 2.1-2.3, while multidimensional series are studied in Section 2.4.
Roald M. Trigub, Eduard S. Bellinsky
Chapter 3. Fourier Integral
Abstract
If f : ℝℶℂ, then its Fourier transform is defined as
$$ \hat f\left( y \right) = \left( {2\pi } \right){ - ^{1/2}}\int_{ - \infty }^\infty {f\left( x \right){e^{ - iyx}}dx} = {\left( {2\pi } \right)^{1/2}}\int_{ - \infty }^\infty {f\left( x \right)\cos yxdx - i{{\left( {2\pi } \right)}^{1/2}}\int_{ - \infty }^\infty {f\left( x \right)\sin yxdx,} } $$
(3.0.1)
provided this integral converges in some sense. This is an analog of the Fourier coefficients of a periodic function.
Roald M. Trigub, Eduard S. Bellinsky
Chapter 4. Discretization. Direct and Inverse Theorems of Approximation Theory
Abstract
What may be referred to as discretization is replacing an integral by its Riemann integral sums, substituting a function by the interpolation series, passing from an infinite-dimensional space (of functions) to a finite-dimensional space (of polynomials, splines, etc.), and the like
Roald M. Trigub, Eduard S. Bellinsky
Chapter 5. Extremal Problems of Approximation Theory
Abstract
The main subject of this chapter is the study of best approximation either to separate functions or to classes of functions by polynomials of given degree as well as by approximants from other subspaces. In Section 5.1, we not only outline, in subsections A, B, and C, the precise setting of the investigated problems but also discuss questions of existence and uniqueness of best approximation and give a criterion of best approximation. In Section 5.2 we introduce a specific definition for the space L p (Ω, μ) , while for C on the compact the same procedure is done in Section 5.3. In Section 5.5 we discuss best approximation to classes of functions by polynomials and by entire functions of exponential type. In Section 5.4 extremal properties of splines (5.4.9 – 5.4.12) are given. We are going to use them further on, in Chapter 10. We also study the properties of polynomials (5.4.1 – 5.4.8 and 5.4.13 – 5.4.14) concerning best approximation to a constant by algebraic polynomials with integral coefficients (5.4.15 – 5.4.16).
Roald M. Trigub, Eduard S. Bellinsky
Chapter 6. Representability of a Function as the Fourier Transform of a Measure
Abstract
In this chapter we set forth classical theorems of N. Wiener (in particular, Tauberian theorem), S. Bochner, S. N. Bernstein, S. Bochner and A. Ya. Khintchin, I. J. Schoenberg, and A. Beurling.
Roald M. Trigub, Eduard S. Bellinsky
Chapter 7. Fourier Multipliers
Abstract
In this chapter we study translation invariant linear operators representable as the convolution of a function and a measure (see, for example, Stein and Weiss [M-1971], Ch. I), more exactly operators generated by a numerical sequence {λk} as
$$ f \sim \sum {{c_k}(f){e_k} \mapsto \sum {{\lambda _k}{c_k}(f){e_k} \sim \Delta f} } $$
Roald M. Trigub, Eduard S. Bellinsky
Chapter 8. Methods of Summability of Fourier Series. Moduli of Smoothness and K-Functionals
Abstract
In this chapter general results on multipliers (Chapter 7) and sufficient conditions for representing a function as the Fourier transform (Chapter 6) are applied (Section 8.1) to the study of regularity (in one or another sense) of summability methods of simple and multiple Fourier series (classical and nonclassical), to defining exact rate of approximation of an individual function (Sections 8.2 – 8.4) via moduli of smoothness (defined in a special way for the multiple case and for p < 1). By this, K-functionals of a couple of spaces of smooth functions on the torus and poly-disk are computed as well (Section 8.3); this play a significant role in the real method of interpolation (see A.8.5).
Roald M. Trigub, Eduard S. Bellinsky
Chapter 9. Lebesgue Constants and Approximation of Classes of Functions with Bounded Derivative by Polynomials
Abstract
The norms of multiplier operators are called the Lebesgue constants. In the case of the space C(𝕋m) (or L(𝕋m)) the Lebesgue constants are represented by the integrals
$$ {(2\pi )^{ - m}}\int\limits_{{T^m}} {\left| {\sum\limits_k {{\lambda _{n,k}}{e^{i(k,x)}}} } \right|dx} $$
.
Roald M. Trigub, Eduard S. Bellinsky
Chapter 10. Widths. Approximation by Trigonometric Polynomials with Free Spectrum
Abstract
The definitions of widths and extremal subspaces were given in Section 5.1.
Roald M. Trigub, Eduard S. Bellinsky
Chapter 11. Classes of Functions with Bounded Mixed Derivative
Abstract
For classes of periodic functions of m real variables, we obtain estimates of ε-entropy in the uniform and integral metrics, and estimates of Kolmogorov and trigonometric widths. These results fall short of the completeness of those in the one-dimensional case, but definitely have a flavor of the multidimensional case.
Roald M. Trigub, Eduard S. Bellinsky
Backmatter
Metadaten
Titel
Fourier Analysis and Approximation of Functions
verfasst von
Roald M. Trigub
Eduard S. Bellinsky
Copyright-Jahr
2004
Verlag
Springer Netherlands
Electronic ISBN
978-1-4020-2876-2
Print ISBN
978-90-481-6641-1
DOI
https://doi.org/10.1007/978-1-4020-2876-2