Uniform Approximations by

Trigonometric Polynomials

By A.I. Stepanets

September 2001

VSP

ISBN: 90-6764-347-5

468 pages

$305.50 hardcover

The theory of approximation of functions is one of the central branches in mathematical analysis and has been developed over a number of decades. This monograph deals with a series of problems related to one of the directions of the theory, namely, the approximation of periodic functions by trigonometric polynomials generated by linear methods of summation of Fourier series. More specific, the following linear methods are investigated: classical methods of Fourier, Fejér, Riesz, and Roginski. For these methods the so-called Kolmogorov-Nikol'skii problem is considered, which consists of finding exact and asymptotically exact qualities for the upper bounds of deviations of polynomials generated by given linear methods on given classes of 2-periodic functions. Much attention is also given to the multidimensional case.The material presented in this monograph did not lose its importance since the publication of the Russian edition (1981). Moreover, new material has been added and several corrections were made. In this field of mathematics numerous deep results were obtained, many important and complicated problems were solved, and new methods were developed, which can be extremely useful for many mathematicians. All principle problems considered in this monograph are given in the final form, i.e. in the form of exact asymptotic equalities, and, therefore, retain their importance and interest for a long time.

Contents:

- Preface to the English Edition
- Preface to the Russian Edition
- Introduction
- CHAPTER 1. SIMPLEST EXTREMAL PROBLEMS
- 1. Modulus of Continuity
- 2. Classes of Continuous Functions
- 3. Simplest Extremal Problem for the Classes H![a,b]
- 4. Scheme of Estimation of One-Dimensional Integrals
- 5. Simplest Extremal Problem for the Classes q(x;a)
- 6. Representation of E(phi,rho,H!(2)(P)) in Terms of Rearrangements
- 7. Case of Symmetric Functions
- 8. Simplest Extremal Problem for the Classes H!(N)(P), N>2
- Bibliographical Notes
- CHAPTER 2. APPROXIMATION OF FUNCTIONS OF ONE VARIABLE BY FOURIER SUMS
- 1. Fourier Sums
- 2. Conjugate Functions and Their Classes
- 3. Asymptotic Relations for Zeros of Integral Sine and Integral Cosine
- 4. Representations of Upper Bounds of Deviations of Fourier Sums on the Classes WrH!
- 5. Asymptotic Estimates for E(H!;Sn)
- 6. Asymptotic Estimates for E(WrH!;Sn)
- 7. Simultaneous Approximation of Periodic Functions and Their Derivatives by Fourier Sums
- 8. Asymptotic Estimates for E(~H!;Sn)
- 9. Asymptotic Estimates for E(WqH!;Sn)
- 10. Simultaneous Approximation of Derivatives of Functions from the Class WqH! by Fourier Sums
- Bibliographical Notes
- CHAPTER 3. APPROXIMATION OF FUNCTIONS OF MANY VARIABLES BY FOURIER SUMS
- 1. Multiple Fourier Sums. Statement of the Problem
- 2. Decomposition of Functions into Simple Functions
- 3. Dirichlet Integral for Simple Functions
- 4. Estimates for Deviations of Fourier Sums on the Class H!(N)
- 5. Extremal Functions and Asymptotic Equalities
- Bibliographical Notes
- CHAPTER 4. FEJÉR SUMS
- 1. Linear Methods of Summation of Fourier Series. General Aspects
- 2. Bernoulli Kernels and Extremal Values of Periodic Functions
- 3. Deviation of Fejér Sums on the Classes Wr(H!)
- 4. Fejér Sums on Classes of Conjugate Functions
- 5. Deviation of Fejér Sums on the Classes H!(N)
- Bibliographical Notes
- CHAPTER 5. SPHERICAL RIESZ SUMS
- 1. Bochner Formula
- 2. Deviation of the Riesz Sums
S_{n}^{1}(f;x) on the Classes H!(N)- 3. Asymptotic Equalities for E ffi n (WrH!)
- 4. Representation of E ffi R _N (H!) and Auxiliary Statements
- 5. Estimates from Above for E ffi R _N (H!)
- 6. Asymptotic Equalities for E ffi R _N (H!)
- Bibliographical Notes
- CHAPTER 6. ROGOSINSKI SUMS
- 1. Definitions and General Remarks
- 2. Distribution of Zeros of Functions Associated with Rogosinski Sums
- 3. Estimates from Above for E (H!; Rn)
- 4. Behavior of the Quantities E(WrH!; Rn)
- 5. Rogosinski Sums on the Classes H!(2)
- Bibliographical Notes
- CHAPTER 7. FAVARD SUMS
- 1. Definitions and Auxiliary Remarks
- 2. Location of Zeros of the Favard Kernel
- 3. Location of Zeros of the First Derivative of the Favard Kernel
- 4. Location of Zeros of the Second Derivative of the Favard Kernel
- 5. Estimates for the Lebesgue Constant of the Favard Method
- 6. Properties of the Function rho-k(t)
- 7. Favard Sums on the Classes H!
- 8. Favard Sums on the Classes H!(2)
- Bibliographical Notes
ReferencesIndexMathematics

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