# Theory of Approximation of Functions of a Real Variable

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Theory of Approximation of Functions of a Real Variable discusses a number of fundamental parts of the modern theory of approximation of functions of a real variable. The material is grouped around the problem of the connection between the best approximation of functions to their structural properties. This text is composed of eight chapters that highlight the relationship between the various structural properties of real functions and the character of possible approximations to them by polynomials and other functions of simple construction. Each chapter concludes with a section containing various problems and theorems, which supplement the main text. The first chapters tackle the Weierstrass's theorem, the best approximation by polynomials on a finite segment, and some compact classes of functions and their structural properties. The subsequent chapters describe some properties of algebraic polynomials and transcendental integral functions of exponential type, as well as the direct theorems of the constructive theory of functions. These topics are followed by discussions of differential and constructive characteristics of converse theorems. The final chapters explore other theorems connecting the best approximations functions with their structural properties. These chapters also deal with the linear processes of approximation of functions by polynomials. The book is intended for post-graduate students and for mathematical students taking advanced courses, as well as to workers in the field of the theory of functions.

## Table of Contents

Editorial Preface

Prefatory Note

Foreword

I. Weierstrass's Theorem

1.1 Approximation of Continuous Functions by Polynomials on a Finite Segment.

The Fundamental Theorem

1.2 Proof of Weierstrass's Theorem

1.3 Generalization and Some Particular Cases. Periodic Functions. Functions of Many Variables

1.4 Mean Approximation of Integrable Functions by Polynomials

1.5 Some Stronger Forms of Weierstrass's Theorem

1.6 Approximation of Continuous Functions on an Infinite Interval. Uniform Approximation by Rational Functions

1.7 Uniform Approximation on the Whole Real Axis by Integral Functions of Finite Degree

1.8 On Weighted Uniform Approximation of Continuous Functions on the Whole Real Axis

1.9 On Mean Approximation of Integrable Functions on An Infinite Interval

1.10 Various Problems and Theorems

II. The Best Approximation

2.1 The Best Approximation by Polynomials on a Finite Segment

2.2 Generalization to a Linear Normed Space and Some Particular Cases

2.3 On the Uniqueness of the Polynomial giving the Best Uniform Approximation. Haar's Theorem

2.4 On the Uniqueness of the Polynomial giving the Best Integral Approximation

2.5 A Sequence of Best Approximations. The Fundamental Characteristic Property

2.6 The Best Approximation on An Infinite Interval

2.7 Properties of Best Polynomial Approximations. A Theorem of Chebyshev

2.8 On Some Properties of Polynomials of Best Integral Approximations

2.9 Polynomials which Deviate Least from Zero

2.10 Estimation of the Best Approximation. De la Vallée Poussin's Theorem

2.11 Some Cases of the Exact Solution of the Problem of the Best Approximation

2.12 On Some Criteria for the Best Approximation in an Infinite Interval

2.13 Various Problems and Theorems

III. Some Compact Classes of Functions and their Structural Characteristics

3.1 Compact Classes of Functions of a Single Variable. Criteria of the Type of Arzelà's Theorem. The Concepts of the ε-Entropy and ε-Capacity of a Compact Set

3.2 Moduli of Continuity and some of their Properties

3.3 Moduli of Smoothness of Various Orders

3.4 On Compact Classes of Functions of Many Variables

3.5 Compact Classes of Differentiable Functions

3.6 Sets of Functions of Bounded Variation

3.7 Some Classes of Functions Analytic on a Finite Segment

3.8 Some Classes of Functions Analytic on the Whole Real Axis

3.9 On Regularly Monotonie Functions

3.10 On Quasi-Analytic Classes of Functions

3.11 Conjugate Classes of Functions Defined on the Whole Real Axis

3.12 Various Problems and Theorems

IV. Some Properties of Algebraic Polynomials and Transcendental Integral Functions of Exponential Type

4.1 Interpolation Formulae for Algebraic Polynomials

4.2 Interpolation Formulae for Trigonometric Polynomials

4.3 Interpolation Formulae for Some Classes of Transcendental Integral Functions of Exponential Type

4.4 On the Phenomenon of Interference in the Behavior of Integral Functions of Finite Degree

4.5 Some Integral Representations for Algebraic and Trigonometric Polynomials

4.6 An Integral Representation for Some Classes of Transcendental Integral Functions of Exponential Type. A Theorem of Wiener and Paley

4.7 Some Interpolation and Integral Identities for the Derivatives of Algebraic Polynomials and Transcendental Integral Functions of Exponential Type

4.8 Some Extremal Properties of Algebraic Polynomials and Transcendental Integral Functions of Finite Degree

4.9 Integral and Interpolation Norms of Integral Functions of Finite Degree in Various Metrics

4.10 The Connection Between Trigonometric Polynomials and Other Integral Functions of the Classes Bσ

4.11 Non-Negative Polynomials and Transcendental Integral Functions of Exponential Type

4.12 Various Problems and Theorems

V. Direct Theorems of the Constructive Theory of Functions

5.1 The Effect of the Differential Properties of Functions on the Rapidity of Decrease to Zero of their Best Approximations. Jackson's Theorem

5.2 A Strong form of Jackson's Theorem on the Best Approximation of Continuous Functions by Algebraic Polynomials on a Finite Segment of the Real Axis

5.3 Some Direct Theorems for Functions of Many Variables

5.4 The Rapidity of Decrease to Zero of the Best Approximation of Analytic Functions

5.5 More Precise Forms of Jackson's Theorems. The Estimates of N. I. Akhiezer, M. G. Krein and J. Favard for Differentiable Periodic Functions

5.6 On the Asymptotic Behavior of the Upper Bounds of the Best Approximations of Classes of Functions, Differentiable a Given finite Number of Times. S. N. Bernstein's Theorem

5.7 The Best Approximation of Functions Analytic in a Strip

5.8 Constructive Properties of Some Quasi-Analytic Classes of Functions

5.9 Estimates of the Best Approximations for Some Conjugate Classes of Functions

5.10 Direct Theorems in Arbitrary Banach Spaces

5.11 Various Problems and Theorems

VI. Converse Theorems. Constructive Characteristics of Some Classes of Functions

6.1 Differential Properties of Functions with a Given Sequence of Best Approximations

6.2 Constructive Characteristics of Some Classes of Continuous Functions Defined on a Finite Segment

6.3 Converse Theorems for Functions of Many Variables

6.4 Differential Properties and Best Approximations of Functions in Various Metrics. On Inclusion Theorems for Some Classes of Functions

6.5 The Analyticity of Functions and their Best Approximations

6.6 On the Constructive Characteristics of Quasi-Analytic Classes of Functions

6.7 Certain Converse Theorems for Conjugate Classes of Functions

6.8 Converse Theorems and Some Constructive Characteristics of Compact Sets in Banach Spaces

6.9 Various Problems and Theorems

VII. Further Theorems Connecting the Best Approximations of Functions with their Structural Properties

7.1 On the Exact Order of Decrease of Best Approximations

7.2 Asymptotic Properties of the Best Uniform Approximation of Some Simple Functions with Singularities

7.3 The Best Uniform Approximation of Functions which have a Discontinuous Derivative of Bounded Variation

7.4 On the Best Mean Approximation of the Simpler Functions with Singularities

7.5 On the Asymptotic Behavior of the Best Uniform Approximation of Some Analytic Functions

7.6 On a Constructive Property of Regularly Monotonie Functions

7.7 The Asymptotic Behavior of the Best Uniform Approximation by Polynomials of Some Transcendental Integral Functions

7.8 Various Problems and Theorems

VIII. Linear Processes of Approximation of Functions by Polynomials and Some Estimates Connected with Them

8.1 On the convergence of Linear Processes of Approximation of Functions by Polynomials

8.2 Lebesgue Constants and Functions

8.3 On Linear Methods of Approximation by Polynomials which Give the Best Order of Approximation

8.4 Approximation of Functions by Arithmetic Means of Partial Sums of Fourier Series

8.5 Further Estimates for Linear Methods of Approximation of Functions by Polynomials

8.6 The Approximative Properties of Orthogonal Expansions

8.7 Various Problems and Theorems

Some Results from the Theory of Functions and Functional Analysis

Bibliography

Index

## Product details

- No. of pages: 644
- Language: English
- Copyright: © Pergamon 1963
- Published: January 1, 1963
- Imprint: Pergamon
- eBook ISBN: 9781483184814

## About the Editors

### I. N. Sneddon

### S. Ulam

### M. Stark

## About the Author

### A. F. Timan

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