Theory of Approximation of Functions of A Real Variable

✍ Scribed by A. F. Timan

• Publisher: Pergamon Press
• Year: 1963
• Tongue: English
• Leaves: 638
• Category: Library

✦ Synopsis

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 Weierstrasss 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

Content:
OTHER TITLES IN THE SERIES ON PURE AND APPLIED MATHEMATICS, Page ii
Front Matter, Page iii
Copyright, Page iv
EDITORIAL PREFACE, Page ix
PREFATORY NOTE, Page ix
FOREWORD, Pages xi-xii
CHAPTER I - WEIERSTRASS'S THEOREM, Pages 1-25
CHAPTER II - THE BEST APPROXIMATION, Pages 26-92
CHAPTER III - SOME COMPACT CLASSES OF FUNCTIONS AND THEIR STRUCTURAL CHARACTERISTICS, Pages 93-169
CHAPTER IV - SOME PROPERTIES OF ALGEBRAIC POLYNOMIALS AND TRANSCENDENTAL INTEGRAL FUNCTIONS OF EXPONENTIAL TYPE, Pages 170-253
CHAPTER V - DIRECT THEOREMS OF THE CONSTRUCTIVE THEORY OF FUNCTIONS, Pages 254-330
CHAPTER VI - CONVERSE THEOREMS. CONSTRUCTIVE CHARACTERISTICS OF SOME CLASSES OF FUNCTIONS, Pages 331-401
CHAPTER VII - FURTHER THEOREMS CONNECTING THE BEST APPROXIMATIONS OF FUNCTIONS WITH THEIR STRUCTURAL PROPERTIES, Pages 402-464
CHAPTER VIII - LINEAR PROCESSES OF APPROXIMATION OF FUNCTIONS BY POLYNOMIALS AND SOME ESTIMATES CONNECTED WITH THEM, Pages 465-586
SOME RESULTS FROM THE THEORY OF FUNCTIONS AND FUNCTIONAL ANALYSIS, Pages 587-601
BIBLIOGRAPHY OF MEMOIRS AND BOOKS REFERRED TO IN THE TEXT, Pages 602-620
INDEX, Pages 621-631