Introduction To The Theory Of Weighted Polynomial Approximation

The book starts with an introduction to the general theory of orthogonal polynomials, proving elementary facts about such polynomials in general. Chapter 2 deals with standard results in the theory of trigonometric approximation, such as the Jackson and Bernstein-Stechkin theorems in a novel way, which is more suitable to understand the techniques used in weighted approximation. In Chapter 3, some applications of the potential theory is discussed. Chapter 4 deals with the processes of approximation: expansions in orthogonal series and interpolation. The theme here is to use the classical techniques when they are not really applicable. The results which are used throughout the book without proof are listed in the appendix. The appendix also includes a discussion of the Bernstein approximation problem, although a thorough development of this subject would take us too far away from the general theme of the book. The chapters can be read independently of each other. Any notation and results introduced in one chapter and used in another are completely reproduced as needed. The book may be used as an introduction to the general ideas.