Approximation by Algebraic Convolution Integrals

Abstract

This chapter discusses the approximation by algebraic convolution integrals. An author raised the question whether it is possible to construct a singular convolution integral of a function f defined on [-1,1], which is an algebric polynimial of degree n and which approximates f uniformly on [-1,1] with order O (n -α ). The chapter illustrates that the integral should not be derived under some (triqonomtric) substitution from a corresponding integral which is a trigonometric polynomial of degree n. This problem has been solved in some form or other by many mathematicians. The natural extension of this problem, posed in the chapter, illustrates that whether an algebraic polynomial of degree n can be constructed which gives uniform approximation to the associate f on the whole [-1,1] with order O (n -l-α ) 0 < α ≤ 1.