Degree of Approximation of Real Functions by Reciprocals of Real and Complex Polynomials

Abstract

Let $E_{on}^c (f;I)(E_{on}^r (f;I))$ denote the error in best uniform approximation of a real continuous function f on a closed interval I by reciprocals of polynomials of degree $ \leqq n$ with complex (real) coefficients. We investigate the rate at which $E_{on}^c (f;I)$ (or $E_{on}^r (f;I)$ provided $f \geqq 0$) can decrease. For example, we prove a Jackson type theorem and also present a class of functions for which reciprocal polynomial approximation is significantly better than polynomial approximation.