A Survey of Degree of Approximation of Continuous Functions

Abstract

The purpose of this paper is to give a survey of results in the study of direct theorems in degree (or order) of “best approximation” $E_n (f)$, of a function $f(x)$, by trigonometric polynomials. We will normally require the function $f(x)$ to be $2\pi $-periodic and integrable in the Lebesgue sense. Further differentiability conditions may be imposed upon f, depending upon the theorems being discussed. Our main purpose is to study approximation by trigonometric series, Fourier series representation of a function and certain summability methods of the Fourier series representation. We will be basically interested in the quantitative aspects of the subject. For a more abstract treatment of the subject, see Shapiro [Topics in Approximation Theory, Springer-Verlag, New York, 1971]. A particularly elegant treatment of “degree of approximation”, up to about 1963, is to be found in Timan’s book [Theory of Approximation of Functions of a Real Variable, Pergamon/Macmillan New York, 1963]. We will not study or discuss the literature in the subject of “converse theorems”, see Meinardus [Approximation of Functions : Theory and Numerical Methods, Springer-Verlag, New York, 1967], Timan [Theory of Approximation of Functions of a Real Variable, Pergamon/Macmillan, New York, 1963]. Also, we will not directly study polynomial approximation to continuous functions except where the connection with trigonometric polynomials is obvious.