Lower bounds for the degree of approximation

Abstract

is the optimal degree of approximation of 2W. In this paper we shall give simple methods which permit to find the order of magnitude of Dn(W) for several important classes ?1: for some classes of analytic functions (?6); for the unit ball AP+a of the space CP+a of functions with continuous derivatives of order p, which satisfy a Lipschitz condition with exponent a, 0 <a< 1 (?5). We also consider approximation of continuous functions (??2, 3) and give results about condensation of singularities (Theorems 2, 7). The main content of this paper consists of results which show that standard means (trigonometric approximation, series of orthogonal polynomials) give the best possible approximation, at least up to a bounded factor. Since estimates of Dn(21) from above follow from classical results [1; 5], we are interested in estimating Dn(2I) from below. Clearly, results of this type are the better, the smaller the norm used in the definition (1). This is why most of our theorems are for the LI-norm. Kolmogorov [2] (see also [6]) discussed D-n(2) in the L2-norm, and gave an asymptotic formula for it when 2 is the class of functionsf on an interval with f(P) bounded in the L2-norm. Originally the present paper was written to improve (for linear approximation) the