The Degree of Approximation for Generalized Polynomials

The classical Mu'ntz theorem and the so-called Jackson-Muntz theorems concern uniform approximation on [0,1 ] by polynomials whose exponents are taken from an increasing sequence of positive real numbers A. Under mild restrictions on the exponents, the degree of approximation for A-poly nomials with real coefficients is compared with the corresponding degree of approximation when the coefficients are taken from the integers. Let C [0,1] be the space of aU continuous real valued functions defined on the interval [0, 1] and || • 11 the supremum norm on [0, 1] (||/|| = sup{|/(x)|: 0 <jc < 1}). It is weU known that the ordinary algebraic polynomials with integral coefficients, i.e. integral polynomials, are dense in the subspace C0[0, 1] = {/£C[0, l]:/(0) =/(l) = 0}. This seems to be due originally to Kakeya [10], but many other authors have also studied this or related problems: Pal [17], Okada [16], Bernstein [2], Fekete [3]. Finally, Hewitt and Zuckerman [9] obtained necessary and sufficient conditions. With every closed real interval of length less than 4, they associate a certain finite subset /. A continuous real function / on the interval is arbitrarily uniformly approximable by integral polynomials if and only if/is equal to some integral polynomial on the set /. In 1931, Kantorovic [11] proved that for any positive integer « and any function fE Co[0,1] there exists an integral polynomiá pn(x) = l^=0bkxk such that (1) \\f-pJ<2En(f) + 0(n-x) for«-»«, holds, where Received by the editors February 19, 1975 and, in revised form, November 4, 1975. AMS (MOS) subject classifications (1970). Primary 41A25, 41A10.