The degree of coconvex polynomial approximation

Abstract

Abstract. Let f ∈ C[−1, 1] change its convexity finitely many times in the interval, say s times, at Ys: −1 <y1 <···<ys <1. We estimate the degree of approximation of f by polynomials of degree n, which change convexity exactly at the points Ys. We show that provided n is sufficiently large, depending on the location of the points Ys, the rate of approximation is estimated by the third Ditzian–Totik modulus of smoothness of f multiplied by a constant C(s), which depends only on s. 1. Introduction an