Comparing the degrees of unconstrained and shape preserving approximation by polynomials

Abstract

Let f∈C[−1,1] and denote by En(f) its degree of approximation by algebraic polynomials of degree <n. Assume that f changes its monotonicity, respectively, its convexity finitely many times, say s≥2 times, in (−1,1) and we know that for q=1 or q=2 and some 1<α≤2, such that qα≠4, we have En(f)≤n−qα,n≥s+q+1. The purpose of this paper is to prove that the degree of comonotone, respectively, coconvex approximation, of f, by algebraic polynomials of degree <n, n≥N, is also ≤c(α,s)n−qα, where the constant N depends only on the location of the extrema, respectively, inflection points in (−1,1) and on α.This answers, affirmatively, questions left open by the authors in papers with Kopotun (in Ukrainian Math. J.) and with Vlasiuk (see the list of references).