Positive results and counterexamples in comonotone approximation II

Abstract

Let En(f) denote the degree of approximation of f∈C[−1,1], by algebraic polynomials of degree <n, and assume that we know that for some α>0 and N≥2, nαEn(f)≤1,n≥N. Suppose that f changes its monotonicity s≥1 times in [−1,1]. We are interested in what may be said about its degree of approximation by polynomials of degree <n that are comonotone with f. In particular, if f changes its monotonicity at Ys≔{y1,…,ys} and the degree of comonotone approximation is denoted by En(f,Ys), we investigate when can one say that nαEn(f,Ys)≤c(α,s,N),n≥N∗, for some N∗. Clearly, N∗, if it exists at all (we prove it always does), depends on α, s and N. However, it turns out that for certain values of α, s and N, N∗ depends also on Ys and in some cases even on f itself. The results extend previous results in the case N=1.