Comparing the degrees of unconstrained and shape preserving approximation by polynomials

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 and Vlasiuk (see the list of references).