Positive results and counterexamples in comonotone approximation II

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, nEn(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 nEn(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.