Positive results and counterexamples in comonotone approximation II

Let _{E n} (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 _{Y s} {colon equals} {_y1, . . . , _{y s}} and the degree of comonotone approximation is denoted by _{E n} (f, _{Y s}) , 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 _{Y s} and in some cases even on f itself. The results extend previous results in the case N = 1.