Uniform Estimates of Monotone and Convex Approximation of Smooth Functions Original Research Article

We obtain uniform estimates for monotone and convex approximation of functions by algebraic polynomials in terms of the weighted Ditzian-Totik moduli of smoothness [formula] where φ(x)≔ [formula], for r ≥ 3 and r ≥ 5 in monotone and convex cases, respectively. Together with known results in the positive and negative directions for the other r this complements the investigation of the rate of shape preserving approximation in terms of ωkφ(f(r), n−1)φr, ∞ in the sense of the orders of these moduli. It is also shown that some extra conditions on the smoothness of f allow direct results in the cases for which the general estimate in terms of ωkφ(f(r), n−1)φr, ∞ is not correct.