Uniform Estimates of Monotone and Convex Approximation of Smooth Functions

AbstractWe 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