UNIFORM ESTIMATES OF MONOTONE AND CONVEX APPROXIMATION OF SMOOTH FUNCTIONS

We obtain uniform estimates for monotone and convex approximation of functions by algebraic polynomials in terms of the weighted Ditzian-Totik moduli of smoothness [GRAPHICS] where phi(x):=root 1-x(2), for r greater than or equal to 3 and r greater than or equal to 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 omega(phi)(k)(f((r)), n(-1))(phi r,infinity) 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 omega(phi)(k)(f((r)), n(-1))(phi r,infinity) is not correct. (C) 1995 Academic Press, Inc.