Weighted Moduli of Smoothness and Sign-Preserving Approximation

Abstract

We consider a continuous function that changes its sign on an interval finitely many times and pose the problem of the approximation of this function by a polynomial that inherits its sign. For this approximation, we obtain (in the case where this is possible) Jackson-type estimates containing modified weighted moduli of smoothness of the Ditzian-Totik type. In some cases, constants in these estimates depend substantially on the location of points where the function changes its sign. We give examples of functions for which these constants are unimprovable. We also prove theorems that are analogous, in a certain sense, to inverse theorems of approximation without restrictions.