Simultaneous Approximation by Generalized n-th Order Blending Operators

Abstract

The present note deals with various aspects concerning n-th order blending operators as introduced by Delvos and Posdorf in [2]. These operators constitute a generalization of the well-known technique of discrete blending interpolation as discussed in [6], among others. The motivation of our contribution is two-fold. First, we stress the fact that applications of this technique should not be restricted to those cases in which the univariate building blocks are projectors. To this end, we introduce so-called generalized n-th order blending operators which share many relevant properties of the classical n-th order blending operators, including their potential for data reduction. Secondly, we shall show that quite elegant upper bounds for the approximation error can be given using mixed moduli of smoothness of appropriate orders and some previous results of the present author. Two applications are added in order to illustrate our more general approach.