Optimal Estimates of Approximation Errors for Strongly Positive Linear Operators on Convex Polytopes

Abstract

In the present investigation, we introduce and study linear operators, which underestimate every strongly convex function. We call them, for brevity, sp􀀀linear (approximation) operators. We will provide their sharp approximation errors. We show that the latter is bounded by the error approximation of the quadratic function. We use the centroidel Voronoi tessellations as a domain partition to construct best sp􀀀linear operators. Finally, numerical examples are presented to illustrate the proposed method.