Approximation by Dilated Averages and K-Functionals

Abstract

AbstractFor a positive finite measure dμ(u ) on ℝd normalized to satisfy , the dilated average of f (x ) is given byIt will be shown that under some mild assumptions on dμ(u ) one has the equivalencewhere means , B is a Banach space of functions for which translations are continuous isometries and P(D) is an elliptic differential operator induced by μ. Many applications are given, notable among which is the averaging operator with where S is a bounded convex set in ℝd with an interior point, m(S) is the Lebesgue measure of S, and 𝒳S(u ) is the characteristic function of S. The rate of approximation by averages on the boundary of a convex set under more restrictive conditions is also shown to be equivalent to an appropriate K-functional.