CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
  location:  PublicationsjournalsCJM
Abstract view

Approximation by Dilated Averages and K-Functionals

Open Access article
  Published:2010-05-20
 Printed: Aug 2010
  • Z. Ditzian,
    Department of Math. and Stat. Sciences, University of Alberta, Edmonton, AB
  • A. Prymak,
    Department of Mathematics, University of Manitoba, Winnipeg, MB
Format:   HTML   LaTeX   MathJax  

Abstract

For a positive finite measure dμ(u) on Rd normalized to satisfy ∫Rddμ(u)=1, the dilated average of f( x) is given by Atf(x)=∫Rdf(x−tu)dμ(u). It will be shown that under some mild assumptions on dμ(u) one has the equivalence ||Atf−f||B ≈ inf{ (||f−g||B+t2 ||P(D)g||B): P(D)g ∈ B} for t > 0, where φ(t) ≈ ψ(t) means c−1 ≤ φ(t)/ψ(t) ≤ c, 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 dμ(u) = (1/m(S))χS(u)du, where S is a bounded convex set in Rd with an interior point, m(S) is the Lebesgue measure of S, and χS(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.
Keywords: rate of approximation, K-functionals, strong converse inequality rate of approximation, K-functionals, strong converse inequality
MSC Classifications: 41A27, 41A35, 41A63 show english descriptions Inverse theorems
Approximation by operators (in particular, by integral operators)
Multidimensional problems (should also be assigned at least one other classification number in this section)
41A27 - Inverse theorems
41A35 - Approximation by operators (in particular, by integral operators)
41A63 - Multidimensional problems (should also be assigned at least one other classification number in this section)
 

© Canadian Mathematical Society, 2018 : https://cms.math.ca/