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Approximation by Dilated Averages and K-Functionals

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
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Abstract

For a positive finite measure $d\mu(\mathbf{u})$ on $\mathbb{R}^d$ normalized to satisfy $\int_{\mathbb{R}^d}d\mu(\mathbf{u})=1$, the dilated average of $f( \mathbf{x})$ is given by $A_tf(\mathbf{x})=\int_{\mathbb{R}^d}f(\mathbf{x}-t\mathbf{u})d\mu(\mathbf{u}).$ It will be shown that under some mild assumptions on $d\mu(\mathbf{u})$ one has the equivalence $\|A_tf-f\|_B\approx \inf \{ (\|f-g\|_B+t^2 \|P(D)g\|_B): P(D)g\in B\}\quad\text{for }t>0,$ where $\varphi(t)\approx \psi(t)$ means $c^{-1}\le\varphi(t)/\psi(t)\le c$, $B$ is a Banach space of functions for which translations are continuous isometries and $P(D)$ is an elliptic differential operator induced by $\mu$. Many applications are given, notable among which is the averaging operator with $d\mu(\mathbf{u})= \frac{1}{m(S)}\chi_S(\mathbf{u})d\mathbf{u}$, where $S$ is a bounded convex set in $\mathbb{R}^d$ with an interior point, $m(S)$ is the Lebesgue measure of $S$, and $\chi_S(\mathbf{u})$ 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
 MSC Classifications: 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)

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