Abstract view
Approximation by Dilated Averages and KFunctionals


Published:20100520
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
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 \[ \A_tff\_B\approx \inf \{
(\fg\_B+t^2 \P(D)g\_B): P(D)g\in B\}\quad\text{for }t>0, \]
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.