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