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 dμ(u) on R^{d}
normalized to satisfy ∫_{Rd}dμ(u)=1, the dilated average of
f( x) is given by
A_{t}f(x)=∫_{Rd}f(x−tu)dμ(u).
It will be shown that under some mild assumptions on dμ(u) one has
the equivalence
A_{t}f−f_{B} ≈ inf{ (f−g_{B}+t^{2} 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))χ_{S}(u)du, where S is a bounded convex set
in R^{d} with an interior point, m(S) is the Lebesgue measure of
S, and χ_{S}(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 Kfunctional.