Approximation by crystal-refinable functions

MOURE, MARÍA DEL CARMEN; QUINTERO, ALEJANDRO; MOLTER, URSULA

GEOMETRIAE DEDICATA

SPRINGER

Año: 2019

0046-5755

Let $Gamma$ be a crystal group in $R^d$. A function $arphi:R^dlongrightarrow C$ is said to be {em crystal-refinable} (or $Gamma-$refinable) if it is a linear combination of finitely many of the rescaled and translated functions $arphi(gamma^{-1}(ax))$, where the {em translations} $gamma$ are taken on a crystal group $Gamma$, and $a$ is an expansive dilation matrix such that $aGamma a^{-1}subsetGamma.$ A $Gamma-$refinable function $arphi: R^d ightarrow C$ satisfies a refinement equation $arphi(x)=sum_{gammainGamma}d_gamma arphi(gamma^{-1}(ax))$ with $d_gamma in C$. Let $mathcal S(arphi)$ be the linear span of ${arphi(gamma^{-1}(x)): gamma in Gamma}$ and $mathcal{S}^h={f(x/h):finmathcal{S(arphi)}}$. One important property of $mathcal S(arphi)$ is, how well it approximates functions in $L^2(R^d)$. This property is very closely related to the {em crystal-accuracy} of $mathcal S(arphi)$, which is the highest degree $p$ such that all multivariate polynomials $q(x)$ of ${m degree}(q)