Approximation by crystal-refinable function

Abstract

Let $\Gamma$ be a crystal group in $\mathbb R^d$. A function $\varphi:\mathbb R^d\longrightarrow \mathbb C$ 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 $\varphi(\gamma^{-1}(ax))$, where the {\em translations} $\gamma$ are taken on a crystal group $\Gamma$, and $a$ is an expansive dilation matrix such that $a\Gamma a^{-1}\subset\Gamma.$ A $\Gamma-$refinable function $\varphi: \mathbb R^d \rightarrow \mathbb C$ satisfies a refinement equation $\varphi(x)=\sum_{\gamma\in\Gamma}d_\gamma \varphi(\gamma^{-1}(ax))$ with $d_\gamma \in \mathbb C$. Let $\mathcal S(\varphi)$ be the linear span of $\{\varphi(\gamma^{-1}(x)): \gamma \in \Gamma\}$ and $\mathcal{S}^h=\{f(x/h):f\in\mathcal{S(\varphi)}\}$. One important property of $\mathcal S(\varphi)$ is, how well it approximates functions in $L^2(\mathbb R^d)$. This property is very closely related to the {\em crystal-accuracy} of $\mathcal S(\varphi)$, which is the highest degree $p$ such that all multivariate polynomials $q(x)$ of ${\rm degree}(q)<p$ are exactly reproduced from elements in $\mathcal S(\varphi)$. In this paper, we determine the accuracy $p$ from the coefficients $d_\gamma$. Moreover, we obtain from our conditions, a characterization of accuracy for a particular lattice refinable vector function $F$, which simplifies the classical conditions.