Approximation by crystal-refinable function

Let $\mathrm{\Gamma }$ be a crystal group in ${\mathbb{R}}^d$. A function $\varphi:\mathbb R^d\longrightarrow \mathbb C$issaidtobe\text{em}crystal-refinable(or$\Gamma-$refinable) if it is a linear combination of finitely many of the rescaled and translated functions $\phi ({\gamma }^{-1}(ax))$, where the {\em translations} $\gamma$ are taken on a crystal group $\mathrm{\Gamma }$, and $a$ is an expansive dilation matrix such that $a\mathrm{\Gamma }{a}^{-1}\subset \mathrm{\Gamma }\mathrm{.}$ A $\mathrm{\Gamma }-$refinable function $\phi :{\mathbb{R}}^d\to \mathbb{C}$ satisfies a refinement equation $\varphi(x)=\sum_{\gamma\in\Gamma}d_\gamma \varphi(\gamma^{-1}(ax))$with$d_\gamma \in \mathbb C$\mathrm{.}Let$\mathcal S(\varphi)$bethelinearspanof${\varphi(\gamma^{-1}(x)): \gamma \in \Gamma}$and$$\mathcal{S}$^h={f(x/h):f\in$\mathcal{S}(\mathcal{\phi })$}$. One important property of $\mathcal{S}(\phi )$ is, how well it approximates functions in $L^2({\mathbb{R}}^d)$. This property is very closely related to the {\em crystal-accuracy} of $\mathcal{S}(\phi )$, which is the highest degree $p$ such that all multivariate polynomials $q(x)$ of $\mathrm{d}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}(q)<p$ are exactly reproduced from elements in $\mathcal{S}(\phi )$. 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.