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