Approximation by crystal-refinable functions

Abstract

Let $$\varGamma $$ be a crystal group in $$\mathbb {R}^d$$ . A function $$\varphi :\mathbb {R}^d\longrightarrow \mathbb {C}$$ is said to be crystal-refinable (or $$\varGamma $$ -refinable) if it is a linear combination of finitely many of the rescaled and translated functions $$\varphi (\gamma ^{-1}(ax))$$ , where the translations $$\gamma $$ are taken on a crystal group $$\varGamma $$ , and a is an expansive dilation matrix such that $$a\varGamma a^{-1}\subset \varGamma .$$ A $$\varGamma $$ -refinable function $$\varphi : \mathbb {R}^d \rightarrow \mathbb {C}$$ satisfies a refinement equation $$\varphi (x)=\sum _{\gamma \in \varGamma }d_\gamma \varphi (\gamma ^{-1}(ax))$$ with $$d_\gamma \in \mathbb {C}$$ . Let $$\mathcal S(\varphi )$$ be the linear span of $$\{\varphi (\gamma ^{-1}(x)): \gamma \in \varGamma \}$$ 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 crystal-accuracy of $$\mathcal S(\varphi )$$ , which is the highest degree p such that all multivariate polynomials q(x) of $$\mathrm{degree}(q)<p$$ 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, which simplifies the classical conditions.