Approximation by crystal-refinable functions

Abstract

Let Γ be a crystal group in Rd. A function φ:Rd⟶C is said to be 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 translationsγ are taken on a crystal group Γ, and a is an expansive dilation matrix such that aΓa−1⊂Γ. A Γ-refinable function φ:Rd→C satisfies a refinement equation φ(x)=∑γ∈Γdγφ(γ−1(ax)) with dγ∈C. Let S(φ) be the linear span of {φ(γ−1(x)):γ∈Γ} and Sh={f(x/h):f∈S(φ)}. One important property of S(φ) is, how well it approximates functions in L2(Rd). This property is very closely related to the crystal-accuracy of S(φ), which is the highest degree p such that all multivariate polynomials q(x) of degree(q)