2017/01/28 by Ursula Molter, Molter, Ursula, María del Carmen Moure +3
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Classical Analysis and ODEs (math.CA) #Digital Filter Design and Implementation #FOS: Mathematics #Mathematical Analysis and Transform Methods
paper · pdf · doi:10.48550/arxiv.1701.08226
openalex publication_date 2017/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Γ be a crystal group in \mathbb Rd. A function φ:\mathbb Rd\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⊂Γ. A Γ-refinable function φ: \mathbb Rd → \mathbb C satisfies a refinement equation φ(x)=∑γ∈Γdγφ(γ-1(ax)) with dγ∈ \mathbb C. Let \mathcal S(φ) be the linear span of \φ(γ-1(x)): γ∈ Γ\ and Sh=\f(x/h):f\inS(φ)\. One important property of \mathcal S(φ) is, how well it approximates functions in L2(\mathbb Rd). 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)