On MRAs and prewavelets based on elliptic splines (Q2038373)

Firstly, \textit{C. Micchelli} et al. [Numer. Algorithms 1, No. 3, 331--351 (1991; Zbl 0765.65024)] have presented a multiresolution analysis (MRA) for elliptic \(d\)-variate splines and a dyadic prewavelet decomposition of \(L^2({\mathbb R}^d)\), \(d\ge 2\). In the paper under review, the authors present sampling functions with less restrictive assumptions on their Fourier transforms. Let \(q\) be a homogeneous, \(d\)-variate polynomial of degree \(m > d\) which vanishes only in the origin of \({\mathbb R}^d\). The authors use localized \(q\)-elliptic splines as scaling functions generating stationary MRAs of \(L^2({\mathbb R}^d)\). Then they construct \(r\)-regular prewavelet systems, where \(r=m - d - 1\). These prewavelets have \(m-1\) vanishing moments. For \(d=2\), some examples of scaling functions and prewavelets are presented.