On MRAs and Prewavelets Based on Elliptic Splines

We consider shift-invariant multiresolution spaces generated by q-elliptic splines in Rd, d≥ 2 , which are tempered distributions characterized by a complex-valued elliptic homogeneous polynomial q of degree m> d. To construct Riesz bases of L2(Rd) , a family of non-separable basic smooth functions are obtained by localizing a fundamental solution of the operator q(D), properly. The construction provides a generalization of some known elliptic scaling functions, the most famous being polyharmonic B-splines. Here, we prove that real-valued q leads to r-regular multiresolution analysis, with r= m- d- 1. In addition, we prove that there exist r-regular non-separable prewavelet systems associated with not necessarily regular multiresolution analyses. These prewavelets have m- 1 vanishing moments and the approximation order of the prewavelet decomposition can be established.