Abstract
We consider shift-invariant multiresolution spaces generated by q-elliptic splines in \({\mathbb {R}^{d}},d\ge 2,\) which are tempered distributions characterized by a complex-valued elliptic homogeneous polynomial q of degree \(m>d\). To construct Riesz bases of \(L^{2} ({\mathbb {R}^{d}}),\) 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.
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The authors are grateful to the anonymous referee for her/his careful reading of the manuscript and for her/his helpful comments. This research has been accomplished within Rete ITaliana di Approssimazione (RITA). The last author is member of the INdAM Research group GNCS.
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Bacchelli, B., Rossini, M. On MRAs and Prewavelets Based on Elliptic Splines. Results Math 76, 40 (2021). https://doi.org/10.1007/s00025-021-01348-y
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DOI: https://doi.org/10.1007/s00025-021-01348-y