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.

Bacchelli, B., Rossini, M. (2021). On MRAs and Prewavelets Based on Elliptic Splines. RESULTS IN MATHEMATICS, 76(1), 1-28 [10.1007/s00025-021-01348-y].

On MRAs and Prewavelets Based on Elliptic Splines

Bacchelli B.;Rossini M.
2021

Abstract

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.
Articolo in rivista - Articolo scientifico
non-separable wavelets; q-elliptic splines; regular multiresolution analysis; regular prewavelets;
English
1
28
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Bacchelli, B., Rossini, M. (2021). On MRAs and Prewavelets Based on Elliptic Splines. RESULTS IN MATHEMATICS, 76(1), 1-28 [10.1007/s00025-021-01348-y].
Bacchelli, B; Rossini, M
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/10281/358351
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