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Computational approach to solvability of refinement equations

Authors: Victor D. Didenko and Bernd Silbermann
Journal: Math. Comp. 78 (2009), 1435-1466
MSC (2000): Primary 65T60; Secondary 42C40, 39B32
Published electronically: March 10, 2009
MathSciNet review: 2501057
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Abstract: The solvability and Fredholm properties of refinement equations in spaces of square-integrable functions are studied. Necessary and jointly necessary and sufficient conditions for the solvability of homogeneous and non-homogeneous refinement equations are established. It is shown that in the space $L_2(\mathbb {R})$ the kernel space of any homogeneous equation with a non-trivial solution is infinite dimensional. Moreover, the solvability problem is reduced to the study of singular values of certain matrix sequences. These sequences arise from Galerkin approximations of auxiliary linear operators. The corresponding constructions use only the coefficients of refinement equations that generate multiresolution analysis, and the coefficients of the refinement equation studied. For the equations with polynomial symbols the most complete results are obtained if the corresponding operator is considered on an appropriate subspace of the space $L_2(\mathbb {R})$.

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Additional Information

Victor D. Didenko
Affiliation: Department of Mathematics, Universiti Brunei Darussalam, Bandar Seri Begawan, BE1410, Brunei

Bernd Silbermann
Affiliation: Faculty of Mathematics, University of Technology Chemnitz, 09107 Chemnitz, Germany

Keywords: Refinement equation, solvability, singular values, splitting
Received by editor(s): November 16, 2007
Received by editor(s) in revised form: August 18, 2008
Published electronically: March 10, 2009
Additional Notes: The first author was supported in part by Universiti Brunei Darussalam Grants PNC2/2/RG/1(66) and PNC2/2/RG/1(72)
Article copyright: © Copyright 2009 American Mathematical Society