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Journal of Operator Theory

Volume 67, Issue 2, Spring 2012  pp. 301-316.

Properties of L2-solutions of refinement equations

Authors Victor D. Didenko
Author institution: Faculty of Science, Universiti Brunei Darussalam, Bandar Seri Begawan, BE 1410, Brunei

Summary:  This paper is devoted to the study of L2-solutions of the operator equations fBMFaF1f=g, where a is the operator of multiplication by a matrix aLm×m(\sRs), m,s\sN, F denotes the Fourier transform, and BM is the dilation operator BMf(x):=f(Mx), x\sRs, generated by a non-singular matrix M\sRm×m. This class of equations contains discrete and continuous refinement equations widely used in wavelet analysis, signal processing, computer graphics and other fields of mathematics and in applications. It is shown that the set of nontrivial solutions of the homogeneous equation is either empty or contains a subset isomorphic to a space L(VM), where VM is a Lebesgue measurable set with positive Lebesgue measure. It follows that the operator IBMFaF1 is Fredholm if and only if it is invertible. Moreover, if the dilation M satisfies some mild conditions, then ker(IBMFaF1)¯\im(IBMFaF1).


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