# Journal of Operator Theory

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

Properties of $L_2$-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 $L_2$-solutions of the operator equations $f-B_M \mathfrak{F} a \mathfrak{F}^{-1} f= g$, where $a$ is the operator of multiplication by a matrix $a\in L_\infty^{m \times m}(\sR^s)$, $m,s\in\sN$, $\mathfrak{F}$ denotes the Fourier transform, and $B_M$ is the dilation operator $B_Mf(x):=f(Mx)$, $x\in \sR^s$, generated by a non-singular matrix $M\in \sR^{m\times m}$. 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 $L_\infty(\mathbb{V}_M)$, where $\mathbb{V}_M$ is a Lebesgue measurable set with positive Lebesgue measure. It follows that the operator $I-B_M \mathfrak{F} a \mathfrak{F}^{-1}$ is Fredholm if and only if it is invertible. Moreover, if the dilation $M$ satisfies some mild conditions, then $\ker \,(I-B_M \mathfrak{F} a \mathfrak{F}^{-1})\subset \overline{\im(I-B_M \mathfrak{F} a \mathfrak{F}^{-1})}$.

**Keywords:**refinement equation, $L_2$-solutions, Fredholm properties

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