This paper is devoted to the study of L2-solutions of the operator equations ? - B_MFαF~(-1) ? = g, where α is the operator of multiplication by a matrix α ∈L_8~(mxm (R~s), m,s ∈ N, F denotes the Fourier transform, and B_M is the dilation operator B_M ?(x):= ? (Mx), x ∈ R~s, generated by a non-singular matrix M ∈ R~(mxm). 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.
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