A Note on Matrix Refinement Equations

Refinement equations involving matrix masks are receiving much attention these days. They can play a central role in the study of refinable finitely generated shift-invariant spaces, multiresolutions generated by more than one function, multiwavelets, splines with multiple knots, and matrix subdivision schemes---including Hermite-type subdivision schemes. Several recent papers on this subject begin with an assumption on the eigenstructure of the mask, pointing out that this assumption is heuristically "natural" or "preferred." In this note, we prove that stability of the shifts of the refinable function requires this assumption.