On Existence and Weak Stability of Matrix Refinable Functions

Abstract.

We consider the existence of distributional (or L ₂ ) solutions of the matrix refinement equation

$\hat\Phi = \mbox{ {\bf P}}(\cdot/2)\hat\Phi(\cdot/2),$

where P is an r×r matrix with trigonometric polynomial entries.

One of the main results of this paper is that the above matrix refinement equation has a compactly supported distributional solution if and only if the matrix P (0) has an eigenvalue of the form 2 ⁿ , $n \in {\Bbb Z}_+$ . A characterization of the existence of L ₂ -solutions of the above matrix refinement equation in terms of the mask is also given.

A concept of L ₂ -weak stability of a (finite) sequence of function vectors is introduced. In the case when the function vectors are solutions of a matrix refinement equation, we characterize this weak stability in terms of the mask.