On existence and weak stability of matrix refinable functions

We consider the existence of distributional (or L-2) solutions of the matrix refinement equation <(Phi)over cap> = P(./2)<(Phi)over cap>(./2), where P is an r x 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 is an element of 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.