Summary: We consider the existence of distributional (or \(L_2\)) solutions of the matrix refinement equation \[ \widehat\Phi= {\mathbf P}(\cdot/2)\widehat\Phi(\cdot/2), \] where \({\mathbf P}\) is an \(r\times r\) 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 \({\mathbf P}(0)\) has an eigenvalue of the form \(2^n\), \(n\in\mathbb{Z}_+\). 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.