Lp Solutions of Refinement Equations

In the recent characterizations of the L_p solution of the refinement equation in terms of the "p-norm joint spectral radius," there are problems in choosing the initial function for iteration [3, 23], or in addition, requiring stability of the refinable function [13,17]. In this article we overcome these difficulties and give a more complete characterization of this nature. The criterion is constructive and can be implemented. It can be used to describe the regularity of the solution without assuming stability. This has significant advantages over the previous work. The corresponding results for vector refinement equations are also discussed.