Multiple Refinable Distributions: Existence and Factorization

摘要

We consider the existence of compactly suported distribution solutions PHI : chemical bounds (PHI_1, -, PHI_r)~T to matrix refinement equations of the form PHI chemical bounds sum_(alpha the point delong to (is member of) the set ZZ~s) a( alpha ) PHI (2 centre dor - alpha) where a is a finitely supported sequence of r x r matrices called the refinement mask. Such multiple refinable distributions occur naturally in the study of multiple wavelets. A neccessary condition for the existence of PHI with PHI (0) not = 0 is that the matrix sum a (alpha)/2 has an eigenvalue 1. However, this condition is not sufficient, which is different from the scalar case. In this paper we provide a necessary and sufficient condition for the existence of a solution PHI subject to PHI (0) not = 0. Such a solution can always be obtained through some infinite matrix product. Our conditions is totally based on the refinement mask. A general factorization of the refinement mask is presented in the univariate case, which gives a way of constructing multiple refinable distributions.