Matrix Reenement Equations: Existence and Uniqueness

Matrix reenement equations are functional equations of the form f(x) = P N k=0 c k f(2x ? k), where the coeecients c k are matrices and f is a vector-valued function. Reenement equations play key roles in wavelet theory and approximation theory. Existence and uniqueness properties of scalar reenement equations (where the coeecients c k are scalars) are known. This paper considers analogous questions for matrix reenement equations. Conditions for existence and uniqueness of compactly supported distributional solutions are given in terms of the convergence properties of an innnite product of the matrix = 1 2 P c k with itself. Fundamental diierences between solutions of matrix equations and scalar reenement equations are examined. In particular, it is shown that \constrained" solutions of the matrix reenement equation can exist even when the innnite product diverges. The existence of constrained solutions is related to the eigenvalue structure of ; solutions are obtained from the convergence of on its 1-eigenspace.