Characterization of Smoothness of Multivariate Refinable Functions and Convergence of Cascade Algorithms of Nonhomogeneous Refinement Equations

where ' = ('1; : : : ; 'r) T is the unknown, M is an s£s dilation matrix with m = jdetM j, g = (g1; : : : ; gr) is a given compactly supported vector-valued functions on IR and a is a ̄nitely supported re ̄nement mask such that each a(®) is an r£ r (complex) matrix. In this paper, we characterize the optimal smoothness of a multiple re ̄nable function associated with homogeneous re ̄nement equations in terms of the spectral radius of the corresponding transition operator restricated to a suitable ̄nite dimensional invariate subspace. Nonhomogeneous re ̄nement equations occur naturally in multiwavelet constructions. Let '0 be an initial vector of functions in the Sobolev space (W k 2 (IR )) (k 2 IN). The corresponding cascade algorithm is given by