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

This paper concerns multivariate homogeneous refinement equations of the form and multivariate nonhomogeneous refinement equations of the form where =(₁,..., _r )^T is the unknown, M is an s×s dilation matrix with m=|detM|, g=(g ₁,...,g _r )^T is a given compactly supported vector-valued function on R ^s , and a is a finitely supported refinement mask such that each a() is an r×r (complex) matrix. In this paper, we characterize the optimal smoothness of a multiple refinable function associated with homogeneous refinement equations in terms of the spectral radius of the corresponding transition operator restricted to a suitable finite-dimensional invariant subspace when M is an isotropic dilation matrix. Nonhomogeneous refinement equations naturally occur in multi-wavelets constructions. Let ₀ be an initial vector of functions in the Sobolev space (W ₂ ^k (R ^s )) ^r (kN). The corresponding cascade algorithm is given by