Abstract
This paper concerns multivariate homogeneous refinement equations of the form $$\varphi (x) = \sum\limits_{\alpha \in \mathbb{Z}^S } {a(\alpha )\varphi (Mx - \alpha ) + {\text{ }}x{\text{ }} \in {\text{ }}\mathbb{R}} ^s ,$$ and multivariate nonhomogeneous refinement equations of the form $$\varphi (x) = \sum\limits_{\alpha \in \mathbb{Z}^S } {a(\alpha )\varphi (Mx - \alpha ) + g(x){\text{ }} \in {\text{ }}\mathbb{R}} ^s ,$$ where ϕ=(ϕ1,...,ϕ r )T is the unknown, M is an s×s dilation matrix with m=|det M|, 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(α) 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 ϕ0 be an initial vector of functions in the Sobolev space (W 2 k (R s )) r (k∈N). The corresponding cascade algorithm is given by $$\varphi _n (x) = \sum\limits_{\alpha \in \mathbb{Z}^S } {a(\alpha )\varphi _n (Mx - \alpha ) + g(x),{\text{ }} \in {\text{ }}\mathbb{R}} ^s ,n = 1,2....$$
Published Version
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