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

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

$$\varphi _n (x) = \sum\limits_{\alpha \in \mathbb{Z}^S } {a(\alpha )\varphi _n (Mx - \alpha ) + g(x),{\text{ }} \in {\text{ }}\mathbb{R}} ^s ,n = 1,2....$$