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

摘要

This paper concerns multivariate homogeneous refinement equations of the form φ(x) = ∑_(α∈Z~s)a(α)φ(Mx-α), x ∈ R~s and multivariate nonhomogeneous refinement equations of the form φ(x) = ∑_(α∈Z~s)a(α)φ(Mx-α)+g(x), x ∈ R~s, where φ = (φ_1,…,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 φ_0 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 φ_n(x) = ∑_(α∈Z~s)a(α)φ_(n-1)(Mx-α)+g(x), x ∈ R~s, n = 1,2,…. We also provide necessary and sufficient conditions for the strong convergence of the cascade algorithm in the Sobolev space (W_2~k(R~s))~r(k ∈ N)) for the case in which M is isotropic.