Characterization of Smoothness of Multivariate Refinable Functions and Convergence of Cascade Algorithms of Nonhomogeneous Refinement Equations期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:

Song Li

Affiliation:

(1) Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, 310027, P.R. China

Abstract:

This paper concerns multivariate homogeneous refinement equations of the form

$varphi (x) = sumlimits_{alpha in mathbb{Z}^S } {a(alpha )varphi (Mx - alpha ) + {text{ }}x{text{ }} in {text{ }}mathbb{R}} ^s ,$

and multivariate nonhomogeneous refinement equations of the form

$varphi (x) = sumlimits_{alpha in mathbb{Z}^S } {a(alpha )varphi (Mx - alpha ) + g(x){text{ }} in {text{ }}mathbb{R}} ^s ,$

where

₁,...,

_r)^T is the unknown, M is an s×s dilation matrix with m=|det thinsp

M|, g=(g₁,...,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( agr

) 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 phiv

₀ be an initial vector of functions in the Sobolev space (W₂^k(R^s))^r(k isin

N). The corresponding cascade algorithm is given by

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

Keywords:

refinement equation refinable function smoothness cascade algorithm Sobolev space Lipschitz space transition operator self-affine tile

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