Vector subdivision schemes in (L p (ℝ s )) r (1⩽p⩽∞) spaces

Abstract

The purpose of this paper is to investigate the refinement equations of the form $$\varphi (x) = \sum\limits_{\alpha \in \mathbb{Z}^s } {a(\alpha )\varphi (Mx - \alpha ), x \in \mathbb{R}^s ,} $$ where the vector of functions ϕ=(ϕ1..., ϕr)T is in (Lp(ℝs))r, 1⩽p⩽∞, a(α), α∈ℤsis a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that lim→∞ M-n = 0. In order to solve the refinement equation mentioned above, we start with a vector of compactly supported functions φ0∈(Lp(ℝs))r and use the iteration schemes fn:=Qanφ0, n=1,2,..., where Qn is the linear operator defined on (Lp(ℝs))r given by $$Q_a \varphi : = \sum\limits_{\alpha \in \mathbb{Z}^s } {a(\alpha )\varphi (M \cdot - \alpha ), } \varphi \in (L_p (\mathbb{R}^s ))^r .$$ This iteration scheme is called a subdivision scheme or cascade algorithm. In this paper, we characterize the Lp-convergence of subdivision schemes in terms of the p-norm joint spectral radius of a finite collection of some linear operators determined by the sequence a and the set B restricted to a certain invariant subspace, where the set B is a complete set of representatives of the distinct cosets of the quotient group ℤs/Mℤs containing 0.