Vector cascade algorithms with infinitely supported masks in weighted L 2-spaces

Abstract

In this paper, we shall study the solutions of functional equations of the form $$\Phi \sum\limits_{\alpha \in \mathbb{Z}^s } {a(\alpha )\Phi (M \cdot - \alpha ),}$$ where Φ = (ϕ 1, ...,ϕ r ) T is an r × 1 column vector of functions on the s-dimensional Euclidean space, $$a: = (a(\alpha ))_{\alpha \in \mathbb{Z}^s }$$ is an exponentially decaying sequence of r×r complex matrices called refinement mask and M is an s × s integer matrix such that limn → ∞ M −n = 0. We are interested in the question, for a mask a with exponential decay, if there exists a solution Φ to the functional equation with each function ϕ j , j = 1, ..., r, belonging to L 2(ℝ s ) and having exponential decay in some sense? Our approach will be to consider the convergence of vector cascade algorithms in weighted L 2 spaces. The vector cascade operator Q a,M associated with mask a and matrix M is defined by $$Q_{a,M} f: = \sum\limits_{\alpha \in \mathbb{Z}^s } {a(\alpha )f(M \cdot - \alpha ), f = \left( {f_1 , \ldots f_r } \right)^T \in \left( {L_{2,\mu } \left( {\mathbb{R}^s } \right)} \right)^r .}$$ The iterative scheme (Q f)n=1,2,... is called a vector cascade algorithm or a vector subdivision scheme. The purpose of this paper is to provide some conditions for the vector cascade algorithm to converge in (L 2 (ℝ s )) r , the weighted L 2 space. Inspired by some ideas in [Jia, R. Q., Li, S.: Refinable functions with exponential decay: An approach via cascade algorithms. J. Fourier Anal. Appl., 17, 1008–1034 (2011)], we prove that if the vector cascade algorithm associated with a and M converges in (L 2(ℝ s )) r , then its limit function belongs to (L 2, μ (ℝ s )) r for some µ > 0.