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

In this paper, we shall study the solutions of functional equations of the form $$Phi sumlimits_{alpha in mathbb{Z}^s } {a(alpha )Phi (M cdot - alpha ),}$$ where Φ = (? ₁, ...,? _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 lim _{n → ∞} 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 ? _j , j = 1, ..., r, belonging to L ₂(? ^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 $$Q_{a,M} f: = sumlimits_{alpha in mathbb{Z}^s } {a(alpha )f(M cdot - alpha ), f = left( {f_1 , ldots f_r } right)^T in left( {L_{2,mu } left( {mathbb{R}^s } right)} right)^r .}$$ The iterative scheme (Q _a,M ⁿ 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 (? ^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(? ^s )) ^r , then its limit function belongs to (L _{2, μ} (? ^s )) ^r for some µ > 0.