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 ....Zsa(.).(M...), where . = (. 1, ...,. r )T is an r . 1 column vector of functions on the s-dimensional Euclidean space, a:=(a(.))..Zs 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 . 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 Qa,Mf:=...Zsa(.)f(M...),f=(f1,.fr)T.(L2,.(Rs))r. The iterative scheme (Q na,M 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, . (.s))r for some µ > 0.