Vector refinement equations with infinitely supported masks

In this paper we investigate the L2L2-solutions of vector refinement equations with exponentially decaying masks and a general dilation matrix. A vector refinement equation with a general dilation matrix and exponentially decaying masks is of the formφ(x)=∑α∈Zsa(α)φ(Mx-α),x∈Rs,where the vector of functions φ=(φ1,…,φr)Tφ=(φ1,…,φr)T is in (L2(Rs))r,(L2(Rs))r,a≕(a(α))α∈Zsa≕(a(α))α∈Zs is an exponentially decaying sequence of r×rr×r matrices called refinement mask and M   is an s×ss×s integer matrix such that limn→∞M-n=0.limn→∞M-n=0. Associated with the mask a and dilation matrix M   is a linear operator QaQa on (L2(Rs))r(L2(Rs))r given byQaf(x)≔∑α∈Zsa(α)f(Mx-α),x∈Rs,f=(f1,…,fr)T∈(L2(Rs))r.The iterative scheme (Qanf)n=1,2,…, is called vector subdivision scheme or vector cascade algorithm. The purpose of this paper is to provide a necessary and sufficient condition to guarantee the sequence (Qanf)n=1,2,… to converge in L2L2-norm. As an application, we also characterize biorthogonal multiple refinable functions, which extends some main results in [B. Han, R.Q. Jia, Characterization of Riesz bases of wavelets generated from multiresolution analysis, Appl. Comput. Harmon. Anal., to appear] and [R.Q. Jia, Convergence of vector subdivision schemes and construction of biorthogonal multiple wavelets, Advances in Wavelet (Hong Kong, 1997), Springer, Singapore, 1998, pp. 199–227] to the general setting.