Vector refinement equations with infinitely supported masks

• Authors: Song Li;Jianbin Yang
• Venue: Journal of Approximation Theory
• Year: 2007

In this paper we investigate the L"2-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 form@f(x)=@?@a@?Z^sa(@a)@f(Mx-@a),x@?R^s,where the vector of functions @f=(@f"1,...,@f"r)^T is in (L"2(R^s))^r,a@?(a(@a))"@a"@?"Z"^"s is an exponentially decaying sequence of rxr matrices called refinement mask and M is an sxs integer matrix such that lim"n"-"~M^-^n=0. Associated with the mask a and dilation matrix M is a linear operator Q"a on (L"2(R^s))^r given byQ"af(x)@?@?@a@?Z^sa(@a)f(Mx-@a),x@?R^s,f=(f"1,...,f"r)^T@?(L"2(R^s))^r.The iterative scheme (Q"a^nf)"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 (Q"a^nf)"n"="1","2","... to converge in L"2-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.