Vector refinement equations with infinitely supported masks

Autores: Song Li, Jianbin Yang
Localización: Journal of approximation theory, ISSN 0021-9045, Vol. 148, Nº 2, 2007, págs. 158-176
Idioma: inglés
DOI: 10.1016/j.jat.2007.03.004
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Resumen
- In this paper we investigate the L2-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 where the vector of functions f=(f1,¿,fr)T is in is an exponentially decaying sequence of r×r matrices called refinement mask and M is an s×s integer matrix such that limn?8M-n=0. Associated with the mask a and dilation matrix M is a linear operator Qa on given by The iterative scheme 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 to converge in L2-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.