Vector subdivision schemes and multiple wavelets

We consider solutions of a system of refinement equations written in the form φ = α∈Z a(α)φ(2 · −α), where the vector of functions φ = (φ1, . . . , φr)T is in (Lp(R))r and a is a finitely supported sequence of r × r matrices called the refinement mask. Associated with the mask a is a linear operator Qa defined on (Lp(R))r by Qaf := α∈Z a(α)f(2 · −α). This paper is concerned with the convergence of the subdivision scheme associated with a, i.e., the convergence of the sequence (Qn a f)n=1,2,... in the Lp-norm. Our main result characterizes the convergence of a subdivision scheme associated with the mask a in terms of the joint spectral radius of two finite matrices derived from the mask. Along the way, properties of the joint spectral radius and its relation to the subdivision scheme are discussed. In particular, the L2-convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain inva...

• Type: Journal
• Year: 1998
• Where: MOC
• Authors: Rong-Qing Jia, Sherman D. Riemenschneider, Ding-Xuan Zhou
• Keywords: Joint Spectral Radius, MOC 1998, Spectral Radius, Subdivision Scheme