Subdivisions with infinitely supported mask

• Authors: Song Li;Yali Pan
• Venue: Journal of Computational and Applied Mathematics
• Year: 2008

In this paper we investigate the convergence of subdivision schemes associated with masks being polynomially decay sequences. Two-scale vector refinement equations are the form@f(x)=@?@a@?Za(@a)@f(2x-@a),x@?R,where the vector of functions @f=(@f"1,...,@f"r)^T is in (L"2(R))^r and a@?(a(@a))"@a"@?"Z is polynomially decay sequence of rxr matrices called refinement mask. Associated with the mask a is a linear operator on (L"2(R))^r given byQ"af(x)@?@?@a@?Za(@a)f(2x-@a),x@?R,f=(f"1,...,f"r)^T@?(L"2(R))^r.By using same methods in [B. Han, R. Q. Jia, Characterization of Riesz bases of wavelets generated from multiresolution analysis, manuscript]; [B. Han, Refinable functions and cascade algorithms in weighted spaces with infinitely supported masks, manuscript]; [R.Q. Jia, Q.T. Jiang, Z.W. Shen, Convergence of cascade algorithms associated with nonhomogeneous refinement equations, Proc. Amer. Math. Soc. 129 (2001) 415-427]; [R.Q. Jia, Convergence of vector subdivision schemes and construction of biorthogonal multiple wavelets, in: Advances in Wavelet, Hong Kong,1997, Springer, Singapore, 1998, pp. 199-227], a characterization of convergence of the sequences (Q"a^nf)"n"="1","2","... in the L"2-norm is given, which extends the main results in [R.Q. Jia, S.D. Riemenschneider, D.X. Zhou, Vector subdivision schemes and multiple wavelets, Math. Comp. 67 (1998) 1533-1563] on convergence of the subdivision schemes associated with a finitely supported mask to the case in which mask a is polynomially decay sequence. As an application, we also obtain a characterization of smoothness of solutions of the refinement equation mentioned above for the case r=1.