Subdivisions with infinitely supported mask

Abstract

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 φ ( x ) = ∑ α ∈ Z a ( α ) φ ( 2 x - α ) , x ∈ R , where the vector of functions φ = ( φ 1 , … , φ r ) T is in ( L 2 ( R ) ) r and a ≕ ( a ( α ) ) α ∈ Z is polynomially decay sequence of r × r matrices called refinement mask. Associated with the mask a is a linear operator on ( L 2 ( R ) ) r given by Q a f ( x ) ≔ ∑ α ∈ Z a ( α ) f ( 2 x - α ) , 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 n f ) 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 .