Convergence rates of vector cascade algorithms in Lp

• Authors: Song Li
• Venue: Journal of Approximation Theory
• Year: 2005

We investigate the solutions of vector refinement equations of the form ϕ= ∑ α ∈ Zs a(α)ϕ(M ċ - α), where the vector of functions ϕ = (ϕ1.....ϕr)T is in (Lp(Rs))r, 1 ≤ p ≤ ∞, a =: (a(α))α ∈ Zs is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that limn → ∞ M-n = 0. Associated with the mask a and M is a linear operator Qa defined on (Lp(Rs))r by Qaψ := ∑β ∈ Zsa(β)ψ(M ċ-β). The iteration scheme (Qanψ)n = 1.2,... is called a cascade algorithm (see [D.R. Chen, R.Q. Jia, S.D. Riemenschneider, Convergence of vector subdivision schemes in Sobolev spaces, Appl. Comput. Harmon. Anal. 12 (2002) 128-149; B. Han, The initial functions in a cascade algorithm, in: D.X. Zhou (Ed.), Proceeding of International Conference of Computational Harmonic Analysis in Hong Kong, 2002; B. Han, R.Q. Jia, Multivariate refinement equations and convergence of subdivision schemes, SIAM J. Math. Anal. 29 (1998) 1177-1199; R.Q. Jia, Subdivision schemes in Lp spaces, Adv. Comput. Math. 3 (1995) 309-341; R.Q. Jia, S.D. Riemenschneider, D.X. Zhou, Vector subdivision schemes and multiple wavelets, Math. Comp. 67 (1998) 1533-1363; S. Li, Characterization of smoothness of multivariate refinable functions and convergence of cascade algorithms associated with nonhomogeneous refinement equations, Adv. Comput. Math. 20 (2004) 311-331; Q. Sun, Convergence and boundedness of cascade algorithm in Besov space and Triebel-Lizorkin space I, Adv. Math. (China) 29 (2000) 507-526]). Cascade algorithm is an important issue to wavelets analysis and computer graphics. Main results of this paper are related to the convergence and convergence rates of vector cascade algorithm in (Lp(Rs))r (1 ≤ p ≤ ∞). We give some characterizations on convergence of cascade algorithm and also give estimates on convergence rates of this cascade algorithm with M being isotropic dilation matrix. It is well known that smoothness is a very important property of a multiple refinable function. A characterization of Lp(1 ≤ p ≤ ∞) smoothness of multiple refinable functions is also presented when M = qIs × s, where Is×s is the s×s identity matrix, and q ≥ 2 is an integer. In particular, the smoothness results given in [R.Q. Jia, S.D. Riemenschneider, D.X. Zhou, Smoothness of multiple refinable functions and multiple wavelets, SIAM J. Matrix Anal. Appl. 21 (1999) 1-28] is a special case of this paper.