Convergence rates of vector cascade algorithms in [formula omitted

Abstract

We investigate the solutions of vector refinement equations of the form ϕ = ∑ α ∈ Z s a ( α ) ϕ ( M · - α ) , where the vector of functions ϕ = ( ϕ 1 , … , ϕ r ) T is in ( L p ( R s ) ) r , 1 ⩽ p ⩽ ∞ , a ≕ ( a ( α ) ) α ∈ Z s is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that lim n → ∞ M - n = 0 . Associated with the mask a and M is a linear operator Q a defined on ( L p ( R s ) ) r by Q a ψ ≔ ∑ β ∈ Z s a ( β ) ψ ( M · - β ) . The iteration scheme ( Q a n ψ ) 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 L p 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 ( L p ( R s ) ) 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 L p ( 1 ⩽ p ⩽ ∞ ) smoothness of multiple refinable functions is also presented when M = qI s × s , where I s × 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.