Convergence of cascade algorithms in Sobolev spaces for perturbed refinement masks

Journal of Approximation Theory

Volumn 119, Issue 2, 2002, Pages 133-155

  Chen, Di Rong ^a,c   Plonka, Gerlind ^b  

^a BEIHANG UNIVERSITY   (China)

^b UNIVERSITY OF DUISBURG ESSEN   (Germany)

^c HUBEI UNIVERSITY FOR NATIONALITIES   (China)

Author keywords

Cascade algorithm; Joint spectral radius; Perturbation of refinable functions; Sobolev space

Indexed keywords

EID: 0036899407     PISSN: 00219045     EISSN: None     Source Type: Journal    
DOI: 10.1006/jath.2002.3682     Document Type: Article

References (31)

1. A. S. Cavaretta, W. Dahmen, and C. A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 93 (1991), 1-186.
2. D. R. Chen, Algebraic properties of subdivision operators with matrix mask and their applications, J. Approx. Theory 97 (1999), 294-310.
3. D. R. Chen, R. Q. Jia, and S. D. Riemenschneider, Convergence of vector subdivision schemes in Sobolev spaces, Appl. Comp. Harmon. Anal. 12 (2002), 128-149.
4. I. Daubechies and Y. Huang, How does truncation of the mask affect a refinable function? Constr. Approx. 11 (1995), 365-380.
5. I. Daubechies and J. Lagarias, Two-scale difference equations I. Existence and global regularity of solutions, SIAM J. Math. Anal. 22 (1991), 1388-1410.
6. T. N. T. Goodman and S. L. Lee, Convergence of cascade algorithms, in "Mathematical Methods for Curves and Surfaces II," (M. Daehlen, T. Lyche, and L. L. Schumaker, Eds.), pp. 191-212, Vanderbilt Univ. Press, Nashville, 1998.
7. B. Han, "Subdivision Schemes, Biorthogonal Wavelets and Image Compression," Ph.D. thesis, University of Alberta, 1998.
8. B. Han, Error estimate of a subdivision scheme with a truncated refinement mask, unpublished manuscript, 1997.
9. B. Han, The initial functions in a subdivision scheme, in "Wavelet Analysis, Twenty Years Developments," (D.-X. Zhou, Ed.), pp. 9-33, World Scientific Press, Singapore, 2002.
10. B. Han and T. A. Hogan, How is a vector pyramid scheme affected by perturbation in the mask? in "Approximation Theory IX," (C. K. Chui and L. L. Schumaker, Eds.), pp. 97-104, Vanderbilt Univ. Press, Nashville, 1998.
11. B. Han and R. Q. Jia, Multivariate refinement equations and convergence of subdivision schemes, SIAM J. Math. Anal. 29 (1998), 1177-1199.
12. C. Heil and D. Collela, Characterizations of scaling functions: continuous solutions, SIAM J. Matrix Anal. Appl 15 (1994), 496-518.
13. C. Heil, Some stability properties of wavelets and scaling functions, in "Wavelets and Their Applications," (J. S. Byrnes et al., Eds.), NATO Advances in Science Institute Series C: Math. Phys. Sci., Vol. 442, pp. 19-38, Kluwer, Dordrecht, 1994.
14. 2-approximation order from shift-invariant spaces, in "Multivariate Approximation and Applications," (N. Dyn, D. Leviatan, D. Levin, and A. Pinkus, Eds.), pp. 73-111, Cambridge Univ. Press, Cambridge, UK, 2001.
15. p spaces, Adv. in Comput. Math. 3 (1995), 309-341.
16. Deleted in proof.
17. R. Q. Jia, Approximation properties of multivariate wavelets, Math. Comput. 67 (1998), 647-665.
18. R. Q. Jia, Shift-invariant subspaces and linear operator equations, Israel J. Math. 103 (1998), 259-288.
19. R. Q. Jia, Q. T. Jiang, and S. L. Lee, Convergence of cascade algorithms in Sobolev spaces and integrals of wavelets, Numer. Math., 91 (2002), 453-473.
20. Deleted in proof.
21. R. Q. Jia, S. D. Riemenschneider, and D. X. Zhou, Vector subdivision schemes and multiple wavelets, Math. Comput. 67 (1998), 1533-1563.
22. Q. Jiang, Multivariate matrix refinable functions with arbitrary matrix dilation, Trans. Amer. Math. Soc. 351 (1999), 2407-2438.
23. W. Lawton, S. L. Lee, and Z. Shen, Convergence of multidimensional cascade algorithm, Numer. Math. 78 (1998), 427-438.
24. C. A. Micchelli and T. Sauer, Regularity of multiwavelets, Adv. in Comput. Math. 7 (1997), 455-545.
25. C. A. Micchelli and T. Sauer, On vector subdivision, Math. Z. 229 (1998), 621-674.
26. C. A. Micchelli and T. Sauer, Sobolev norm convergence of stationary subdivision schemes, in "Surface Fitting and Multiresolution Methods," (A. Le Méhauté, C. Rabut, and L. L. Schumaker, Eds.), pp. 245-260, Vanderbilt Univ. Press, Nashville, 1997.
27. Z. Shen, Refinable function vectors, SIAM J. Math. Anal. 29 (1998) 35-250.
28. G. Strang, Eigenvalues of (↓ 2)H and convergence of the cascade algorithm, IEEE Signal Process. 44 (1996), 233-238.
29. Q. Sun, Convergence and boundedness of cascade algorithm in Besov spaces and Triebel-Lizorkin spaces: Part II. Adv. in Math. (China) 30 (2001), 22-36.
30. L. F. Villemoes, Wavelet analysis of refinement equations, SIAM J. Math. Anal. 25 (1994), 1433-1460.
31. D.-X. Zhou, Norms concerning subdivision sequences and their applications in wavelets, Appl. Comput. Hamornic Anal. 11 (2001), 329-346.