Convergence of cascade algorithms and smoothness of refinable distributions

Chinese Annals of Mathematics. Series B

Volumn 24, Issue 3, 2003, Pages 367-386

  Sun, Qiyu ^a,b  

^a University of Houston   (United States)

^b NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

Author keywords

Cascade algorithm; Cascade operator; Fractional sobolev space; Linear independent shifts; Refinable distribution; Shift invariant space; Stable shifts

Indexed keywords

EID: 0141991898     PISSN: 02529599     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0252959903000372     Document Type: Article

References (26)

1. d, Proc. Amer. Math. Soc., 130(2002), 2641-2654.
2. Cabrelli, C., Heil, C. & Molter, U., Self-similarity and multiwavelets in higher dimensions, Mem. Amer. Math. Soc., to appear.
3. Cavaretta, A. S., Dahmen, W. & Micchelli, C. A., Stationary subdivision, Mem. Amer. Math. Soc., 93(1991), 1-186.
4. Cohen, A., Wavelet methods in numerical analysis, in Handbook of Numerical Analysis, Vol. VII, P. G. Ciarlet and J. L. Lions (eds.), Elsevier, Amsterdam, 2000.
5. Cohen, A., Daubechies, I. & Plonka, G., Regularity of refinable function vectors, J. Fourier Anal. Appl., 3(1997), 295-323.
6. Cohen, A., Gröchenig, K. & Villemoes, L., Regularity of multivariate refinable functions, Constr. Approx., 15(1999), 241-255.
7. Dahmen, W., Wavelet and multiscale methods for operator equations, Acta Numericana, 6(1997), 55-228.
8. Daubechies, I., Ten lectures on wavelets, SIAM, Philadelphia, PA, 1992.
9. Eirola, T., Sobolev characterization of solutions of dilation equations, SIAM J. Math. Anal., 23(1992), 1015-1030.
10. Jia, R. Q., Approximation properties of multivariate wavelets, Math. Comp., 67(1998), 647-665.
11. p spaces, Adv. Comput. Math., 3(1995), 309-341.
12. Jia, R. Q., Characterization of smoothness of multivariate refinable functions in Sobolev space, Trans. Amer. Math. Soc., 351(1999), 4089-4112.
13. Jia, R. Q., Jiang, Q. & Lee, S. L., Convergence of cascade algorithm in Sobolev spaces and integrals of wavelets, Numer. Math., 91(2002), 453-473.
14. p solutions of vector refinement equations, J. Fourier Anal. Appl., 7(2001), 143-167.
15. Jia, R. Q. & Micchelli, C. A., Using the refinement equations for the construction of pre-wavelets II, Powers of two, in Curves and Surfaces, P. J. Laurent, A. Le Mehaute and L. L. Schumaker (eds.), Academic Press, Boston, MA, 1991, 209-246.
16. Han, B. & Jia, R. Q., Multivariate refinement equations and subdivisions, SIAM J. Math. Anal., 29(1998), 1177-1199.
17. Ma, B. & Sun, Q., Compactly supported refinable distributions in Triebel-Lizorkin spaces and Besov spaces, J. Fourier Anal. Appl., 5(1999), 87-104.
18. Ron, A., Wavelets and their associated operators, in Approximation Theory IX, C. K. Chui and L. L. Schumaker (eds.), Vanderbilt University, 1998, 1-35.
19. Ron, A. & Shen, Z., The Sobolev regularity of refinable functions, J. Approx. Th., 106(2000), 185-225.
20. Stein, E. M., Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N. J., 1970.
21. Sun, Q., Convergence and boundedness of cascade algorithm in Besov spaces and Triebel-Lizorkin spaces: Part I and II, Adv. Math. (China], 29(2000), 507-526; 30(2001), 22-36.
22. Sun, Q., Convergence and boundedness of cascade algorithm in Besov spaces and Triebel-Lizorkin spaces: Part I and II, Adv. Math. (China], 29(2000), 507-526; 30(2001), 22-36.
23. Sun, Q., Stability of the shifts of global supported distributions, J. Math. Anal. Appl., 261(2001), 113-125.
24. Sun, Q., Linear dependence of the shifts of vector-valued compactly supported distributions, preprint.
25. Triebel, H., Theory of function spaces II, Birkhauser, 1992.
26. Villemoes, L. F., Wavelet analysis of refinement equations, SIAM J. Math. Anal., 25(1994), 1433-1460.