Vector cascade algorithms and refinable function vectors in Sobolev spaces

Journal of Approximation Theory

Volumn 124, Issue 1, 2003, Pages 44-88

  Han, Bin ^a  

^a UNIVERSITY OF ALBERTA   (Canada)

Author keywords

Canonical mask; Convergence; Error estimate; Hermite interpolant; Initial function vector; Rate of convergence; Refinable function vector; Smoothness; Vector cascade algorithm and subdivision scheme

Indexed keywords

EID: 0242508421     PISSN: 00219045     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0021-9045(03)00120-5     Document Type: Article

References (47)

1. d), Constr. Approx. (1998) 631-652.
2. C. Cabrelli, C. Heil, U. Molter, Accuracy of lattice translates of several multidimensional refinable functions, J. Approx. Theory 95 (1998) 5-52.
3. A.S. Cavaretta, W. Dahmen, C.A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 453 (1991).
4. 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.
5. D.R. Chen, G. Plonka, Convergence of cascade algorithms in Sobolev spaces for perturbed refinement masks, J. Approx. Theory 119 (2) (2002) 133-155.
6. D. R. Chen, X. Zheng, Stability implies convergence of cascade algorithms in Sobolev spaces, J. Math. Anal. Appl. 268 (2002) 41-52.
7. A. Cohen, I. Daubechies, G. Plonka, Regularity of refinable function vectors, J. Fourier Anal. Appl. 3 (1997) 295-324.
8. A. Cohen, K. Gröchenig, L. Villemoes, Regularity of multivariate refinable functions, Constr. Approx. 15 (1999) 241-255.
9. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Series in Applied Mathematics, SIAM, Philadelphia, 1992.
10. I. Daubechies, Y. Huang, How does truncation of the mask affect a refinable function?, Constr. Approx. 11 (1995) 365-380.
11. I. Daubechies, J. C. Lagarias, Two-scale difference equations: II. Local regularity, infinite products of matrices and fractals, SIAM J. Math. Anal. 23 (1992) 1031-1079.
12. N. Dyn, D. Levin, Analysis of Hermite-interpolatory subdivision schemes, in: S. Dubuc, G. Deslauriers, (Eds.), Spline Functions and the Theory of Wavelets, Vol. 18, CRM Proceedings and Lectures Notes, Amer. Math. Soc., Providence, RI, 1999, pp. 105-113.
13. T.N.T. Goodman, S.L. Lee, Convergence of cascade algorithms, in: Mathematical Methods for Curves and Surfaces II, Innovative Applications in Mathematics, Vanderbilt University Press, Nashville, TN, 1998, pp. 191-212.
14. B. Han, Error estimate of a subdivision scheme with a truncated refinement mask, unpublished manuscript, 1997.
15. B. Han, Subdivision schemes, biorthogonal wavelets and image compression, PhD Thesis, University of Alberta, 1998.
16. B. Han, Analysis and construction of optimal multivariate biorthogonal wavelets with compact support, SIAM J. Math. Anal. 31 (2000) 274-304.
17. B. Han, Approximation properties and construction of Hermite interpolants and biorthogonal multiwavelets, J. Approx. Theory 110 (2001) 18-53.
18. B. Han, The initial functions in a cascade algorithm, in: D.X. Zhou (Ed.), Wavelet Analysis: Twenty Years' Developments, Series in Analysis, Vol. 1, World Scientific Publishing, New Jersey, 2002, pp. 154-178.
19. B. Han, Computing the smoothness exponent of a symmetric multivariate refinable function, SIAM J. Matrix Anal. Appl. 24 (3) (2003) 693-714.
20. B. Han, T.A. Hogan, How is a vector pyramid scheme affected by perturbation in the mask?, in: C.K. Chui, L.L. Schumaker (Eds.), Approximation Theory IX, Vanderbilt University Press, Nashville, TN, 1998, pp. 97-104.
21. B. Han, R.Q. Jia, Multivariate refinement equations and convergence of subdivision schemes, SIAM J. Math. Anal. 29 (1998) 1177-1199.
22. B. Han, Q. Mo, Multiwavelet frames from refinable function vectors, Adv. Comput. Math. 18 (2003) 211-245.
23. B. Han, T.P.-Y. Yu, B. Piper, Multivariate refinable Hermite interpolants, Math. Comp., to appear.
24. C. Heil, D. Colella, Characterizations of scaling functions: continuous solutions, SIAM J. Matrix Anal. Appl. 15 (1994) 496-518.
25. p, spaces, Adv. Comput. Math. 3 (1995) 309-341.
26. R.Q. Jia, Characterization of smoothness of multivariate refinable functions in Sobolev spaces, Trans. Amer. Math. Soc. 351 (1999) 4089-4112.
27. R.Q. Jia, Approximation with scaled shift-invariant spaces by means of quasi-projection operator, preprint, 2002.
28. R.Q. Jia, Q.T. Jiang, Spectral analysis of the transition operators and its applications to smoothness analysis of wavelets, SIAM J. Matrix. Anal. Appl. 24 (2003) 1071-1109.
29. R.Q. Jia, Q.T. Jiang, S.L. Lee, Convergence of cascade algorithms in Sobolev spaces and integral of wavelets, Numer. Math. 91 (2002) 453-473.
30. R.Q. Jia, C.A. Micchelli, On linear independence of integer translates of a finite number of functions, Proc. Edinburgh Math. Soc. (1992) 69-85.
31. R.Q. Jia, S.D. Riemenschneider, D.X. Zhou, Vector subdivision schemes and multiple wavelets, Math. Comp. 67 (1998) 1533-1563.
32. 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.
33. Q.T. Jiang, Multivariate matrix refinable functions with arbitrary matrix dilation, Trans. Amer. Math. Soc. 351 (1999) 2407-2438.
34. W. Lawton, S.L. Lee, Z.W. Shen, Convergence of multidimensional cascade algorithm, Numer. Math. 78 (1998) 427-438.
35. S. Li, Convergence of cascade algorithms in Sobolev spaces associated with multivariate refinement equations, J. Math. Anal. Appl. 257 (2001) 154-169.
36. r(1≤p≤∞), Sci. China (Ser. A ) 33 (2003) 1-12.
37. J.L. Merrien, A family of Hermite interpolants by bisection algorithms, Numer. Algorithm 2 (1992) 187-200.
38. C.A. Micchelli, T. Sauer, Regularity of multiwavelets, Adv. Comput. Math. 7 (1997) 455-545.
39. C.A. Micchelli, T. Sauer, On vector subdivision, Math. Z. 229 (1998) 621-674.
40. C.A. Micchelli, T. Sauer, Sobolev norm convergence of stationary subdivision schemes, in: Surface Fitting and Multiresolution Methods, Vanderbilt University Press, Nashville, TN, 1997, pp. 245-260.
41. G. Plonka, A. Ron, A new factorization technique of the matrix mask of univariate refinable functions, Numer. Math. 87 (2001) 555-595.
42. A. Ron, Z.W. Shen, The Sobolev regularity of refinable functions, J. Approx. Theory 106 (2000) 185-225.
43. Z.W. Shen, Refinable function vectors, SIAM J. Math. Anal. 29 (1998) 235-250.
44. L.F. Villemoes, Wavelet analysis of refinement equations, SIAM J. Math. Anal. 25 (1994) 1433-1460.
45. T. Yu, Parametric families of Hermite subdivision schemes in dimension 1, preprint.
46. D.X. Zhou, Multiple refinable Hermite interpolants, J. Approx. Theory 102 (2000) 46-71.
47. D.X. Zhou, Norms concerning subdivision sequences and their applications in wavelets, Appl. Comput. Harmon. Anal. 11 (2001) 329-346.