Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix

Advances in Computational Mathematics

Volumn 24, Issue 1-4, 2006, Pages 375-403

  Han, Bin ^a  

^a UNIVERSITY OF ALBERTA   (Canada)

Author keywords

Cascade algorithm; Rate of convergence; Refinable function vector; Smoothness; Sobolev space; Vector refinement equation

Indexed keywords

EID: 33645770045     PISSN: 10197168     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10444-004-7615-2     Document Type: Article

References (32)

1. C. Cabrelli, C. Heil and U. Molter, Self-similarity and multiwavelets in high dimensions, Mem. Amer. Math. Soc., to appear.
2. A.S. Cavaretta, W. Dahmen and C.A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 453 (1991).
3. D.R. Chen, R.Q. Jia and S.D. Riemenschneider, Convergence of vector subdivision schemes in Sobolev spaces, Appl. Comput. Harmon. Anal. 12 (2002) 128-149.
4. A. Cohen, I. Daubechies and G. Plonka, Regularity of refinable function vectors, J. Fourier Anal. Appl. 3(1997) 295-324.
5. A. Cohen, K. Gröchenig and L. Villemoes, Regularity of multivariate refinable functions, Constr. Approx. 15 (1999) 241-255.
6. D. Colella and C. Heil, Characterization of scaling functions: Continuous solutions, SIAM J. Matrix Anal. Appl. 15 (1994) 496-518.
7. B. Han, Analysis and construction of optimal multivariate biorthogonal wavelets with compact support, SIAM J. Math. Anal. 31 (2000) 274-304.
8. B. Han, Approximation properties and construction of Hermite interpolants and biorthogonal multiwavelets, J. Approx. Theory 110 (2001) 18-53.
9. B. Han, Computing the smoothness exponent of a symmetric multivariate refinable function, SIAM J. Matrix Anal. Appl. 24(3) (2003) 693-714.
10. B. Han, Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix, J. Comput. Appl. Math. 155 (2003) 43-67.
11. B. Han, Vector cascade algorithms and refinable function vectors in Sobolev spaces, J. Approx. Theory 124 (2003) 44-88.
12. B. Han and R.Q. Jia, Multivariate refinement equations and convergence of subdivision schemes, SIAM J. Math. Anal. 29 (1998) 1177-1199.
13. B. Han, T. Yu and Y. Xue, Non-interpolatory Hermite subdivision schemes, Math. Comp., to appear. Available at http://www.ualberta.ca/~bhan/publ.htm
14. p spaces, Adv. Comput. Math. 3 (1995) 309-341.
15. R.Q. Jia, Approximation properties of multivariate wavelets, Math. Comp. 67 (1998) 647-665.
16. R.Q. Jia, Characterization of smoothness of multivariate refinable functions in Sobolev spaces, Trans. Amer. Math. Soc. 351 (1999) 4089-4112.
17. R.Q. Jia and Q.T. Jiang, Spectral analysis of the transition operator and its applications to smoothness analysis of wavelets, SIAM J. Matrix Anal. Appl. 24 (2003) 1071-1109.
18. R.Q. Jia, Q.T. Jiang and S.L. Lee, Convergence of cascade algorithms in Sobolev spaces and integral of wavelets, Numer. Math. 91 (2002) 453-473.
19. p solutions of refinement equations, J. Fourier Anal. Appl. 7 (2001) 143-167.
20. R.Q. Jia, S.D. Riemenschneider and D.X. Zhou, Vector subdivision schemes and multiple wavelets, Math. Comp. 67 (1998) 1533-1563.
21. R.Q. Jia, S.D. Riemenschneider and D.X. Zhou, Smoothness of multiple refinable functions and multiple wavelets, SIAM J. Matrix Anal. Appl. 21 (1999) 1-28.
22. R.Q. Jia and S.R. Zhang, Spetral properties of the transition operator associated to a multivariate refinement equation, Linear Algebra Appl. 292 (1999) 155-178.
23. p-solutions for the two-scale dilation equations, SIAM J. Math. Anal. 26 (1995) 1018-1046.
24. B. Ma and Q. Sun, Compactly supported refinable distributions in Triebel-Lizorkin spaces and Besov spaces, J. Fourier Anal. Appl. 5 (1999) 87-104.
25. C.A. Micchelli and H. Prautzsch, Uniform refinement of curves, Linear Algebra Appl. 114/115 (1989) 841-870.
26. C.A. Micchelli and T. Sauer, Regularity of multiwavelets, Adv. Comput. Math. 7 (1997) 455-545.
27. A. Ron and Z.W. Shen, The Sobolev regularity of refinable functions, J. Approx. Theory 106 (2000) 185-225.
28. Q. Sun, Convergence and boundedness of cascade algorithm in Besov spaces and Tribel-Lizorkin spaces: Part I and II, Adv. Math (China) 29 (2000) 507-526;
29. Q. Sun, Convergence and boundedness of cascade algorithm in Besov spaces and Tribel-Lizorkin spaces: Part I and II, Adv. Math (China) 30 (2001) 22-36.
30. Q. Sun, Convergence of cascade algorithms and smoothness of refinable distributions, Chinese Ann. of Math. 24 (2003) 367-386.
31. D.X. Zhou, The p-norm joint spectral radius for even integers, Methods Appl. Anal. 5 (1998) 39-54.
32. D.X. Zhou, Norms concerning subdivision sequences and their applications in wavelets, Appl. Comput. Harmon. Anal. 11 (2001) 329-346.