Approximation Properties and Construction of Hermite Interpolants and Biorthogonal Multiwavelets

Journal of Approximation Theory

Volumn 110, Issue 1, 2001, Pages 18-53

  Han, Bin ^a  

^a UNIVERSITY OF ALBERTA   (Canada)

Author keywords

Biorthogonal multivariate multiwavelets; Hermite interpolants; Hermite interpolatory mask; accuracy order; sum rules; refinable function vectors

Indexed keywords

EID: 0041407181     PISSN: 00219045     EISSN: None     Source Type: Journal    
DOI: 10.1006/jath.2000.3545     Document Type: Article

References (42)

1. d). Constr. Approx. 14:1998;631-652.
2. Cabrelli C., Heil C., Molter U. Accuracy of lattice translates of several multidimensional refinable functions. J. Approx. Theory. 95:1998;5-52.
3. C. Cabrelli, C. Heil, and, U. Molter, Accuracy of several multidimensional refinable distributions, preprint, 1998.
4. D. R. Chen, Algebraic properties of subdivision operators with matrix mask and their applications, J. Approx. Theory, in press.
5. Chen D. R., Han B., Riemenschneider S. D. Construction of multivariate biorthogonal wavelets with arbitrary vanishing moments. Adv. Comput. Math. 13:2000;131-165.
6. Chui C. K., Lian J. A study of orthonormal multi-wavelets. Appl. Numer. Math. 20:1996;273-298.
7. Cohen A., Daubechies I., Feauveau J. C. Biorthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math. 45:1992;485-560.
8. Dahmen W., Han B., Jia R. Q., Kunoth A. Biorthogonal multiwavelets on the interval: cubic Hermite splines. Constr. Approx. 16:2000;221-259.
9. Dahmen W., Micchelli C. A. Biorthogonal wavelet expansions. Constr. Approx. 13:1997;293-328.
10. Daubechies I. Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41:1988;906-996.
11. Daubechies I. Ten Lectures on Wavelets. 1992;SIAM, Philadelphia.
12. Donovan G. C., Geronimo G., Hardin D. P., Massopust P. Construction of orthogonal wavelets using fractal interpolation function. SIAM J. Math. Anal. 27:1996;1158-1192.
13. Dyn N., Levin D. Interpolating subdivision schemes for the generation of curves and surfaces. Multivariate Approximation and Interpolation. Internat. Ser. Numer. Math. 94:1990;Birkhäuser, Basel. p. 91-106.
14. S. S. Goh, Q. Jiang, and, T. Xia, Construction of biorthogonal multiwavelets using the lifting scheme, preprint, 1998.
15. Goodman T. N. T., Lee S. L., Tang W. S. Wavelets in wondering subspaces. Trans. Amer. Math. Soc. 338:1993;639-654.
16. Han B. On dual wavelet tight frames. Appl. Comput. Harm. Anal. 4:1997;380-413.
17. Han B. Analysis and construction of optimal multivariate biorthogonal wavelets with compact support. SIAM J. Math. Anal. 31:2000;274-304.
18. Han B., Jia R. Q. Multivariate refinement equations and convergence of subdivision schemes. SIAM J. Math. Anal. 29:1998;1177-1199.
19. Han B., Jia R. Q. Optimal interpolatory subdivision schemes in multidimensional spaces. SIAM J. Numer. Anal. 36:1998;105-124.
20. B. Han, and, R. Q. Jia, Quincunx fundamental refinable functions and quincunx biorthogonal wavelets, Math. Comp, posted on October 17, 2000, PII S0025-5718(00)01300-4, in press.
21. W. He, and, M. J. Lai, Construction of bivariate nonseparable compactly supported orthonormal multiwavelets with arbitrarily high regularity, preprint, 1998.
22. Heil C., Colella D. Matrix refinement equations and subdivision schemes. J. Fourier Anal. Appl. 2:1996;363-377.
23. Heil C., Strang G., Strela V. Approximation by translates of refinable functions. Numer. Math. 73:1996;75-94.
24. 2-approximation order from shift-invariant spaces, preprint, 1999.
25. Ji H., Riemenschneider S., Shen Zuowei. Multivariate compactly supported fundamental refinable functions, duals and biorthogonal wavelets. Stud. Appl. Math. 102:1999;173-204.
26. Jia R. Q. Approximation properties of multivariate wavelets. Math. Comp. 67:1998;647-665.
27. Jia R. Q. Shift-invariant spaces and linear operator equations. Israel J. Math. 103:1998;259-288.
28. Jia R. Q., Micchelli C. A. On linear independence of integer translates of a finite number of functions. Proc. Edinburgh Math. Soc. 36:1992;69-85.
29. Jia R. Q., Riemenschneider S., Zhou D. X. Approximation by multiple refinable functions and multiple functions. Canad. J. Math. 49:1997;944-962.
30. Jia R. Q., Riemenschneider S., Zhou D. X. Smoothness of multiple refinable functions and multiple wavelets. SIAM J. Matrix Anal. 21:1999;1-28.
31. Jiang Q. T. Multivariate matrix refinable functions with arbitrary matrix dilation. Trans. Amer. Math. Soc. 351:1999;2407-2438.
32. J. Kovčević, and, W. Sweldens, Wavelets family of increasing order in arbitrary dimensions, preprint, 1997.
33. Logar A., Sturmfels B. Algorithms for the Quillen-Suslin theorem. J. Algebra. 145:1992;231-239.
34. Micchelli C. A., Sauer T. Regularity of multiwavelets. Adv. Comput. Math. 7:1997;455-545.
35. Nicolaides R. A. On a class of finite elements generated by Lagrange interpolation. SIAM J. Numer. Anal. 9:1972;435-445.
36. Plonka G. Approximation order provided by refinable function vectors. Constr. Approx. 13:1997;221-244.
37. Plonka G., Strela V. Construction of multi-scaling functions with approximation and symmetry. SIAM J. Math. Anal. 29:1998;481-510.
38. A. Ron, and, Zuowei, Shen, The Sobolev regularity of refinable functions, preprint, 1997.
39. Shen Zuowei. Refinable function vectors. SIAM J. Math. Anal. 29:1998;235-250.
40. Strela V. A note on construction of biorthogonal multi-scaling functions. Aldroubi A., Lin E. B. Contemporary Mathematics. 1998;149-157 Amer. Math. Soc. Providence.
41. Sweldens W. The lifting scheme: A custom-design construction of biorthogonal wavelets. Appl. Comput. Harmon. Anal. 3:1996;186-200.
42. Zhou D. X. Multiple refinable Hermite interpolants. J. Approx. Theory. 102:2000;46-71.