Nonstationary tight wavelet frames, I: Bounded intervals

Applied and Computational Harmonic Analysis

Volumn 17, Issue 2 SPEC. ISS., 2004, Pages 141-197

  Chui, Charles K ^a,b   He, Wenjie ^a   Stöckler, Joachim ^c  

^a UNIVERSITY OF MISSOURI ST LOUIS   (United States)

^b STANFORD UNIVERSITY   (United States)

^c TU DORTMUND UNIVERSITY   (Germany)

Author keywords

Approximate dual; Bounded interval frame; Nonstationary frame; Tight frame

Indexed keywords

APPROXIMATION THEORY; BOUNDARY CONDITIONS; COMPUTER AIDED DESIGN; COMPUTER AIDED MANUFACTURING; FOURIER TRANSFORMS; POLYNOMIALS; REDUNDANCY;

MATHEMATICAL FORMULATION; MULTIRESOLUTION; SYMMETRIC GENERATORS; VANISHING MOMENT RECOVERY (VMR);

WAVELET TRANSFORMS;

EID: 4344660740     PISSN: 10635203     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.acha.2004.02.004     Document Type: Article

References (34)

1. de Boor C. A Practical Guide to Splines. revised ed. 2002;Springer-Verlag, Berlin.
2. de Boor C., Fix G.J. Spline approximation by quasi-interpolants. J. Approx. Theory. 8:1973;19-45.
3. Chui C.K. Multivariate splines. CBMS-NSF Reg. Conf. Series in Applied Mathematics, vol. 54 :1988;SIAM, Philadelphia.
4. Chui C.K. Wavelets: A Mathematical Tool for Signal Analysis. 1997;SIAM, Philadelphia.
5. Chui C.K., De Villiers J.M. Spline-wavelets with arbitrary knots on a bounded interval: Orthogonal decomposition and computational algorithms. Commun. Appl. Anal. 2:(4):1998;457-486.
6. Chui C.K., He W. Compactly supported tight frames associated with refinable functions. Appl. Comput. Harmon. Anal. 8:2000;293-319.
7. Chui C.K., He W., Stöckler J. Compactly supported tight and sibling frames with maximum vanishing moments. Appl. Comput. Harmon. Anal. 13:2002;224-262.
8. C.K. Chui, W. He, J. Stöckler, Nonstationary tight wavelet frames, II: unbounded intervals, Appl. Comput. Harmon. Anal., submitted for publication.
9. Chui C.K., He W., Stöckler J., Sun Q. Compactly supported tight affine frames with integer dilations and maximum vanishing moments. Adv. Comput. Math. 18:2003;159-187.
10. Chui C.K., Lian J.A. Nonstationary wavelets and refinement sequences of nonuniform B -splines. Chui C.K., Schumaker L.L., Stöckler J. Approximation Theory X: Wavelets, Splines, and Applications. 2001;207-229 Vanderbilt Univ. Press.
11. Chui C.K., Stöckler J. Recent development of spline-wavelet frames with compact support. Welland G.V. Beyond Wavelets. 2003;151-214 Elsevier.
12. Ciesielski Z. The B -spline basis in a space of algebraic polynomials. Ukrainian Math. J. 38:1986;311-315.
13. ∞ manifolds. Stud. Math. 76:1983;1-58. 96-136.
14. Cohen A., Daubechies I., Vial P. Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon. Anal. 1:(1):1993;54-81.
15. Dahmen W., Micchelli C.A. Banded matrices with banded inverses, II: Locally finite decomposition of spline spaces. Constr. Approx. 9:1993;263-281.
16. Daubechies I. Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41:1988;909-996.
17. Daubechies I., Guskov I., Sweldens W. Commutation for irregular subdivision. Constr. Approx. 17:2001;479-514.
18. Daubechies I., Han B., Ron A., Shen Z.W. Framelets: MRA-based constructions of wavelet frames. Appl. Comput. Harmon. Anal. 14:2003;1-46.
19. DeVore R.A., Lorentz G.G. Constructive Approximation. 1993;Springer-Verlag, New York.
20. H.W. Gould, Combinatorial Identities, Morgantown, WV, 1972.
21. B. Han, Q. Mo, Tight wavelet frames generated by three symmetric B -spline functions with high vanishing moments, preprint.
22. Jetter K., Stöckler J. An identity for multivariate Bernstein polynomials. Comput. Aided Geom. Design. 20:2003;563-577.
23. K. Jetter, J. Stöckler, New polynomial preserving operators on simplices, Preprint 242, University of Dortmund, Dept. of Mathematics, November 2003.
24. R. Q. Jia, Approximation with scaled shift-invariant spaces by means of quasi-projection operators, in preparation.
25. Jüttler B. The dual basis functions for the Bernstein polynomials. Adv. Comput. Math. 8:1998;345-352.
26. Q. Mo, Compactly supported symmetric MRA wavelet frames, Ph.D. thesis, Dept. of Mathematics and Statistical Sciences, University of Alberta, Edmonton, 2003.
27. Piegl L., Tiller W. The NURBS Book. second ed. 1997;Springer-Verlag, Berlin/Heidelberg.
28. d): The analysis of the analysis operator. J. Funct. Anal. 148:(2):1997;408-447.
29. d): Dual systems. J. Fourier Anal. Appl. 3:1997;617-637.
30. Sablonnière P., Sbibih D. Spline integral operators exact on polynomials. Approx. Theory Appl. 10:1994;56-73.
31. Schoenberg I.J., Curry H.B. On Pólya frequency functions IV: The fundamental spline functions and their limits. J. Anal. Math. 17:1966;71-107.
32. Schumaker L.L. Spline Functions: Basic Theory. 1981;Wiley, New York.
33. 2 -spline projector is bounded independently of the knot sequence: A proof of de Boor's conjecture. Acta Math. 187:2001;59-137.
34. Wu D.B. Dual bases of a Bernstein polynomial basis on simplices. Comput. Aided Geom. Design. 10:1993;483-489.