On Approximation with Spline Generated Framelets

Constructive Approximation

Volumn 20, Issue 2, 2004, Pages 207-232

  Gribonval, Rémi ^a   Nielsen, Morten ^b  

^a CAMPUS DE BEAULIEU   (France)

^b AALBORG UNIVERSITY   (Denmark)

Author keywords

Adaptive sparse expansions; Besov spaces; Biorthogonal spline wavelet; Direct and inverse theorems; Framelets; Nonlinear approximation; Redundant system; Spline multiresolution; Wavelet frame

Indexed keywords

EID: 1842715640     PISSN: 01764276     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00365-003-0532-7     Document Type: Article

References (28)

1. C. BENNETT, R. SHARPLEY (1988): Interpolation of Operators. Boston, MA: Academic Press.
2. P. CASAZZA (1998): Every frame is a sum of three (but not two) orthonormal bases - and other frame representations. J. Fourier Anal. Appl., 4(6):727-732.
3. S. CHEN, D. L. DONOHO (1999): Atomic decomposition by basis pursuit. SIAM J. Sci. Comput., 20(1):33-61.
4. C. K. CHUI (1992): An Introduction to Wavelets. Boston, MA: Academic Press.
5. C. K. CHUI (1992): On cardinal spline-wavelets. In: Wavelets and Their Applications. Boston, MA: Jones and Bartlett, pp. 419-438.
6. C. K. CHUI, J.-Z. WANG (1992): On compactly supported spline wavelets and a duality principle. Trans. Amer. Math. Soc., 330(2):903-915.
7. A. COHEN, W. DAHMEN, R. DEVORE (2000): Multiscale decompositions on bounded domains. Trans. Amer. Math. Soc., 352(8):3651-3685.
8. A. COHEN (1992): Biorthogonal wavelets. In: Wavelets: A Tutorial in Theory and Applications, Vol. 2 (C. K. Chui, ed.). Boston, MA: Academic Press, pp. 123-152.
9. I. DAUBECHIES (1992): Ten Lectures on Wavelets. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM).
10. I. DAUBECHIES, B. HAN, A. RON, Z. SHEN (2001): Framelets: MRA-based constructions of wavelet frames. Preprint.
11. G. DAVIS, S. MALLAT, M. AVELLANEDA (1997): Adaptive greedy approximations. Constr. Approx., 13(1):57-98.
12. R. DE VORE, B. JAWERTH, B. J. LUCIER (1992): Image compression through wavelet transform coding. IEEE Trans. Inform. Theory, 38:719-746.
13. R. A. DEVORE, S. V. KONYAGIN, V. N. TEMLYAKOV (1998): Hyperbolic wavelet approximation. Constr. Approx., 14(1): 1-26.
14. R. A. DEVORE (1998): Nonlinear approximation. In: Acta Numerica, 1998. Cambridge: Cambridge University Press, pp. 51-150.
15. R. A. DEVORE, B. JAWERTH, V. POPOV (1992): Compression of wavelet decompositions. Amer. J. Math., 114(4):737-785.
16. R. A. DEVORE, G. G. LORENTZ (1993): Constructive Approximation. Berlin: Springer-Verlag.
17. M. FRAZIER, B. JAWERTH (1985): Decomposition of Besov spaces. Indiana Univ. Math. J., 34(4):777-799.
18. R. GRIBONVAL, M. NIELSEN (to appear): Nonlinear approximation with dictionaries. I. Direct estimates. J. Fourier Anal. Appl.
19. L. K. JONES (1997): The computational intractability of training sigmoidal neural networks. IEEE Trans. Inform. Theory, 43(1):167-173.
20. Y. MEYER (1990): Ondelettes et opérateurs. H. Paris: Hermann. Opérateurs de Calderón-Zygmund. [Calderón-Zygmund operators].
21. Y. MEYER (1992): Wavelets and Operators. Cambridge: Cambridge University Press. Translated from the 1990 French original by D. H. Salinger.
22. P. PETRUSHEV (2000): Bases consisting of rational functions of uniformly bounded degrees or more general functions. J. Funct. Anal., 174(1):18-75.
23. P. P. PETRUSHEV (1988) : Direct and converse theorems for spline and rational approximation and Besov spaces. In: Function Spaces and Applications (Lund, 1986), Berlin: Springer-Verlag, pp. 363-377.
24. d). II. Dual systems. J. Fourier Anal. Appl., 3(5):617-637. Dedicated to the memory of Richard J. Duffin.
25. d): The analysis of the analysis operator. J. Funct. Anal., 148(2):408-447.
26. A. RON, Z. SHEN (2001): The wavelet dimension function is the trace of a shift-invariant system. Preprint.
27. H. TRIEBEL (2001): The Structure of Functions. Basel: Birkhäuser Verlag.
28. 2(R). J. Fourier Anal. Appl., 5(1):73-85.