Entropy, Universal Coding, Approximation, and Bases Properties

Constructive Approximation

Volumn 20, Issue 1, 2004, Pages 1-37

  Kerkyacharian, Gérard ^a,b   Picard, Dominique ^b  

^a UNIVERSITÉ PARIS OUEST NANTERRE LA DÉFENSE   (France)

^b SORBONNE UNIVERSITÉ   (France)

Author keywords

Coding; Entropy; Linear and nonlinear approximation; m Term approximation; Quasi greedy bases; Unconditional bases; Wavelet bases

Indexed keywords

EID: 1142300685     PISSN: 01764276     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00365-003-0556-z     Document Type: Article

References (41)

1. p. Mat. Sb., 73:295-317.
2. C. B. CARL (1981): Entropy numbers, s-numbers, and eigenvalue problems. J. Funct. Anal., 41:290-306.
3. A. COHEN (2000): Wavelets in numerical analysis. In: Handbook of Numerical Analysis, Vol. VII (P.-G. Ciarlet and J.-L. Lions, eds.). Amsterdam: Elsevier.
4. A. COHEN, I. DAUBECHIES, P. VIAL (1993): Wavelet bases on the interval and fast algorithms. Appl. Comput. Harmonic Anal. 1:54-81.
5. A. COHEN, R. DEVORE, R. HOCHMUTH (2000): Restricted nonlinear approximation. Constr. Approx., 16:85-113.
6. A. COHEN, W. DAHMEN, I. DAUBECHIES, R. DEVORE (2001): Tree approximation and encoding. Appl. Comput. Harmonic Anal., 11:192-226.
7. A. COHEN, R. DEVORE, G. KERKYACHARIAN, D. PICARD (2001): Saturation spaces associated with nonlinear statistical methods. Appl. Comput. Harmonic Anal., 11:167-191.
8. R. DEVORE (1998): Nonlinear approximation. Acta Numerica, 7:51-150.
9. R. DEVORE, B. JAWERTH, V. A. POPOV (1992): Compression of wavelet decomposition. Amer. J. Math., 114:737-785.
10. R. DEVORE, G. LORENTZ (1993): Constructive Approximation. Grundlehren, Vol. 303. Heidelberg: Springer-Verlag.
11. R. DEVORE, V. A. POPOV (1988): Interpolation space and non-linear approximation. In: Function Space and Applications (M. Cwikel et al., eds.). Lecture Notes in Mathematics, Vol. 1302. Berlin: Springer-Verlag, pp. 191-205.
12. R. DEVORE, S. V. KONYAGIN, V. N. TEMLYAKOV (1998): Hyperbolic wavelet approximation. Constr. Approx., 14:1-26.
13. S. J. DILWORTH, N. J. KALTON, D. KUTZAROVA, V. N. TEMLYAKOV (2001): The thresholding greedy algorithm, greedy bases, and duality. IMI Preprint series, 01:23.
14. D. L. DONOHO (1993): Unconditional bases are optimal bases for data compression and statistical estimation. Appl. Comput. Harmonic Anal., 1:100-115.
15. D. L. DONOHO (1996): Unconditional bases and bit-level compression. Appl. Comput. Harmonic Anal., 3:388-392.
16. D. FEFFERMAN, E. STEIN (1971): Some maximal inequalities. Amer. J. Math., 93:107-115.
17. M. FRAZIER, B. JAWERTH (1990): A discrete transform and decomposition of distribution spaces. J. Funct. Anal., 93:34-170.
18. R. GRIBONVAL, M. NIELSEN (2001): Some remarks on nonlinear approximation with Schauder bases. Preprint IMI. University of South Carolina.
19. W. HOEFFDING (1963): Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc., 58:13-30.
20. Y. KATZNELSON (1968): An Introduction to Harmonic Analysis. New York: Wiley.
21. J. P. KAHANE, K. DE LEEUW, Y. KATZNELSON (1977): Sur les coefficients de Fourier des fonctions continues. C. R. Acad. Sci. Paris Sér. A, 285:1001-1003.
22. G. KERKYACHARIAN, D. PICARD (2000): Thresholding algorithms, maxisets and well-concentrated bases, with Discussion. Test, 9:283-345.
23. G. KERKYACHARIAN, D. PICARD (2001): Replicant compression coding in Besov spaces. Preprint. Laboratoire de probabilités et Modèles aléatoires. Universités Paris 6-Paris 7.
24. A. N. KOLMOGOROV, V. M. TIKHOMIROV (1959): ε-entropy and ε-capacity. Uspekhi Mat. Nauk, 14:3-86. (English transl.: Amer. Math. Soc. Transl., Ser. 2, Vol. 17, pp. 277-364.)
25. A. N. KOLMOGOROV, V. M. TIKHOMIROV (1959): ε-entropy and ε-capacity. Uspekhi Mat. Nauk, 14:3-86. (English transl.: Amer. Math. Soc. Transl., Ser. 2, Vol. 17, pp. 277-364.)
26. S. V. KONYAGIN, V. N. TEMLYAKOV (1999): A remark on greedy approximation in Banach spaces. East J. Approx., 5:1-15.
27. J. LINDENSTRAUSS, L. TZAFIRI (1977): Classical Banach spaces, Vol. 1. New York: Springer-Verlag.
28. G. G. LORENTZ (1950): Some new functional spaces. Ann. of Math., 51:37-55.
29. G. O. LORENTZ (1966): Metric entropy and approximation. Bull. Amer. Math. Soc., 72:903-937.
30. G. G. LORENTZ, M. VON GOLITSHEK, J. MAKOVOZ (1996): Constructive Approximation: Advanced Problems. New York: Springer-Verlag.
31. S. MALLAT (1998): A Wavelet Tour of Signal Processing. New York: Academic Press.
32. Y. MEYER (1990): Ondelettes et Opérateurs. Paris: Hermann.
33. J. SCHAUDER (1927): Zur Theorie stetiger Abbildungen in Funktionalräumen. Math. Z., 26:47-65.
34. E. SCHMIDT (1907): Zur Theorie der linearen Integralgleichungen. Math. Ann., 63:433-476.
35. S. B. STECHKIN (1955): On absolute convergence of orthogonal series. Dokl, 102:37-40.
36. N. V. TEMLYAKOV (1998): Nonlinear Kolmogorov's width. Math. Zametki, 63:891-902, 4:87-106.
37. N. V. TEMLYAKOV (1998): Nonlinear Kolmogorov's width. Math. Zametki, 63:891-902, 4:87-106.
38. N. V. TEMLYAKOV (1998): The best m-term approximation and greedy algorithms. Adv. in Comput. Math., 8:249-265.
39. N. V. TEMLYAKOV (1998): Nonlinear m-term approximation with regard to the multivariate Hoar system. East J. Approx. Theory, 4:87-106.
40. N. V. TEMLYAKOV (2001): Nonlinear methods of approximation. IMI Preprint series, 9, 01:57.
41. P. WOJTASZCZYK (2000): Greedy algorithm for general biorthogonal system. J. Approx. Theory, 107:293-314.