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Democracy functions and optimal embeddings for approximation spaces

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Abstract

We prove optimal embeddings for nonlinear approximation spaces \(\mathcal{A}^{\alpha}_q\), in terms of weighted Lorentz sequence spaces, with the weights depending on the democracy functions of the basis. As applications we recover known embeddings for N-term wavelet approximation in L p, Orlicz, and Lorentz norms. We also study the “greedy classes” \({\mathcal{G}_{q}^{\alpha}}\) introduced by Gribonval and Nielsen, obtaining new counterexamples which show that \({\mathcal{G}_{q}^{\alpha}}\not=\mathcal{A}^{\alpha}_q\) for most non-democratic unconditional bases.

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References

  1. Aimar, H.A., Bernardis, A.L., Martín-Reyes, F.J.: Multiresolution approximation and wavelet bases of weighted Lebesgue spaces. J. Fourier Anal. Appl. 9(5), 497–510 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bennett, C., Sharpley, R.: Interpolation of operators. Academic Press Inc (1988)

  3. Bergh, J., Löfström, J.: Interpolation spaces. An introduction, no. 223. Springer-Verlag, New York (1976)

    Book  MATH  Google Scholar 

  4. Carro, M.J., Raposo, J., Soria, J.: Recent developments in the theory of lorentz spaces and weighted inequalities. Mem. Am. Math. Soc. 877, 187 (2007)

    MathSciNet  Google Scholar 

  5. DeVore, R.A.: Nonlinear approximation. Acta Numer. 7, 51–150 (1998)

    Article  MathSciNet  Google Scholar 

  6. DeVore, R., Konyagin, S., Temlyakov, V.: Hyperbolic wavelet approximation. Constr. Approx. 14, 1–26 (1998)

    Article  MathSciNet  Google Scholar 

  7. DeVore, R., Popov, V.A.: Interpolation spaces and nonlinear approximation. Function Spaces and Applications (Lund, 1986). Lecture Notes in Math., vol. 1302. Springer, Berlin, pp. 191–205 (1988)

    Google Scholar 

  8. DeVore, R.A., Temlyakov, V.N.: Some remarks on greedy algorithms. Adv. Comput. Math. 5(2–3), 113–187 (1996)

    MathSciNet  Google Scholar 

  9. Dilworth, S.J., Kalton, N.J., Kutzarova, D., Temlyakov, V.N.: The thresholding greedy algorithm, greedy bases, and duality. Constr. Approx. 19, 575–597 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  10. Frazier, M., Jawerth, B.: A discrete transform and decomposition of distribution spaces. J. Funct. Anal. 93, 34–170 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. Garrigós, G., Hernández, E.: Sharp Jackson and Bernstein inequalities for n-term approximation in sequence spaces with applications. Indiana Univ. Math. J. 53, 1739–1762 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Garrigós, G., Hernández, E., Martell, J.M.: Wavelets, Orlicz spaces and greedy bases. Appl. Comput. Harmon. Anal. 24, 70–93 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gribonval, R., Nielsen, M.: Some remarks on non-linear approximation with Schauder bases. East. J. Approx. 7(2), 1–19 (2001)

    MathSciNet  Google Scholar 

  14. Hernández, E., Martell, J.M., de Natividade, M.: Quantifying democracy of wavelet bases in Lorentz spaces. Constr. Approx. 33, 1–14 (2011). doi:10.1007/s00365-010-9113-8

    Article  MathSciNet  MATH  Google Scholar 

  15. Hernández, E., Weiss, G.: A first course on wavelets. CRC Press, Boca Raton, FL (1996)

    Book  MATH  Google Scholar 

  16. Hsiao, C., Jawerth, B., Lucier, B.J., Yu, X.M.: Near optimal compression of almost optimal wavelet expansions. Wavelets: Mathemathics and Applications, Stud. Adv. Math., vol. 133, pp. 425–446. CRC, Boca Raton, FL (1994)

    Google Scholar 

  17. Jawerth, B., Milman, M.: Wavelets and best approximation in Besov spaces. In: Interpolation Spaces and Related Topics (Haifa, 1990), pp. 107–112, Israel Math. Conf. Proc., vol. 5. Bar-Ilan University, Ramat Gan (1992)

    Google Scholar 

  18. Jawerth, B., Milman, M.: Weakly rearrangement invariant spaces and approximation by largest elements. In: Interpolation Theory and Applications, pp. 103–110, Contemp. Math., vol. 445. Amer. Math. Soc., Providence, RI (2007)

    Chapter  Google Scholar 

  19. Kamont, A., Temlyakov, V.N.: Greedy approximation and the multivariate Haar system. Stud. Math. 161(3), 199–223 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kerkyacharian, G., Picard, D.: Entropy, universal coding, approximation, and bases properties. Constr. Approx. 20, 1–37 (2004). doi:10.1007/s00365-003-0556-z

    Article  MathSciNet  MATH  Google Scholar 

  21. Kerkyacharian, G., Picard, D.: Nonlinear approximation and muckenhoupt weights. Constr. Approx. 24, 123–156 (2006). doi:10.1007/s00365-005-0618-5

    Article  MathSciNet  MATH  Google Scholar 

  22. Kyriazis, G. Multilevel characterization of anisotropic function spaces. SIAM J. Math. Anal. 36, 441–462 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Konyagin, S.V., Temlyakov, V.N.: A remark on greedy approximation in Banach spaces. East. J. Approx. 5, 365–379 (1999)

    MathSciNet  MATH  Google Scholar 

  24. Krein, S., Petunin, J., Semenov, E.: Interpolation of Linear Operators. Translations Math. Monographs, vol. 55. Amer. Math. Soc., Providence (1992)

    Google Scholar 

  25. Merucci, C.: Applications of interpolation with a function parameter to Lorentz, Sobolev and Besov spaces. Interpolation Spaces and Allied Topics in Analysis. Lecture Notes in Math., vol. 1070, pp. 183–201. Springer, Berlin (1984)

    Google Scholar 

  26. Meyer, Y.: Ondelettes et Operateurs. I: Ondelettes. Hermann, Paris, (1990). [English translation: Wavelets and Operators. Cambridge University Press (1992)]

  27. de Natividade, M.: Best approximation with wavelets in weighted Orlicz spaces. Monatsh. Math. 164, 87–114 (2011). doi:10.1007/s00605-010-0244-6

    Article  MathSciNet  MATH  Google Scholar 

  28. Oswald, P.: Greedy algorithms and best m-term approximation with respect to biorthogonal systems. J. Fourier Anal. Appl. 7(4), 325–341 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  29. Pietsch, A.: Approximation spaces. J. Approx. Theory 32, 113–134 (1981)

    Article  MathSciNet  Google Scholar 

  30. Rochberg, R., Taibleson, M.: An averaging operator on a tree. In: Harmonic analysis and partial differential equations (El Escorial, 1987), pp. 207–213, Lecture Notes in Math., vol. 1384. Springer, Berlin (1989)

    Chapter  Google Scholar 

  31. Stechkin, S.B.: On absolute convergence of orthogonal series. Dokl. Akad. Nauk SSSR 102, 37–40 (1955)

    MathSciNet  MATH  Google Scholar 

  32. Soardi, P.: Wavelet bases in rearrangement invariant function spaces. Proc. Am. Math. Soc. 125(12), 3669–3973 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  33. Temlyakov, V.N.: The best m-term approximation and greedy algorithms. Adv. Comput. Math. 8, 249–265 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  34. Temlyakov, V.N.: Nonlinear m-term approximation with regard to the multivariate Haar system. East. J. Approx. 4, 87–106 (1998)

    MathSciNet  MATH  Google Scholar 

  35. Temlyakov, V.N.: Nonlinear methods of approximation. Found. Comput. Math. 3(1), 33–107 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. Temlyakov, V.N.: Greedy approximation. Acta Numer., vol. 17, pp. 235–409, Cambridge University Press (2008)

  37. Wojstaszczyk, P.: The Franklin system is an unconditional basis in H 1. Ark. Mat. 20, 293–300 (1982)

    Article  MathSciNet  Google Scholar 

  38. Wojstaszczyk, P.: Greedy algorithm for general biorthogonal systems. J. Approx. Theory 107, 293–314 (2000)

    Article  MathSciNet  Google Scholar 

  39. Wojstaszczyk, P.: Greediness of the Haar system in rearrangement invariant spaces. Banach Cent. Publ., Warszawa 72, 385–395 (2006)

    Article  Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Gustavo Garrigós, Eugenio Hernández & Maria de Natividade

Authors
  1. Gustavo Garrigós
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  2. Eugenio Hernández
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  3. Maria de Natividade
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Correspondence to Eugenio Hernández.

Additional information

Communicated by Volodya Temlyakov.

Research supported by Grants MTM2007-60952 and MTM2010-16518 (Spain). Research of M. de Natividade supported by Instituto Nacional de Bolsas de Estudos de Angola, INABE.

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Garrigós, G., Hernández, E. & de Natividade, M. Democracy functions and optimal embeddings for approximation spaces. Adv Comput Math 37, 255–283 (2012). https://doi.org/10.1007/s10444-011-9197-0

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