Revista Matemática Complutense

, Volume 30, Issue 2, pp 369–392 | Cite as

Lebesgue inequalities for the greedy algorithm in general bases

  • Pablo M. Berná
  • Óscar Blasco
  • Gustavo GarrigósEmail author


We present various estimates for the Lebesgue type inequalities associated with the thresholding greedy algorithm, in the case of general bases in Banach spaces. We show the optimality of the involved constants in some situations. Our results recover and slightly improve various estimates appearing earlier in the literature.


Thresholding greedy algorithm Quasi-greedy basis Conditional basis 

Mathematics Subject Classification

41A65 41A46 46B15 



we wish to thank F. Albiac, J.L. Ansorena and E. Hernández for many useful conversations about these topics. We also thank the anonymous referee for the careful reading and useful comments.


  1. 1.
    Albiac, F., Ansorena, J.L.: Characterization of 1-almost greedy bases. Rev. Mat. Complut. 30(1), 13–24 (2017). doi: 10.1007/s13163-016-0204-3
  2. 2.
    Albiac, F., Wojtaszczyk, P.: Characterization of 1-greedy bases. J. Approx. Theory 138, 65–86 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dilworth, S.J., Kalton, N.J., Kutzarova, D.: On the existence of almost greedy bases in Banach spaces. Studia Math. 159(1), 67–101 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dilworth, S.J., Kutzarova, D., Odell, E., Schlumprecht, Th, Zsák, A.: Renorming spaces with greedy bases. J. Approx. Theory 188, 39–56 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dilworth, S.J., Kutzarova, D., Oikhberg, T.: Lebesgue constants for the weak greedy algorithm. Rev. Mat. Complut. 28(2), 393–409 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dilworth, S.J., Soto-Bajo, M., Temlyakov, V.N.: Quasi-greedy bases and Lebesgue-type inequalities. Studia Math 211, 41–69 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Garrigós, G., Hernández, E., Oikhberg, T.: Lebesgue-type inequalities for quasi-greedy bases. Constr. Approx. 38(3), 447–470 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Garrigós, G., Wojtaszczyk, P.: Conditional quasi-greedy bases in Hilbert and Banach spaces. Indiana Univ. Math. J. 63(4), 1017–1036 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hajek, P., Montesinos Santalucía, V., Vanderwerff, J., Zizler, V.: Biorthogonal Systems in Banach Spaces. Springer-Verlag, Berlin (2008)zbMATHGoogle Scholar
  12. 12.
    Hsiao, C., Jawerth, B., Lucier, B.J., Yu, X.M.: Near optimal compression of almost optimal wavelet expansions. Wavelet: mathematics and applications, Stud. Adv. Math., CRC, Boca Raton, FL, (133), 425–446 (1994)Google Scholar
  13. 13.
    Konyagin, S.V.: On the Littlewood problem. Izv. Akad. Nauk. USSR 45, 243–265 (1981)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Konyagin, S.V., Temlyakov, V.N.: A remark on greedy approximation in Banach spaces. East J. Approx. 5, 365–379 (1999)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces, vol. I. Springer-Verlag, Berlin (1977)CrossRefzbMATHGoogle Scholar
  16. 16.
    McGehee, O.C., Pigno, L., Smith, B.: Hardy’s inequality and \(L^1\) norm of exponential sums. Ann. Math. 113(3), 613–618 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Oswald, P.: Greedy algorithms and best m-term approximation with respect to biorthogonal systems. J. Fourier Anal. Appl. 7, 325–341 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Stein, E.M., Shakarchi, R.: Fourier Analysis: An Introdution. Princeton University Press, Princeton (2003)zbMATHGoogle Scholar
  19. 19.
    Temlyakov, V.N.: Greedy algorithm and m-term trigonometric approximation. Const. Approx. 14, 569–587 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Temlyakov, V.N.: Nonlinear \(m\)-term approximation with regard to the multivariate Haar system. East J. Approx. 4, 87–106 (1998)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Temlyakov, V.N.: Greedy Approximation. Cambridge University Press, Cambridge (2011)CrossRefzbMATHGoogle Scholar
  22. 22.
    Temlyakov, V.N.: Sparse approximation with bases. In: Tikhonov, S. (ed.) Advanced Courses in Mathematics - CRM Barcelona. Birkhäuser-Springer (2015)Google Scholar
  23. 23.
    Temlyakov, V.N., Yang, M., Ye, P.: Lebesgue-type inequalities for greedy approximation with respect to quasi-greedy bases. East J. Approx. 17, 127–138 (2011)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Wojtaszczyk, P.: Greedy algorithm for general biorthogonal systems. J. Approx. Theory 107, 293–314 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Wojtaszczyk, P.: Greedy type bases in Banach spaces. In: Bojanov, B. (ed.) Constructive Theory of Functions, Varna 2002, pp. 136–155. DARBA, Sofia (2003)Google Scholar

Copyright information

© Universidad Complutense de Madrid 2017

Authors and Affiliations

  • Pablo M. Berná
    • 1
  • Óscar Blasco
    • 2
  • Gustavo Garrigós
    • 3
    Email author
  1. 1.Instituto Universitario de Matemática Pura y AplicadaUniversitat Politècnica de ValènciaValenciaSpain
  2. 2.Departamento de Análisis MatemáticoUniversidad de ValenciaValenciaSpain
  3. 3.Departamento de MatemáticasUniversidad de MurciaMurciaSpain

Personalised recommendations