Skip to main content
SpringerLink
Log in

Greedy approximation with regard to non-greedy bases

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

The main goal of this paper is to understand which properties of a basis are important for certain direct and inverse theorems in nonlinear approximation. We study greedy approximation with regard to bases with different properties. We consider bases that are tensor products of univariate greedy bases. Some results known for unconditional bases are extended to the case of quasi-greedy bases.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Cohen, A., DeVore, R.A., Hochmuth, R.: Restricted nonlinear approximation. Constr. Approx. 16, 85–113 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. DeVore, R.A.: Nonlinear approximation. In: Acta Numerica, vol. 7, pp. 51–150. Cambridge University Press (1998)

  3. DeVore, R.A., Temlyakov, V.N.: Nonlinear approximation by trigometric sums. J. Fourier Anal. Appl. 2(1), 29–48 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. 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 

  7. Kerkyacharian, G., Picard, D.: Entropy, universal coding, approximation, and bases properties. Constr. Approx. 20, 1–37 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Kerkyacharian, G., Picard, D., Temlyakov, V.N.: Some inequalities for the tensor product of greedy basis and weight-greedy basis. East. J. Approx. 4, 103–118 (2006)

    MathSciNet  Google Scholar 

  9. Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces I. Springer, Berlin (1977)

    MATH  Google Scholar 

  10. Smela, K.: Subsequences of the Haar basis consisting of full levels in H p for 0 < p < ∞. Proc. Am. Math. Soc. 135 1709–1716 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Stechkin, S.B.: On absolute convergence of orthogonal series. Kokl. Akad. Nauk SSSR 102, 37–40 (1995) (in Russian)

    Google Scholar 

  12. Temlyakov, V.N.: Greedy algorithm and m-term trigonometric approximation. Constr. Approx. 14, 569–587 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  15. Temlyakov, V.N.: Nonlinear approximation with regard to bases. In: Approximation Theory, vol. X, pp. 373–402. Vanderbilt University Press, Nashville, TN (2002)

    Google Scholar 

  16. Temlyakov, V.N.: Universal bases and greedy algorithms for anisotropic function classes. Constr. Approx. 18, 529–550 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

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

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    V. N. Temlyakov & Mingrui Yang

  2. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, China

    Peixin Ye

Authors
  1. V. N. Temlyakov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mingrui Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Peixin Ye
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mingrui Yang.

Additional information

Communicated by Qiyu Sun.

The research of P. Ye was supported by the Natural Science Foundation of China (Grant No. 10971251).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Temlyakov, V.N., Yang, M. & Ye, P. Greedy approximation with regard to non-greedy bases. Adv Comput Math 34, 319–337 (2011). https://doi.org/10.1007/s10444-010-9155-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-010-9155-2

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation