Skip to main content
SpringerLink
Log in

Lower Bounds for n-Term Approximations of Plane Convex Sets and Related Topics

  • Published:
Mathematical Notes Aims and scope Submit manuscript

Abstract

In this paper, we establish lower bounds for n-term approximations in the metric of L 2(I2) of characteristic functions of plane convex subsets of the square I 2 with respect to arbitrary orthogonal systems. It is shown that, as n→∞, these bounds cannot decrease more rapidly than \(1/n\).

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. R. A. De Vore, “Nonlinear approximation,” Acta Numerica, 7 (1998), 51-150.

    Google Scholar 

  2. V. N. Temlyakov, Nonlinear Methods of Approximations, IMI Preprint 2001:09, University of South Carolina, 2001.

  3. B. S. Kashin, “On lower estimates for n-term approximation in Hilbert space,” in: Approximation Theory. Volume Dedicated to B. Sendov, Darba, Sofia, 2002, pp. 241-257.

    Google Scholar 

  4. B. S. Kashin, “On the approximation properties of complete orthonormal systems,” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 172 (1985), 187-191.

    Google Scholar 

  5. A. Cohen, R. De Vore, P. Petrushev, and H. Xu, “Nonlinear approximation and the space BV (ℝ2),” Amer. J. Math., 121 (1999), 587-628.

    Google Scholar 

  6. D. L. Donoho, “Sparse components of images and optimal atomic decompositions,” Constr. Approx., 17 (2001), no. 3, 353-382.

    Google Scholar 

  7. I. Daubechies, Ten Lectures on Wavelets [Russian translation], RKhD, Moscow-Izhevsk, 2001.

    Google Scholar 

  8. B. S. Kashin and T. Yu. Kulikova, “A note on the approximation properties of frames of general form,” Mat. Zametki [Math. Notes], 72 (2002), no. 2, 312-315.

    Google Scholar 

  9. B. S. Kashin and A. A. Saakyan, Orthogonal Series [in Russian], 2nd edition, AFTs, Moscow, 1999.

    Google Scholar 

  10. R. P. Agnew, “On double orthogonal series,” Proc. London Math. Soc. Ser. 2, 33 (1932), no. 6, 420-434.

    Google Scholar 

  11. F. Moricz, “Multiparameter strong laws of large numbers. I,” Acta Sci. Math., 40 (1978), 143-156.

    Google Scholar 

  12. B. S. Kashin, “On the coefficients of the expansion of a class of functions in complete systems,” Sibirsk. Mat. Zh. [Siberian Math. J.], 18 (1977), no. 1, 122-131.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. V. A. Steklov Mathematics Institute, Russian Academy of Sciences, Russia

    B. S. Kashin

Authors
  1. B. S. Kashin
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kashin, B.S. Lower Bounds for n-Term Approximations of Plane Convex Sets and Related Topics. Mathematical Notes 72, 473–478 (2002). https://doi.org/10.1023/A:1020528210992

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1020528210992

Search

Navigation