Skip to main content
SpringerLink
Log in

A weak-type inequality for uniformly bounded trigonometric polynomials

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

This paper is devoted to refining the Bernstein inequality. Let D be the differentiation operator. The action of the operator Λ = D/n on the set of trigonometric polynomials T n is studied: the best constant is sought in the inequality between the measures of the sets {x ∈ T: |Λt(x)| > 1} and {x ∈ T: |t(x)| > 1}. We obtain an upper estimate that is order sharp on the set of uniformly bounded trigonometric polynomials T C n = {t ∈ T n : ‖t‖ ≤ C}.

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. S. Bernstein, Sur l’ordre de la mailleure approximation des fonctions continues par des polynômes de degré donné (Hayez, Bruxelles, 1912), Mém. Acad. R. Belg., Cl. Sci., Ser. 2, 4.

    Google Scholar 

  2. S. N. Bernstein, “On the Best Approximation of Continuous Functions by Polynomials of Given Degree,” in Collected Works (Akad. Nauk SSSR, Moscow, 1952), Vol. 1, pp. 11–104 [in Russian].

    Google Scholar 

  3. V. V. Arestov, “Sharp Inequalities for Trigonometric Polynomials with Respect to Integral Functionals,” Tr. Inst. Mat. Mekh., Ural. Otd. Ross. Akad. Nauk 16(4), 38–53 (2010) [Proc. Steklov Inst. Math. 273 (Suppl. 1), S21–S36 (2011)].

    Google Scholar 

  4. A. G. Babenko, “Weak-Type Inequalities for Trigonometric Polynomials,” Tr. Inst. Mat. Mekh., Ural. Otd. Ross. Akad. Nauk 2, 34–41 (1992).

    MathSciNet  MATH  Google Scholar 

  5. V. V. Arestov and A. S. Mendelev, “Trigonometric Polynomials of Least Deviation from Zero in Measure and Related Problems,” J. Approx. Theory 162(10), 1852–1878 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Riesz, “Formule d’interpolation pour la dérivée d’un polynome trigonométrique,” C. R. Acad. Sci. Paris 158, 1152–1154 (1914).

    MATH  Google Scholar 

  7. M. Riesz, “Eine trigonometrische Interpolationsformel und einige Ungleichungen für Polynome,” Jahresber. Dtsch. Math.-Ver. 23, 354–368 (1914).

    MATH  Google Scholar 

  8. S. M. Nikol’skii, Approximation of Functions of Several Variables and Embedding Theorems (Nauka, Moscow, 1977); Engl. transl. of the 1st ed.: Approximation of Functions of Several Variables and Embedding Theorems (Springer, Berlin, 1975).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Evernote Corporation, ul. Barklaya 6, stroenie 5, Moscow, 121087, Russia

    E. D. Livshits

Authors
  1. E. D. Livshits
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Original Russian Text © E.D. Livshits, 2013, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2013, Vol. 280, pp. 215–226.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Livshits, E.D. A weak-type inequality for uniformly bounded trigonometric polynomials. Proc. Steklov Inst. Math. 280, 208–219 (2013). https://doi.org/10.1134/S0081543813010148

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543813010148

Keywords

Search

Navigation