Skip to main content
SpringerLink
Log in

Jacobi translation and the inequality of different metrics for algebraic polynomials on an interval

  • Mathematics
  • Published:
Doklady Mathematics Aims and scope Submit manuscript

Abstract

The sharp inequality of different metrics (Nikol’skii’s inequality) for algebraic polynomials in the interval [−1, 1] between the uniform norm and the norm of the space L (α,β) q , 1 ≤ q < ∞, with Jacobi weight ϕ(α,β)(x) = (1 − x)α(1 + x)β α ≥ β > −1, is investigated. The study uses the generalized translation operator generated by the Jacobi weight. A set of functions is described for which the norm of this operator in the space L (α,β) q , 1 ≤ q < ∞, \(\alpha > \beta \geqslant - \frac{1}{2}\), is attained.

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. M. Nikol’skii, Tr. Mat. Inst. Akad. Nauk SSSR 38, 244–278 (1951).

    Google Scholar 

  2. G. Szego and A. Zygmund, J. Anal. Math 3 (1), 225–244 (1953).

    Article  Google Scholar 

  3. G. V. Milovanovic, D. S. Mitrinovic, and Th. M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros (World Sci., Singapore, 1994).

    Book  MATH  Google Scholar 

  4. V. Arestov and M. Deikalova, Comput. Methods Funct. Theory 15 (4), 689–708 (2015).

    Article  MathSciNet  Google Scholar 

  5. A. Lupas, Univ. Beograd. Publ. Electrotehn. Fak. Sep. Mat. Fiz. 461/497, 241–243 (1974).

    Google Scholar 

  6. V. Babenko, V. Kofanov, and S. Pichugov, in Approximation Theory: A Volume Dedicated to Blagovest Sendov, Ed. by B. Bojanov (DARBA, Sofia, 2002), pp. 24–53.

  7. D. V. Gorbachev, Proc. Steklov Inst. Math., Suppl. no. 2, S117–S138 (2004).

    Google Scholar 

  8. I. E. Simonov and P. Yu. Glazyrina, J. Approx. Theory 192, 69–81 (2015).

    Article  MathSciNet  Google Scholar 

  9. V. M. Tikhomirov, Certain Questions of Approximation Theory (Mosk. Gos. Univ., Moscow, 1976) [in Russian].

    Google Scholar 

  10. V. V. Arestov and M. V. Deikalova, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk 19 (2), 34–47 (2013).

    Google Scholar 

  11. G. Szego, Orthogonal Polynomials (Am. Math. Soc., New York, 1959; Inostrannaya Literatura, Moscow, 1961).

    MATH  Google Scholar 

  12. G. N. Watson, Treatise on the Theory of Bessel Functions (Cambridge Univ. Press, Cambridge, 1944; Inostrannaya Literatura, Moscow, 1949), Part1.

    MATH  Google Scholar 

  13. G. Gasper, Ann. Math. Ser. 2 93 (1), 112–118 (1971).

    Article  Google Scholar 

  14. H. Bavinck, J. Math. Anal. Appl. 37, 767–797 (1972).

    Article  MathSciNet  Google Scholar 

  15. T. Koornwinder, SIAM J. Math. Anal. 5, 125–137 (1974).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics and Computer Science, Ural Federal University, Yekaterinburg, 620000, Russia

    V. V. Arestov & M. V. Deikalova

  2. Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences, ul. S. Kovalevskoi 16, Yekaterinburg, 620990, Russia

    V. V. Arestov & M. V. Deikalova

Authors
  1. V. V. Arestov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. V. Deikalova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. V. Arestov.

Additional information

Original Russian Text © V.V. Arestov, M.V. Deikalova, 2017, published in Doklady Akademii Nauk, 2017, Vol. 472, No. 3, pp. 243–247.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arestov, V.V., Deikalova, M.V. Jacobi translation and the inequality of different metrics for algebraic polynomials on an interval. Dokl. Math. 95, 21–25 (2017). https://doi.org/10.1134/S1064562417010100

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1064562417010100

Search

Navigation