Constructive Approximation

, Volume 49, Issue 1, pp 1–27 | Cite as

On the Markov Inequality in the \(L_2\)-Norm with the Gegenbauer Weight

  • G. NikolovEmail author
  • A. Shadrin


Let \(w_\lambda (t) := (1-t^2)^{\lambda -1/2}\), where \(\lambda > -\,\frac{1}{2}\), be the Gegenbauer weight function; let \(\Vert \cdot \Vert _{w_\lambda }\) be the associated \(L_2\)-norm,
$$\begin{aligned} \Vert f\Vert _{w_\lambda } = \left\{ \int _{-1}^1 |f(x)|^2 w_\lambda (x)\,dx\right\} ^{1/2}; \end{aligned}$$
and denote by \(\mathcal{P}_n\) the space of algebraic polynomials of degree \(\le \,n\). We study the best constant \(c_n(\lambda )\) in the Markov inequality in this norm
$$\begin{aligned} \Vert p_n'\Vert _{w_\lambda } \le c_n(\lambda ) \Vert p_n\Vert _{w_\lambda },\qquad p_n \in \mathcal{P}_n, \end{aligned}$$
namely the constant
$$\begin{aligned} c_n(\lambda ) := \sup _{p_n \in \mathcal{P}_n} \frac{\Vert p_n'\Vert _{w_\lambda }}{\Vert p_n\Vert _{w_\lambda }}. \end{aligned}$$
We derive explicit lower and upper bounds for the Markov constant \(c_n(\lambda )\), which are valid for all n and \(\lambda \).


Markov type inequalities Gegenbauer polynomials Matrix norms 

Mathematics Subject Classification




This research was done during a three week stay of the authors in the Oberwolfach Mathematical Institute in April 2016, within the Research in Pairs Program. The authors thank the Institute for hospitality and the perfect research conditions. The research of the first-named author is supported by the Bulgarian National Research Fund through Contract DN 02/14, and by the Sofia University Research Fund under Contract 80-10-11. The second-named author is supported by a research grant from Pembroke College, Cambridge.


  1. 1.
    Aleksov, D., Nikolov, G., Shadrin, A.: On the Markov inequality in the \(L_2\) norm with the Gegenbauer weight. J. Approx. Theory 208, 9–20 (2016)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bojanov, B.: Markov-type inequalities for polynomials and splines. In: Chui, C.K., Schumaker, L.L., Stoeckler, J. (eds.) Approximation Theory X. Abstract and Classical Analysis, pp. 31–90. Vanderbilt University Press, Vanderbilt (2002)Google Scholar
  3. 3.
    Böttcher, A.: Best constants for Markov type inequalities in Hilbert space norms. In: Recent Trends in Analysis, Proceedings of the Conference in Honor of Nikolai Nikolski, Bordeaux 2011, pp. 73–83, Theta, Bucharest (2013)Google Scholar
  4. 4.
    Böttcher, A., Dörfler, P.: Weighted Markov-type inequalities, norms of Volterra operators, and zeros of Bessel functions. Math. Nachr. 283, 357–367 (2010)zbMATHGoogle Scholar
  5. 5.
    Böttcher, A., Dörfler, P.: On the best constant in Markov-type inequalities involving Gegenbauer norms with different weights. Oper. Matr. 161, 40–57 (2010)zbMATHGoogle Scholar
  6. 6.
    Böttcher, A., Dörfler, P.: On the best constant in Markov-type inequalities involving Laguerre norms with different weights. Monatsh. Math. 5, 261–272 (2011)zbMATHGoogle Scholar
  7. 7.
    Dörfler, P.: New inequalities of Markov type. SIAM J. Math. Anal. 18, 490–494 (1987)CrossRefzbMATHGoogle Scholar
  8. 8.
    Dörfler, P.: Asymptotics of the best constant in a certain Markov-type inequality. J. Approx. Theory 114, 84–97 (2002)CrossRefzbMATHGoogle Scholar
  9. 9.
    Konyagin, S.: Estimates for derivatives of polynomials. Dokl. Akad. Nauk SSSR 243, 1116–1118 (1978). (Russian)Google Scholar
  10. 10.
    Kroó, A.: On the exact constant in the \(L_2\) Markov inequality. J. Approx. Theory 151, 208–211 (2008)CrossRefzbMATHGoogle Scholar
  11. 11.
    Markov, A.A.: On a question of D.I. Mendeleev. Zapiski Petersb. Akad. Nauk 62, 1–24 (1889) (in Russian). Available also at:
  12. 12.
    Markov, V.A.: On functions which deviate least from zero in a given interval, Saint-Petersburg University (1892) (in Russian); German translation: Math. Ann. 77, 213–258 (1916). Available also at:
  13. 13.
    Milovanović, G.V., Mitrinović, D.S., Rassias, ThM: Topics in Polynomials: Extremal Problems, Inequalities, Zeros. World Scientific, Singapore (1994)CrossRefzbMATHGoogle Scholar
  14. 14.
    Nikolov, G.: Markov-type inequalities in the \(L_2\)-norms induced by the Tchebycheff weights. Arch. Inequal. Appl. 1(3–4), 361–375 (2003)zbMATHGoogle Scholar
  15. 15.
    Nikolov, G., Shadrin, A.: On the \(L_2\) Markov inequality with Laguerre weight. In: Govil, N.K., et al. (eds.) Progress in Approximation Theory and Applicable Complex Analysis. Springer Optimization and Its Applications, vol. 117, pp. 1–17. Springer, Cham (2017). CrossRefGoogle Scholar
  16. 16.
    Nikolov, G., Shadrin, A.: Markov \(L_2\)–inequality with the Laguerre weight. arXiv:1705.03824v1 [math.CA]
  17. 17.
    Rahman, Q.I., Schmeisser, G.: Analytic Theory of Polynomials. Clarendon Press, Oxford (2002)zbMATHGoogle Scholar
  18. 18.
    Schmidt, E.: Über die nebst ihren Ableitungen orthogonalen Polynomensysteme und das zugehörige Extremum. Math. Ann. 119, 165–204 (1944). (in German) CrossRefzbMATHGoogle Scholar
  19. 19.
    Shadrin, A.: Twelve proofs of the Markov inequality. In: Dimitrov, D.K., Nikolov, G., Uluchev, R. (eds.), Approximation Theory: A volume dedicated to Borislav Bojanov, pp. 233–298. Professor Marin Drinov Academic Publishing House, Sofia (2004). Available also at:
  20. 20.
    Turán, P.: Remark on a theorem of Ehrhard Schmidt. Mathematica (Cluj) 2, 373–378 (1960)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of Mathematics and InformaticsUniversity of SofiaSofiaBulgaria
  2. 2.Department of Applied Mathematics and Theoretical Physics (DAMTP)Cambridge UniversityCambridgeUK

Personalised recommendations