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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
Article
  • 224 Downloads

Abstract

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 \(\lambda \).

Keywords

Markov type inequalities Gegenbauer polynomials Matrix norms 

Mathematics Subject Classification

41A17 

Notes

Acknowledgements

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.

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Copyright information

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Authors and Affiliations

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

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