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2014 On a Theorem by Bojanov and Naidenov Applied to Families of Gegenbauer-Sobolev Polynomials
V. G. Paschoa, D. Pérez, Y. Qintana
Commun. Math. Anal. 16(2): 9-18 (2014).

Abstract

Let $\{Q_{n,\lambda}^{(\alpha)}\}_{n\ge 0}$ be the sequence of monic orthogonal polynomials with respect the Gegenbauer-Sobolev inner product $$\langle f,g\rangle s:=\int_{-1}^1 f(x)g(x)(1-x^2)^{\alpha-\frac{1}{2}} dx+\lambda \int_{-1}^1 f'(x)g'(x)(1-x^2)^{\alpha-\frac{1}{2}}dx,$$ where $\alpha \gt -\frac{1}{2}$ and $\lambda \ge 0$. In this paper we use a recent result due to B.D. Bojanov and N. Naidenov [3], in order to study the maximization of a local extremum of the $k$th derivative $\frac{d^k}{dx^k}$ in $[-M_{n,\lambda},M_{n,\lambda}]$, where $M_{n,\lambda}$ is a suitable value such that all zeros of the polynomial $Q_{n,\lambda}^{(\alpha)}$ are contained in $[-M_{n,\lambda},M_{n,\lambda}]$ and the function $\left|Q_{n,\lambda}^{(\alpha)}\right|$ attains its maximal value at the end-points of such interval. Also, some illustrative numerical examples are presented.

Citation

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V. G. Paschoa. D. Pérez. Y. Qintana. "On a Theorem by Bojanov and Naidenov Applied to Families of Gegenbauer-Sobolev Polynomials." Commun. Math. Anal. 16 (2) 9 - 18, 2014.

Information

Published: 2014
First available in Project Euclid: 20 October 2014

zbMATH: 1321.33014
MathSciNet: MR3270574

Subjects:
Primary: 33C45 , 41A17

Keywords: extremal properties , Gegenbauer-Sobolev polynomials , orthogonal polynomials , oscillating polynomials , Sobolev orthogonal polynomials

Rights: Copyright © 2014 Mathematical Research Publishers

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