Communications in Mathematical Analysis

On a Theorem by Bojanov and Naidenov Applied to Families of Gegenbauer-Sobolev Polynomials

V. G. Paschoa, D. Pérez, and Y. Qintana

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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.

Article information

Source
Commun. Math. Anal., Volume 16, Number 2 (2014), 9-18.

Dates
First available in Project Euclid: 20 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.cma/1413810435

Mathematical Reviews number (MathSciNet)
MR3270574

Zentralblatt MATH identifier
1321.33014

Subjects
Primary: 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions] 41A17: Inequalities in approximation (Bernstein, Jackson, Nikol s kii-type inequalities)

Keywords
Orthogonal polynomials Sobolev orthogonal polynomials Gegenbauer-Sobolev polynomials oscillating polynomials extremal properties

Citation

Paschoa, V. G.; Pérez, D.; Qintana, Y. On a Theorem by Bojanov and Naidenov Applied to Families of Gegenbauer-Sobolev Polynomials. Commun. Math. Anal. 16 (2014), no. 2, 9--18. https://projecteuclid.org/euclid.cma/1413810435


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