On a Theorem by Bojanov and Naidenov applied to families of Gegenbauer-Sobolev polynomials

Abstract

Let $\{Q_{n,\lambda }^{(\alpha )}{\}}_{n\ge 0}$ be the sequence of monic orthogonal polynomials with respect the Gegenbauer-Sobolev inner product $$⟨f,g{⟩}_S:={\int }_{-1}^{1}f(x)g(x)(1-x^2{)}^{\alpha -\frac{1}{2}}dx+\lambda {\int }_{-1}^1{f}^{\prime }(x)g^{\prime }(x)(1-x^2{)}^{\alpha -\frac{1}{2}}dx,$$ where $\alpha >-\frac{1}{2}$ and $\lambda \ge 0$. In this paper we use a recent result due to B.D. Bojanov and N. Naidenov \cite{BN2010}, in order to study the maximization of a local extremum of the $k$th derivative $\frac{d^k}{d{x}^k}{Q}_{n,\lambda }^{(\alpha )}$ 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.

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