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

2014/03/27 by Vanessa G. Paschoa, Paschoa, Vanessa G., Dilcia Pérez +3
Mathematics · #33C45 #41A17 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:33C45 #msc:41A17

paper · pdf · doi:10.48550/arxiv.1403.6927

2 figures

arxiv created 2014/03/28 · arxiv updated 2014/03/31

Abstract

Let \Q^(α)_n,λ\_{n≥ 0} be the sequence of monic orthogonal polynomials with respect the Gegenbauer-Sobolev inner product ⟨ f,g⟩_S:=∫_-1¹f(x)g(x)(1-x²)^α-(1)/(2)dx+λ∫_-1¹f'(x)g'(x)(1-x²)^α-(1)/(2) dx, where α>-(1)/(2) and λ≥ 0. In this paper we use a recent result due to B.D. Bojanov and N. Naidenov \citeBN2010, in order to study the maximization of a local extremum of the kth derivative (d^k)/(dx^k)Q^(α)_n,λ in [-M_n,λ, M_n,λ], where M_n,λ is a suitable value such that all zeros of the polynomial Q^(α)_n,λ are contained in [-M_n,λ, M_n,λ] and the function |Q^(α)_n,λ| attains its maximal value at the end-points of such interval. Also, some illustrative numerical examples are presented.

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