On oscillating polynomials

Abstract

Extremal problems of Markov type are studied, concerning maximization of a local extremum of the derivative in the class of real polynomials of bounded uniform norm and with maximal number of zeros in [−1, 1]. We prove that if a symmetric polynomial f , with all its zeros in [−1, 1], attains its maximal absolute value at the end-points, then | f ′ | attains maximal value at the end-points too. As an application of the method developed here, we show that the classic Zolotarev polynomials have maximal derivative at one of the end-points.

On the Largest Critical Value of $T_n^(k)$

“…Our proof follows the same approach. An alternative proof of Theorem 1.1 can be obtained using some results of Bojanov and Naidenov in [2]. 2.…”

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

“…(2) Supported in part by a grant from Ministerio de Economía y Competitividad, Dirección General de Investigación Científica y Técnica (MTM2012-36732-C03-01), Spain. therefore they belong to other important class of algebraic polynomials, namely the oscillating polynomials [3,19].…”

“…The main result of [3] allows to guarantee the maximal absolute value of higher derivatives of a symmetric oscillating polynomial on a finite interval are attained in the end-points of such interval, whenever the maximal absolute value of the polynomial is attained in the end-points of that interval. Then, [3, Section 4] contains a brief explanation about applications of previous result to orthogonal polynomials on the real line associated to symmetric weights.…”

“…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 kth derivative, where M n,λ is a suitable value such that all zeros of the polynomialn,λ attains its maximal value at the end-points of such interval. Also, some illustrative numerical examples are presented.…”

“…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 kth derivative…”