On the weighted trigonometric Bojanov--Chebyshev extremal problem

B. Nagy ; Sz. Gy. Révész

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 29 (2023) no. 4, pp. 193-216

Résumé

We investigate the weighted Bojanov–Chebyshev extremal problem for trigonometric polynomials, that is, the minimax problem of minimizing $\|T\|_{w,C(\mathbb{T})}$, where $w$ is a sufficiently nonvanishing, upper bounded, nonnegative weight function, the norm is the corresponding weighted maximum norm on the torus $\mathbb{T}$, and $T$ is a trigonometric polynomial with prescribed multiplicities $\nu_1,\ldots,\nu_n$ of root factors $|\sin(\pi(t-z_j))|^{\nu_j}$. If the $\nu_j$ are natural numbers and their sum is even, then $T$ is indeed a trigonometric polynomial and the case when all the $\nu_j$ are 1 covers the Chebyshev extremal problem. Our result will be more general, allowing, in particular, so-called generalized trigonometric polynomials. To reach our goal, we invoke Fenton's sum of translates method. However, altering from the earlier described cases without weight or on the interval, here we find different situations, and can state less about the solutions.

Détail

Keywords: minimax and maximin problems, kernel function, sum of translates function, majorization.
Mots-clés : vector of local maxima, equioscillation