On the weighted trigonometric Bojanov-Chebyshev extremal problem

TL;DR

This paper studies a weighted extremal problem for trigonometric polynomials with prescribed roots, extending classical Chebyshev problems by incorporating weights and generalized polynomials, using Fenton’s method.

Contribution

It generalizes the weighted Bojanov-Chebyshev extremal problem to include generalized trigonometric polynomials and nonvanishing weights, broadening the scope of classical extremal polynomial theory.

Findings

Established conditions for extremal solutions with weights and prescribed roots.

Extended classical Chebyshev extremal problem to generalized trigonometric polynomials.

Applied Fenton's sum of translates method to a more general setting.

Abstract

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.