On the weighted trigonometric Bojanov-Chebyshev extremal problem

2023/09/12 by Béla Nagy, Nagy, Béla, Szilárd Gy. Révész +1
Mathematics · #26A51 #26D07 #49K35 #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2309.06083

openalex publication_date 2023/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ν₁,…,ν_n of root factors |sin(π(t-z_j))|^ν_j. If the ν_j are natural numbers and their sum is even, then T is indeed a trigonometric polynomial and the case when all the ν_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.

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