On the weighted trigonometric Bojanov - Chebyshev extremal problem

Nagy, Béla and Révész, Szilárd (2023) On the weighted trigonometric Bojanov - Chebyshev extremal problem. TRUDY INSTITUTA MATEMATIKI I MEKHANIKI URO RAN, 29 (4). pp. 193-216. ISSN 0134-4889

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Abstract

We investigate the weighted Bojanov-Chebyshev extremal problem for trigonometric polynomials, that is, the minimax problem of minimizing kT kw,C(T) , where w is a sufficiently nonvanishing, upper bounded, nonnegative weight function, the norm is the corresponding weighted maximum norm on the torus T, and T is a trigonometric polynomial with prescribed multiplicities ν1, ... , νn of root factors | sin(π(t − zj ))| νj . If the νj are natural numbers and their sum is even, then T is indeed a trigonometric polynomial and the case when all the νj 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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