MSC: 26A51, 26D07, 49K35
DOI: 10.21538/0134-4889-2023-29-4-193-216
This research of Béla Nagy was supported by project TKP2021-NVA-09. Project no. TKP2021-NVA-09 has been implemented with the support provided by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund, financed under the TKP2021-NVA funding scheme. The work of Sz. Gy. Révész was supported in part by Hungarian National Research, Development and Innovation Fund project # K-132097.
Полный текст статьи (Full text)
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.
Keywords: minimax and maximin problems, kernel function, sum of translates function, vector of local maxima, equioscillation, majorization
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Received August 24, 2023
Revised October 18, 2023
Accepted November 6, 2023
Funding Agency: This research of Béla Nagy was supported by project TKP2021-NVA-09. Project no. TKP2021-NVA-09 has been implemented with the support provided by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund, financed under the TKP2021-NVA funding scheme. The work of Sz. Gy. Révész was supported in part by Hungarian National Research, Development and Innovation Fund project # K-132097.
Béla Nagy, Department of Analysis, Bolyai Institute, University of Szeged Aradi vértanuk tere 1, 6720 Szeged, Hungary, e-mail: nbela@math.u-szeged.hu
Szilárd Gy. Révész, HUN-REN Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, 1053 Budapest, Hungary, e-mail: revesz.szilard@renyi.hu
Cite this article as: Béla Nagy and Szilárd Gy. Révész. On the weighted trigonometric Bojanov–Chebyshev extremal problem. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 4, pp. 193–216.
Русский
Б. Надь, С.Д. Ревеc. О весовой тригонометрической экстремальной задаче Боянова — Чебышева
Исследуется весовая экстремальная задача Боянова — Чебышева для тригонометрических полиномов, т. е. минимаксная задача минимизации $\|T\|_{w,C(\mathbb{T})}$, в которой $w$ — достаточно ненулевая ограниченная сверху неотрицательная весовая функция, в качестве нормы рассмотрена соответствующая взвешенная максимальная норма на торе $\mathbb{T}$, и $T$ — тригонометрический полином с заданными кратностями $\nu_1,\ldots,\nu_n$ корневых множителей $|\sin (\pi(t-z_j))|^{\nu_j}$. Если $\nu_j$ — натуральные числа с четной суммой, то $T$ действительно является тригонометрическим полиномом, и случай, когда все $\nu_j$ равны 1, охватывает экстремальную задачу Чебышева. Наш результат будет более общим, допускающим, в частности, так называемые обобщенные тригонометрические полиномы. Для достижения этой цели используется метод суммы сдвигов Фентона. Однако, в отличие от ранее описанных случаев без веса или на промежутке, здесь рассмотрены другие ситуации, а о решениях получено меньше информации.
Ключевые слова: задачи на минимакс и максимин, ядерная функция, функция суммы сдвигов, вектор локальных максимумов, эквиколебание, мажорирование
