chebЧебышевский сборникChebyshevskii Sbornik2226-8383Tula State Lev Tolstoy Pedagogical University10.22405/2226-8383-2019-20-3-394-400cheb-730Research ArticleКраткие сообщенияОб экстремальных задачах типа Никольского – Бернштейна и Турана для преобразования ДанкляExtremal Nikolskii – Bernstein- and Turán-type problems for Dunkl transformГорбачевДмитрий ВикторовичGorbachevDmitry Viktorovichdvgmail@mail.ruДобровольскийНиколай НиколаевичDobrovol’skyNikolai Nikolaevichnikolai.dobrovolsky@gmail.com201905032020203394400Copyright © Горбачев Д.В., Добровольский Н.Н., 20202020Горбачев Д.В., Добровольский Н.Н.Gorbachev D.V., Dobrovol’sky N.N.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://www.chebsbornik.ru/jour/article/view/730

Изучается взаимосвязь между экстремальными задачами типа Турана и Никольского – Бернштейна на Rd с весом Данкля. Задача Турана состоит в нахождении супремума заданного момента положительно определенной (относительно преобразования Данкля) функции с носителем в евклидовом шаре и фиксированным значением в нуле. В точном L1-неравенстве Никольского–Бернштейна оценивается супремум-норма лапласиана Данкля целой функции экспоненциального сферического типа с единичной L1-нормой. Также отмечается связь с экстремальными задачами типа Фейера и Бомана. Преобразование Данкля покрывает случай классического преобразования Фурье в случае единичного веса.

Неравенства Никольского - -Бернштейна являются классическими в теории приближений, а задачи типа Турана имеют приложения в метрической геометрии. Тем не менее мы доказываем, что они имеют один и тот же ответ, который явно выписывается. Простое доказательство опирается на наши старые результаты из теории решения экстремальных задач для преобразования Данкля.

We study the interrelation between the extremal Turán-type problems and Nikolskii – Bernstein problems for nonnegative functions on Rd with the Dunkl weight. The Turán problem is to find the supremum of a given moment of a positive definite (with respect to the Dunkl transform) function with a support in the Euclidean ball and a fixed value at zero. In the sharp L1-Nikolskii–Bernstein inequality, the supremum norm of the Dankl Laplacian of an entire function of exponential spherical type with the unit L1-norm is estimated. Extremal Feuér and Beaumann problems is also mentioned. The Dunkl transform covers the case of the classical Fourier transform in the case of unit weight.

Nikolskii–Bernstein inequalities are classical in approximation theory, and the Turán-type problems have applications in metric geometry. Nevertheless, we prove that they have the same answer, which is given explicitly. The easy proof is relied on our old results from the theory of solving extremal problems to the Dunkl transform.

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The authors declare that there are no conflicts of interest present.