УДК 517.5
MSC: 41A44, 41A46, 33C10
DOI: 10.21538/0134-4889-2023-29-1-259-279
Полный текст статьи (Full text)
In this paper, we have solved several extremal problems of the best mean-square approximation of function $f$, on the semiaxis with a power-law weight, which can be used to solve various problems. Sharp Jackson–Stechkin type inequalities are obtained on some classes of functions in which the values of the best approximations are estimated from above through moduli of smoothness of the $k$-th order.
Keywords: exact constants in Jackson–Stechkin inequality, moduli of smoothness, best approximations, Bessel function
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Received April 6, 2022
Revised January 31, 2023
Accepted February 5, 2023
Tilektes Ereshovich Tileubayev, Faculty of Mechanics and Mathematics L.N. Gumilyov Eurasian National University, 2 Satpaeva st., Astana, 010008 Republic of Kazakhstan, e-mail: Tileubaev@mail.ru
Cite this article as: T.E. Tileubayev. Exact constants in Jackson–Stechkin inequality in $L^2$ with a power-law weight, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 1, pp. 259–279.
Русский
Т.Е. Тилеубаев. Точные константы в неравенстве Джексона — Стечкина в $L^2$ со степенным весом
В статье решено несколько экстремальных задач наилучшего среднеквадратичного приближения функции $f$ на полуоси со степенным весом. Полученные результаты можно использовать для решения различных задач. Доказаны точные неравенства типа Джексона — Стечкина на некоторых классах функций, в которых значения наилучших приближений оцениваются сверху через модули гладкости $k$-го порядка.
Ключевые слова: точные константы в неравенстве Джексона — Стечкина, модули гладкости, наилучшие приближения, функция Бесселя
