In this work, we obtain exact Jackson–Stechkin type inequalities in the Hardy space $H_{q,\rho}$ ($1\le q\le\infty$, $0<\rho\le R$), in which the values of the best polynomial approximations are estimated from above in terms of the $\mathcal{K}$-functionals of the $r$th derivatives. For function classes defined by the mentioned characteristics, exact values of Bernstein and Kolmogorov $n$-widths in the space $H_{q,\rho}$ are calculated.
Keywords: Jackson–Stechkin type inequality, best polynomial approximation, $\mathcal{K}$-functional, $n$-widths
Received September 2, 2024
Revised November 12, 2024
Accepted November 18, 2024
Mirgand Shabozovich Shabozov, Dr. Phys.-Math. Sci., Prof., Tajik National University, Dushanbe, 734025 Tajikistan, e-mail: shabozov@mail.ru
Ravshan Azam Karimzoda, National University, Dushanbe, 734025 Tajikistan, e-mail: ravshan.karimov.93@mail.ru
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Cite this article as: M.Sh. Shabozov, R.A. Karimzoda. $\mathcal{K}$-Functionals and exact values of $n$-widths for some classes of functions in the Hardy space. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 4, pp. 301–308.
