For periodic functions differentiable in the sense of Weyl and belonging to the space $L_2$, sharp inequalities of Jackson–Stechkin type are obtained for a special $m$th-order modulus of continuity generated by the Steklov operator (function). Similar characteristics of smoothness of functions were considered earlier by V. A. Abilov, F. V. Abilova, V. M. Kokilashvili, S. B. Vakarchuk, V. I. Zabutnaya, K. Tukhliev, etc. For classes of functions defined in terms of these characteristics, we solve a number of extremal problems of polynomial approximation theory.
Keywords: best approximation, periodic function, special modulus of continuity, Jackson–Stechkin inequalities, extremal problems
Received August 20, 2019
Revised October 31, 2019
Accepted November 11, 2019
Mirgand Shabozovich Shabozov, Dr. Phys.-Math. Sci., Prof., Tajik National University, Dushanbe, 734025 Republic of Tajikistan, e-mail: shabozov@mail.ru
Adolat Azamovna Shabozova, Tajik National University, Dushanbe, 734025 Republic of Tajikistan, e-mail: shabozova91@mail.ru
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Cite this article as: M.Sh.Shabozov, A.A.Shabozova. Sharp inequalities of Jackson–Stechkin type for periodic functions in $L_2$ differentiable in the Weyl sense, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 4, pp. 255–264.
