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On refinement of Jackson type inequalities for approximation of differentiable periodic functions by linear operators. (English. Russian original) Zbl 0846.42005

Russ. Math. 39, No. 2, 75-78 (1995); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1995, No. 2(393), 79-82 (1995).
The aim of this paper is the following
Theorem. If \(p\in \{1, \infty\}\), \(n\in \mathbb{N}:= \{1, 2, \dots \}\) then for any odd \(r\in \{1, 3, 5, \dots\}\) and \(f\in L^r_p\) there exists \(d_r>0\) such that \[ |f- F_{n,r} (f) |_p\leq K_r n^{-r} \{2^{-1} d_r \omega_{p,1} (f^{(r)}; \pi n^{-1})+ 4^{-1} (1-d_r) \omega_{p,2} (f^{(r)}; \pi n^{-1})\} \] where \(F_{n,r} (f, .)\) and \(K_r\) are the operator and the constant of Favard, \(L^r_p\) is the set of all \(2\pi\)-priodic functions \(f\) whose \((r-1)\)-th derivatives are locally absolutely continuous on the whole axis with \(f^{(r)}\in L_p\) and \(\omega_{p,k} (f;.)\) is the modulus of smoothness of order \(k\in 1, 2,\dots\) in the space \(L_p\).
Reviewer: I.Badea (Craiova)

MSC:

42A10 Trigonometric approximation
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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