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On sequences of linear functionals and some operators of the class \(S_{2m}\). (English. Russian original) Zbl 0963.41014
Sib. Math. J. 41, No. 2, 199-203 (2000); translation from Sib. Mat. Zh. 41, No. 2, 247-252 (2000).
Suppose that \(W_f\subset C_{2\pi}\) is some set of continuous \(2\pi\)-periodic functions, \(W_L\) is a set of linear operators \(L\colon C_{2\pi}\to C_{2\pi}\), and \(\|\cdot \|\) stands for the Chebyshev norm on \(C_{2\pi}\). The authors prove several approximation theorems of qualitative type that are assertions representable schematically as follows \[ f\in W_f,\;L_n\in W_L, \text{ and hypotheses} \Rightarrow \Bigl(\exists \alpha_n (f)\to 0: \|L_n\bigl(f(t),x\bigr)-f(x)\|\leq\alpha_n \Bigr). \] A pioneering contribution to the field was made by P. P. Korovkin [Dokl. Akad. Nauk SSSR, n. Ser. 90, 961–964 (1953; Zbl 0050.34005)]. In [Trans. Mosc. Math. Soc. 15, 61–77 (1966); translation from Tr. Mosk. Mat. Obshch. 15, 55–69 (1966; Zbl 0161.11501)], V. S. Klimov, M. A. Krasnosel’skiĭ, and E. A. Lifshits observed that the classical Korovkin’s approximation theorem is a consequence of a rather simple theorem about smooth points. The main idea of the authors of the paper under review is to define the notion of a smooth point in a form different from the conventional. They also present some applications of the proven theorems. The paper is a continuation of [Yu. G. Abakumov and {V. G. Banin}, Sov. Math. 35, No. 11(354), 3–6 (1991); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1991, No. 11(354), 3–6 (1991; Zbl 0771.41023)].
MSC:
41A35 Approximation by operators (in particular, by integral operators)
41A36 Approximation by positive operators
47A58 Linear operator approximation theory
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References:
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