On sequences of linear functionals and some operators of the class \(S_{2m}\)
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 \textit{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)], \textit{V. S. Klimov}, \textit{M. A. Krasnosel'skiĭ, and \textit{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 [\textit{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)].}
approximation properties of linear operators
scientific article; zbMATH DE number 1534472
scientific article