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)].

Reviewer: S.S.Kutateladze (Novosibirsk)

##### MSC:

41A35 | Approximation by operators (in particular, by integral operators) |

41A36 | Approximation by positive operators |

47A58 | Linear operator approximation theory |

##### Keywords:

Korovkin-type theorems; approximations by operators; approximation properties of linear operators
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\textit{Yu. G. Abakumov} et al., Sib. Math. J. 41, No. 2, 247--252 (2000; Zbl 0963.41014); translation from Sib. Mat. Zh. 41, No. 2, 247--252 (2000)

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##### References:

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[2] | Korovkin P. P., ”On convergence of positive linear operators in the space of continuous functions,” Dokl. Akad. Nauk SSSR,90, No. 6, 961–964 (1953). |

[3] | Videnskiî V. S., Positive Linear Operators of Finite Rank [in Russian], Leningrad Ped. Inst., Leningrad (1985). |

[4] | Klimov V. S., Krasnosel’skiî M. A., andLifshits E. A., ”Smooth points of a cone and convergence of positive functionals and operators,” Trudy Moskovsk. Mat. Obshch.,15, 55–69 (1966). |

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[8] | Abakumov Yu. G. andBanin V. G., ”One approach to studying approximation properties of linear operators,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 11, 3–6 (1991). · Zbl 0771.41023 |

[9] | Korovkin P. P., ”Convergent sequences of linear operators,” Uspekhi Mat. Nauk,17, No. 4, 147–152 (1962). |

[10] | Baskakov V. A., ”On a method for constructing operators of the classS 2m ,” in: The Theory of Functions and Approximations. Lagrange Interpolation [in Russian], Saratov, 1984, pp. 19–25. |

[11] | Vassiliev R. K., ”On an exact order of approximation of the differentiable functions by a sequence of operators of classS 2m ,” Suppl. ai Rend. del Circolo Matematico di Palermo2, No. 33, 490–507 (1993). (Proc. Second Intern. Conf. on Functional Analysis and Aproximation Theory, Acquafredda di Maratea (POTENZA), September 14–19, 1992.) |

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