# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Labsker L. G., Test Sets in Banach Space [in Russian], MIPKKhM, Moscow (1989). · Zbl 0918.46011 [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). [5] Rubinov A. M., ”On one theorem by V. S. Klimov, Krasnosel’skiî M. A., and Lifshits E. A.,” in: Optimization, Novosibirsk, 1971, No. 3, pp. 154–158. [6] Kutateladze S. S. andRubinov A. M., Minkowski Duality and Its Applications [in Russian], Nauka, Novosibirsk (1976). [7] Abakumov Yu. G. and Banin V. G., ”Sequences of linear functionals positive on some cones and analogs of Korovkin’s theorems,” submitted to VINITI on 11.11.86, No. 7695. [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.)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.