Approximation of some classes of functions by Landau type operators

Abstract

This paper aims to highlight a class of integral linear and positive operators of Landau type which have affine functions as fixed points. We focus to reveal approximation properties both in \(L_p\) spaces and in weighted \(L_p\) spaces (1≤p<∞). Also, we give an extension of the operators to approximate real-valued vector functions. In this case, the study pursues the approximation of continuous functions on convex compacts. The evaluation of the rate of convergence in one and multidimensional cases is performed by using adequate moduli of smoothness.

Authors

Octavian Agratini
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Romania
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy, Romania

Ali Aral
Kirikkale University, Turkey

Keywords

Landau operator; weighted space; Korovkin theorem, modulus of smoothness

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Cite this paper as:

O. Agratini, A. Aral, Approximation of some classes of functions by Landau type operators, Results in Mathematics, 76 (2021) art. no. 12, https://doi.org/10.1007/s00025-020-01319-9

About this paper

Journal

Results in Mathematics

Publisher Name

Springer

Print ISSN

1422-6383

Online ISSN

1420-9012

Google Scholar Profile

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[1] Agratini, O., Gal, S.G., On Landau type appproximation operators. Mediterr. J. Math. (sent to publication)
[2]Altomare, F., Campiti, M., Korovkin-Type Approximation Theory and its Applications, de Gruyter Series Studies in Mathematics, vol. 17. Walter de Gruyter & Co., Berlin, New York (1994), Book Google Scholar
[3] Censor, E., Quantitative results for positive linear approximation operators. J. Approx. Theory 4, 442–450 (1971), MathSciNet Article Google Scholar
[4] Chen, Zhanben, Shih, Tsimin, A new class of generalized Landau linear positive operator sequence and its properties of approximation. Chin. Q. J. Math. 13(1), 29–43 (1998), Google Scholar
[5] De Vore, R.A., Lorentz, G.G., Constructive Approximation, A Series of Comprehensive Studies in Mathematics, vol. 303. Springer, Berlin, Heidelberg (1993) Google Scholar
[6] Gadjiev, A.D., Aral, A., Weighted Lp-approximation with positive linear operators on unbounded sets. Appl. Math. Lett. 20(10), 1046–1051 (2007), MathSciNet Article Google Scholar
[7] Gal, S.G., Iancu, I.: Quantitative approximation by nonlinear convolution operators of Landau-Choquet type. Carpathian J. Math. 37, Issue 1 (to appear) (2021)
[8] Gao, J.B., Approximation properties of a kind of generalized discrete Landau operator, (Chinese). J. Huazhong Univ. Sci. Tech. 12(5), 1–4 (1984), MathSciNet Google Scholar
[9] Hardy, G.H., Littlewood, J.E., Pólya, G., Inequalities. Cambridge University Press, Cambridge Mathematical Library (1988), MATH Google Scholar
[10] Jackson, D., A proof of Weierstrass theorem. Am. Math. Mon. 41(5), 309–312 (1934), MathSciNet Article Google Scholar
[11] Landau, E., Über die Approximation einer stetingen Funktion durch eine ganze rationale Funktion. Rend. Circ. Mat. Palermo 25, 337–345 (1908), Article Google Scholar
[12] Mamedov, R.G., Approximation of functions by generalized linear Landau operators, (Russian). Dokl. Akad. Nauk SSSR 139(1), 28–30 (1961), MathSciNet Google Scholar
[13] Mamedov, R.G., On the order and on the asymptotic value of the approximation of functions by generalized linear Landau operators, (Russian), Akad. Nauk Azerbadzan SSR, Trudy Inst. Mat. Meh., 2(10), 49–65 (1963)
[14] Pendina, T.P., Iterations of positive linear operators of exponential type and of Landau polynomials, (Russian). In: Geometric Problems of the Theory of Functions and Sets (Russian), Kalinin. Gos. Univ., Kalinin, 105–111, (1987)
[15] Shashkin, Y.A.: Korovkin systems in spaces of continuous functions, (Russian), Izv. Akad. Nauk SSSR Ser. Mat., 26, Issue 4, 495–512 (1962), translated in Am. Math. Soc. Transl., Series 2, 54, 125–144 (1996)
[16] Shisha, O., Mond, B., The degree of convergence of linear positive operators. Proc. Natl. Acad. Sci. USA 60, 1196–1200 (1968), MathSciNet Article Google Scholar
[17] Sikkema, P.C., Approximation formulae of Voronovskaya type for certain convolution operators. J. Approx. Theory 26(1), 26–45 (1979), MathSciNet Article Google Scholar
[18] Veselinov, V., Certain estimates to the approximation of functions by de la Vallée–Poussin and Landau operators, (Russian). Ann. Univ. Sofia Fac. Math. 66, 153–158 (1974)Google Scholar
[19] Volkov, V.I., On the convergence of sequences of linear positive operators in the space of continuous functions of two variables. Dokl. Akad. Nauk SSSR (N.S.), 115, 17–19 (1957) (Russian)
[20] Yuksel, I., Ispir, N., Weighted approximation by a certain family of summation integral-type operators. Comput. Math. Appl. 52, 1463–1470 (2006), MathSciNet Article Google Scholar

2021

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