A sequence of positive linear operators associated with an approximation process

2 years ago

(original), Approximation Theory, Linear and positive approximation operators, not-at-ICTP, paper

Abstract

Considering a general class of discrete linear positive operators, by using a one-to-one function, we associate to the class mentioned above a new sequence of operators. Our aim is to establish the transfer of approximation properties on this construction. The study is carried out in a weighted space and our results are materialized in obtaining both a convergence the Theorem of Korovkin type and an inequality for the approximation error expressed in terms of a certain weighted modulus of smoothness. Two particular cases are analyzed.

Authors

Octavian Agratini
Department of Mathematics, Babes-Bolyai University, Cluj-Napoca, Romania

Keywords

Approximation process; Markov type operator; Weighted space; Weighted approximation; Modulus of smoothness.

Paper coordinates

O. Agratini, A sequence of positive linear operators associated with an approximation process, Applied Mathematics and Computation, 269 (2015), pp. 23-28. https://doi.org/10.1016/j.amc.2015.07.043

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About this paper

Journal

Applied Mathematics and Computation

Publisher Name

Elsevier

DOI

https://doi.org/10.1016/j.amc.2015.07.043

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[1] O. Agratini, T. Andrica, Discrete approximation processes of King’s type, in: P.P. Pardalos, T.M. Rassias, A.A. Khan (Eds.), Nonlinear Analysis and Variational Problems, Springer Optimization and Its Applications, vol. 35, Springer, 2010, pp.3–12.

[2] F. Altomare, M.M. Cappelletti, V. Leonessa, I. Rasa, On Markov operators preserving polynomials, J. Math. Anal. Appl. 415(2014) 477–495.

[3] A. Aral, D. Inoan, I. Rasa, On the generalized Szász–Mirakyan operators, Results Math. 65(2014) 441–452.

[4] V.A. Baskakov, An example of a sequence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk SSSR 113(1957) 249–251.

[5] M. Becker, Global approximation theorems for Szász–Mirakjan and Baskakov operators in polynomial weighted spaces, Indiana Univ. Math. J. 27(1978), 127–142.

[6] M. Birou, A class of Markov type operators which preserve ej, j≥1, Appl. Math. Comput. 250(2015), 1–11.

[7] G. Bleimann, P.L. Butzer, L. Hahn, Bernstein-type operators approximating continuous functions on the semiaxis, Indagat. Math. 42(1980), 256–262.

[8] P.C. Consul, G.C. Jain, A generalization of the Poisson distribution, Technometrics 15 (1973) 791–799.

[9] J. Dieudonné, Éléments d’analyse, in: Tome1: Fondements de l’Analyse Moderne, Gauthiers-Villars, Paris, 1968.

[10] Z. Ditzian, V. Totik, Moduli of Smoothness, in: Springer Series in Computational Mathematics, vol. 9, Springer-Verlag, NewYork, 1987.

[11] A.D. Gadzhiev, Theorems of Korovkin type, Math. Notes 20(5) (1976) 995–998.

[12] A.D. Gadzhiev, A. Aral, The estimates of approximation by using a new type of weighted modulus of continuity, Comput. Math. Appl. 54(2007) 127–135.

[13] A. Guterman, B. Shapiro, On linear operators preserving the set of positive polynomials, J. Fixed Point Theory Appl. 3 (2) (2008), 411–429.

[14] A. Holhos, Quantitative estimates for positive linear operators in weighted spaces, Gen. Math. 16 (4) (2008) 99–110.

[15] G.C. Jain, Approximation of functions by a new class of linear operators, J. Aust. Math. Soc. 13 (1972), 271–276.

[16] A. Olgun, F. Tasdelen, A. Erencin, A generalization of Jain’s operators, Appl. Math. Comp. 266 (2015), 6–11.

[17] O. Szász, Generalization of S. Bernstein’s polynomials to the infinite interval, J. Res. Nat. Bur. Stand., Sect. B 45 (1950), 239–245.

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