On an approximation process of integral type


In this paper we study a class of integral type positive linear operators depending on a parameter \(\beta,0\leq\beta<1\). Approximation properties of this class are explored: the rate of convergence in terms of the usual moduli of smoothness is given, the uniform approximation over unbounded intervals is established, the convergence in certain weighted spaces is investigated. For particular case \(b=\) \(0\) some previous results are recaptured.


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


Linear approximation process; Korovkin-type theorem; Modulus of smoothness; Weighted space; Uniform convergence

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O. Agratini, On an approximation process of integral type, Applied Mathematics and Computation, 236 (2014), pp. 195-201. https://doi.org/10.1016/j.amc.2014.03.052


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[1] J.A. Adell, J. de la Cal, Bernstein–Durrmeyer operators, Comput. Math. Appl. 30 (1995) 1–14.

[2] O. Agratini, Approximation properties of a class of linear operators, Math. Methods Appl. Sci. 36 (17) (2013) 2353–2358.

[3] F. Altomare, Korovkin-type theorems and approximation by positive linear operators, Surv. Approx. Theory 5 (2010) 92–164.

[4] F. Altomare, M. Campiti, Korovkin-type approximation theory and its applications, Walter de Gruyter Studies in Mathematics, vol. 17, de Gruyter & Co., Berlin, 1994.

[5] H. Bohman, On approximation of continuous and of analytic functions, Ark. Mat. 2 (1952) 43–56.

[6] J. de la Cal, J. Cárcamo, On uniform approximation by some classical Bernstein-type operators, J. Math. Anal. Appl. 279 (2003) 625–638.

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

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

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

[10] P.P. Korovkin, On convergence of linear positive operators in the space of continuous functions (Russian), Dokl. Akad. Nauk SSSR (N.S.) 90 (1953) 961–964.

[11] B. Lerner, A. Lone, M. Rao, On generalized Poisson distributions, Probab. Math. Stat. 17 (1997), Fasc. 2, 377–385.

[12] G.G. Lorentz, Bernstein Polynomials, 2nd ed., Chelsea Publ. Comp, New York, 1986.

[13] S.M. Mazhar, V. Totik, Approximation by modified Szász operators, Acta Sci. Math. 49 (1985) 257–269.

[14] G.M. Mirakjan, Approximation of continuous functions with the aid of polynomials (Russian), Dokl. Akad. Nauk SSSR 31 (1941) 201–205.

[15] R.S. Phillips, An inversion formula for Laplace transforms and semi-groups of linear operators, Ann. Math. Second Ser. 59 (1954) 325–356.

[16] T. Popoviciu, On the proof of Weierstrass theorem using interpolation polynomials (Romanian), Lucrarile Sesiunii Gen. St. Acad. Române, 2–12 iunie 1950, Editura Academiei P.R.P., 1951, pp. 1554–1667 (Translated in English by Daniela Kacsó in East Journal on Approximations, 4 (1998), fasc. 1, 107–110).

[17] O. Shisha, B. Mond, The degree of convergence of linear positive operators, Proc. Natk. Acad. Sci. USA 60 (1968) 1196–1200.

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

[19] S. Tarabie, On Jain-Beta linear operators, Appl. Math. Inf. Sci. 6 (2) (2012) 213–216.

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