The Kantorovich form of some extensions for the Szász-Mirakjan operators

Dan Bărbosu; Ovidiu T. Pop; Dan Miclăuş

doi:10.33993/jnaat391-916

Authors

Dan Bărbosu North University of Baia Mare, Romania
Ovidiu T. Pop National College "Mihai Eminescu", Satu Mare, Romania
Dan Miclăuş North University of Baia Mare, Romania

DOI:

https://doi.org/10.33993/jnaat391-916

Keywords:

Szász-Mirakjan operators, Kantorovich operators, Bohman-Korovkin theorem, modulus of continuity, Shisha-Mond theorem, degree of approximation, parametric extension, Korovkin theorem for the bivariate case, bivariate modulus of continuity

Abstract views: 263

Abstract

Recently, C. Mortici defined a class of linear and positive operators depending on a certain function \(\varphi\). These operators generalize the well known Szász-Mirakjan operators. A convergence theorem for the defined sequence by the mentioned operators was given.Other interesting approximation properties of these generalized Szász-Mirakjan operators and also their bivariate form were obtained by D. Bărbosu, O. T. Pop and D. Miclăuș.In the present paper we are dealing with the Kantorovich form of the generalized Szász-Mirakjan operators. We construct the Kantorovich associated operators and then we establish a convergence theorem for the defined operators. The degree of approximation is expressed in terms of the modulus of continuity. Next, we construct the bivariate and respectively the GBS corresponding operators and we establish some of their approximation properties.

Downloads

Download data is not yet available.

References

Agratini, O., Approximation by linear operators, Presa Universitară Clujeană, Cluj-Napoca, 2000 (in Romanian).

Altomare, F. and Campiti, M., Korovkin-type Approximation Theory and Its Applications, de Gruyter Series in Mathematics, 17, Walter de Gruyter & Co., Berlin, New York, 1994. DOI: https://doi.org/10.1515/9783110884586

Badea, I., Modulus of continuity in the Bogel sense and some applications in approximation by an operator of Bernstein type, Studia Univ. "Babeş-Bolyai", Ser. Math. Mech., 4 (2), pp. 69-78, 1973 (in Romanian).

Badea, C. and Cottin, C., Korovkin-type Theorems for Generalized Boolean Sum Operators, Colloquia Mathematica Societatis "Janos Bolyai", Approximation Theory, Kecskemét (Hungary), 58, pp. 51-67, 1990.

Badea, C., Badea, I. and Gonska, H. H., A test function theorem and approximation by pseudopolynomials, Bull. Australl. Math. Soc., 34, pp. 53-64, 1986. https://doi.org/10.1017/s0004972700004494 DOI: https://doi.org/10.1017/S0004972700004494

Badea, C., Badea, I., Cottin, C. and Gonska, H. H., Notes on the degree of approximation of B-continuous and B-differentiable functions, J. Approx. Theory Appl., 4, pp. 95-108, 1988.

Bărbosu, D., The functions approximation of more variables by boolean sums of linear interpolation type operators, Ed. Risoprint, Cluj-Napoca, 2002 (in Romanian).

Bărbosu, D., Pop, O. T. and Miclăuş, D., On some extensions for the Szász-Mirakjan operators, Annals of Oradea University, to appear.

Becker, M., Global approximation theorems for Szász-Mirakjan and Baskakov operators in polynomial weight spaces, Indiana Univ. Math. J., 27 (1), pp. 127-142, 1978. DOI: https://doi.org/10.1007/978-3-0348-7180-8_28

Bögel, K., Mehrdimensionalle Differentiation von Funktionen mehrer Verändlicher, J. R. Angew. Math., 170, pp. 197-217, 1937. DOI: https://doi.org/10.1515/crll.1934.170.197

Ciupa, A. and Gavrea, I., On a Favard-Szász type operator, Studia Univ. "Babeş-Bolyai", Mathematica, 34, pp. 39-46, 1994.

Delvos, F. J. and Schempp, W., Boolean Methods in Interpolation and Approximation, Longman Scientific and Technical, 1989.

Ditzian, Z. and Totik, V., Moduli of Smoothness, Springer Verlag, Berlin, 1987. DOI: https://doi.org/10.1007/978-1-4612-4778-4

Favard, J., Sur les multiplicateurs d'interpolation, Journal Pures Appl., 23 (9), pp. 219-247, 1944.

Jakimovski, A. and Leviatan, D., Generalized Szász operators for the approximation in the infinite interval, Mathematica (Cluj), 34, pp. 97-103, 1969.

Kantorovich, L. V., Sur certain développements suivant les polynômes de la forme de S. Bernstein, I, II, C. R. Acad. URSS, pp. 563-568, 595-600, 1930.

Miclăuş, D., The Voronovskaja type theorem for the Szász-Mirakjan-Kantorovich operators, Journal of Science and Arts, 2 (13), pp. 257-260, 2010.

Mirakjan, G. M., Approximation of continuous functions with the aid of polynomials, Dokl. Acad. Nauk SSSR, 31, pp. 201-205, 1941.

Mortici, C., An Extension of the Szász-Mirakjan Operators, An. Şt. Univ. Ovidius Constanţa, 17 (1), pp. 137-144, 2009.

Pop, O. T., Bărbosu, D. and Miclăuş, D., The Voronovskaja type theorem for an extension of Szász-Mirakjan operators, Demonstratio Mathematica, to appear.

Shisha, O. and Mond, B., The degree of convergence of linear positive operators, Proc. Nat. Acad. Sci. U.S.A., 60, pp. 1196-1200, 1968. https://doi.org/10.1073/pnas.60.4.1196 DOI: https://doi.org/10.1073/pnas.60.4.1196

Stancu, D. D., Approximation of function by a new class of linear polynomial operators, Rev. Roum. Math. Pures et Appl., 13, pp. 1173-1194, 1968.

Stancu, D. D., Coman, Gh., Agratini, O. and Tr ambiţaş, R., Numerical Analysis and Approximation Theory, I, Presa Universitară Clujeană, Cluj-Napoca, 2001 (in Romanian).

Stancu, F., On the remainder term in approximation formula by univariate and bivariate Mirakjan operators, An. Şt. Univ. "Al. I. Cuza" Iaşi, XIV, pp. 415-422, 1968 (in Romanian).

Szasz, O., Generalization of Bernstein's polynomials to the infinite intervals, J. Res. Nat. Bur. Standards, 45, pp. 239-245, 1950. https://doi.org/10.6028/jres.045.024 DOI: https://doi.org/10.6028/jres.045.024

Totik, V., Uniform approximation by positive operators on infinite intervals, Analysis Mathematica, 10, pp. 163-182, 1984. https://doi.org/10.1007/bf02350525 DOI: https://doi.org/10.1007/BF02350525