Skip to content
BY-NC-ND 4.0 license Open Access Published by De Gruyter Open Access May 24, 2017

Approximation Properties of Ibragimov-Gadjiev-Durrmeyer Operators on Lp (ℝ+)

  • Gülsüm Ulusoy EMAIL logo and Ali Aral
From the journal Demonstratio Mathematica

Abstract

We deal with the approximation properties of a new class of positive linear Durrmeyer type operators which offer a reconstruction of integral type operators including well known Durrmeyer operators. This reconstruction allows us to investigate approximation properties of the Durrmeyer operators at the same time. It is first shown that these operators are a positive approximation process in Lp(ℝ+) . While we are showing this property of the operators we consider the Ditzian-Totik modulus of smoothness and corresponding K-functional. Then, weighted norm convergence, whose proof is based on Korovkin type theorem on Lp(ℝ+), is given. At the end of the paper we show several examples of classical sequences that can be obtained from the Ibragimov-Gadjiev-Durrmeyer operators.

MSC 2010: 41A36; 41A25; 41A35

References

[1] Ditzian Z., Ivanov K., Bernstein-type operators and their derivatives, J. Approx. Theory, 1989, 56, 72-9010.1016/0021-9045(89)90134-2Search in Google Scholar

[2] Heilmann M., Direct and converse results for operators of Baskakov-Durrmeyer type, Approx. Theory Appl., 1989, 5 (1), 105-12710.1080/01630568908816295Search in Google Scholar

[3] Agrawal P. N., Mohammad A. J., On Lp-approximation by a linear combination of a new sequence of linear positive operators, Turk. J. Math, 2003, 27, 389-405Search in Google Scholar

[4] Dzjadyk V. K., Approximation of functions by positive linear operators and singular integrals, (Russ.) Mat. Sb. (N,S), 1966, 112 (70), 508-517Search in Google Scholar

[5] Gadjiev A. D., Aral A., Weighted Lp -approximation with positive linear operators on unbounded sets, Appl. Math. Lett., 2007, 20 (10), 1046-105110.1016/j.aml.2006.12.007Search in Google Scholar

[6] Gadjiev A. D., Ibragimov I.I., Ibragimov I.I., On a sequence of linear positive operators, 1970, 11, 1092-1095Search in Google Scholar

[7] Aral A., Approximation by Ibragimov-Gadjiyev operators in polynomial weighted space, Proc. of IMM of NAS of Azerbaijan, 2003, XIV, 35-44Search in Google Scholar

[8] Coskun T., On a Construction of positive linear operators for approximation of continuous functions in the weighted spaces, J. Comp. Anal. and Appl., 2011, 13 (4), 756-770Search in Google Scholar

[9] Dogru O., On a certain family of linear positive operators, Turkish J. Math., 1997, 21 (4), 387-399Search in Google Scholar

[10] Dogru, O., On the order of approximation of unbounded functions by the family of generalized linear positive operators, Commun Fac. Sci. Univ. Ankara, Series A1, 1997, 46, 173-18110.1501/Commua1_0000000435Search in Google Scholar

[11] Gadjiev A. D., Ispir N., On a sequence of linear positive operators in weighted spaces, Proc. of IMM of Azerbaijan AS, 1999, Vol. XI(XIX), 45-56Search in Google Scholar

[12] Aral A., Acar T., Modern mathematical methods and high performance computing in science and technology, 1-15, Springer Proc. Math. Stat., 171, (Springer, Singapore, 2016)10.1007/978-981-10-1454-3_1Search in Google Scholar

[13] Ditzian, Z., Totik, V., Moduli of Smoothness, Springer Series in Computational Mathematics 9, (Springer-Verlag, Berlin, Heidelberg, Newyork, 1987)10.1007/978-1-4612-4778-4Search in Google Scholar

[14] Bergh J., Löfström J., Interpolation Spaces, An Introduction, Springer-Verlag, Berlin, heidelberg, New York, 197610.1007/978-3-642-66451-9Search in Google Scholar

[15] Mazhar S. M., Totik V., Approximation by modified Szasz operators, Acta Sci. Math., 1985, 49 , 257-269Search in Google Scholar

[16] Sahai A., Prasad G., On simultaneous approximation by modified Lupas operators, J. Approx. Theory, 1985, 45 (12), 122-128 10.1016/0021-9045(85)90039-5Search in Google Scholar

Received: 2015-05-04
Accepted: 2016-04-27
Published Online: 2017-05-24
Published in Print: 2017-04-25

© 2017 Gülsüm Ulusoy and Ali Aral

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Downloaded on 28.3.2024 from https://www.degruyter.com/document/doi/10.1515/dema-2017-0017/html
Scroll to top button