On the Chlodowsky variant of Jakimovski-Leviatan-Păltănea Operators

Year 2021, , 812 - 833, 01.09.2021

Özge Dalmanoğlu , Mediha Örkcü

https://doi.org/10.35378/gujs.794810

Abstract

In the present paper, our purpose is to generalize the Jakimovski-Leviatan-Păltănea operators in the sense of Chlodowsky. After introducing the new operators we first obtain the moments of these operators in order to establish the convergency properties with the help of Korovkin's theorem. After that, we give the local approximation result and the Voronovskaya type theorem. We also examine the convergence properties of the operators in the weighted space of functions. Lastly we determine the rate of convergence of the operators with the aid of the weighted modulus of continuity

Keywords

Weighted modulus of continuity, Păltănea operators, Chlodowsky operators, Voronovskaya theorem, Convergence rate

References

• [1] Jakimovski, A. and Leviatan, D., “Generalized Sźasz operators for the approximation in the infinite interval”, Mathematica (Cluj), 11:97-103, (1969)
• [2] Szász, O., “Generalization of S. Bernsteins polynomials to the infinite interval”, J. Res. Natl. Bur. Stand. 97:239-245, (1950).
• [3] Atakut, Ç. and Büyükyazıcı, İ., “Approximation by modified integral type Jakimovski-Leviatan operators”, Filomat, 30(1): 29-39, (2016).
• [4] Büyükyazıcı, İ., Tanberkan, H., Serenbay, S.K., Atakut, Ç., “Approximation by Chlodowsky type Jakimovski--Leviatan operators”, Journal of Computational and Applied Mathematics, 259:153-163, (2014).
• [5] Dalmanoglu, Ö. and Serenbay, S.K., Approximation by Chlodowsky type q-Jakimovski-Leviatan operators, Communications Faculty of Sciences University of Ankara Series A1-Mathematics and Statistics, 65(1):157-169, (2016.).
• [6] Gupta, P., Agrawal, P.N., “Jakimovski-Leviatan operators of Durrmeyer type involving Appell polynomials”, Turkish Journal of Mathematics, 42(3): 1457-1470, (2018).
• [7] Karaisa, A., “Approximation by Durrmeyer type Jakimoski Leviatan operators”, Mathematical Methods and Applied Sciences, 39(9): 2401-2410, (2016).
• [8] Phillips, R.S., “An inversion formula for Laplace transforms and semi-groups of linear operators”, Annals of Mathematics, 325-356, (1954).
• [9] Păltănea, R., “Modified Szász-Mirakjan operators of integral form”, Carpathian Journal of Mathematics, 24(3): 378-385, (2018).
• [10] Verma, D.K., Gupta, V., “Approximation for Jakimovski-Leviatan-Păltănea operators”, Annali dell' Università di Ferrara, 61(3): 367-380, (2015).
• [11] Mursaleen, M., AL-Abeid, A.A.H., Ansari, K.J., “Approximation by Jakimovski-Leviatan-Păltănea operators involving Sheffer polynomials”, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 113(2): 1251-1265, (2019).
• [12] Gadjiev, A. D., Efendiev, R. O., İbikli, E., “Generalized Bernstein-Chlodowsky polynomials”, Rocky Mountain Journal of Mathematics, 28(4): 1267-1277, (1998).
• [13] Mursaleen, M., Ansari, K.J., “On Chlodowsky variant of Szász operators by Brenke type polynomials”, Applied Mathematics and Computation, 271: 991-1003, (2015).
• [14] Mursaleen, M., AL-Abied A., Ansari, K.J., “Rate of convergence of Chlodowsky type Durrmeyer Jakimovski-Leviatan operators”, Tbilisi Mathematical Journal, 10(2): 173-184, (2017).
• [15] Neer, T., Acu, A.M., Agrawal, P.N., “Degree of approximation by Chlodowsky variant of Jakimovski-Leviatan-Durrmeyer type operators”, Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 113, 3445-3459, (2019).
• [16] DeVore R. A., Lorentz, G. G., “Constructive Approximation”, Springer-Verlag, Berlin, (1993).
• [17] Gadjiev, A.D., “The convergence problem for a sequence of positive linear operators on bounded sets and theorems analogous to that of P.P. Korovkin”, Dokl. Akad. Nauk SSSR, 218(5): 1433-1436, (1974). (transl.in Sov. Math. Dokl. 15(5), 1974)
• [18] Altomare, F., Campiti, M., “Korovkin-type Approximation Theory and its Applications”, De Gruyter Studies in Mathematics 17, W. De Gruyter, Berlin-New York, (1994).
• [19] Yüksel, İ., İspir, N., “Weighted Approximation by a Certain Family of Summation Integral-Type Operators”, Computers and Mathematics with Applications, 52(10-11): 1463-1470, (2006).