Adaptation of the Kantorovich Type Integral to the Dunkl Operator

Year 2022, , 334 - 346, 30.09.2022

Gurhan İçöz , Esma Gökmen

https://doi.org/10.54287/gujsa.1148199

Abstract

The purpose of this article is to show the adaptation of the Kantorovich type integral to the Dunkl operator. This article gives a sequence of operators to get an approximation result. The variant of the operator which is the Kantorovich type integral has been given and examined the approximation ratio by the first and second order modulus of continuity. The approximation order of the operators is shown by the first order modulus of continuity and the Lipschitz class functions.

Keywords

Dunkl Analogue, Kantorovich-type Integral, First and Second Modulus of Continuity, Lipschitz Class Function

References

• Altomare, F., & Campiti, M. (1994). Korovkin-type Approximation Theory and its Applications. Walter de Gruyter.
• Ben Cheikh, Y., & Gaied, M. (2007). Dunkl-Appell d-orthogonal polynomials. Integral Transforms and Special Functions, 18(8), 581-597. doi:10.1080/10652460701445302
• Bernstein, S. N. (1912). Demonstration of the Weierstrass theorem based on the calculation of probabilities. Common Soc. Math. Charkow Ser. 2t, 13, 1-2.
• DeVore, R. A., & Lorentz, G. G. (1993). Constructive approximation. Springer Berlin, Heidelberg.
• Gadzhiev, A. D. (1974). The convergence problem for a sequence of positive linear operators on unbounded sets, and theorems analogous to that of PP Korovkin. Dokl. Akad. Nauk SSSR, 218(5), 1001-1004.
• İçöz, G., & Çekim, B. (2016). Stancu-type generalization of Dunkl analogue of Szász-Kantorovich operators. Mathematical Methods in the Applied Sciences, 39(7), 1803-1810. doi:10.1002/mma.3602
• İçöz, G., Varma, S., & Sucu, S. (2016). Approximation by operators including generalized Appell polynomials. Filomat, 30(2), 429-440. doi:10.2298/FIL1602429I
• Jakimovski, A., & Leviatan, D. (1969). Generalized Szász operators for the approximation in the infinite interval. Mathematica (Cluj), 11(34), 97-103.
• Kanat, K., & Sofyalıoğlu, M. (2018). Approximation by (p,q)- Lupaş–Schurer–Kantorovich operators. Journal of Inequalities and Applications, 263. doi:10.1186/s13660-018-1858-9
• Kanat, K., & Sofyalıoğlu, M. (2019). On Stancu type generalization of (p, q)-Baskakov-Kantorovich operators. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(2), 1995-2013. doi:10.31801/cfsuasmas.478852
• Korovkin, P. P. (1953). Convergence of linear positive operators in the spaces of continuous functions. Dokl. Akad. Nauk. SSSR (N.S.), 90, 961-964.
• Lorentz, G. G. (1953). Bernstein Polynomials. University of Toronto Press.
• Mursaleen, M., Rahman, S., & Alotaibi, A. (2016). Dunkl generalization of q-Szász-Mirakjan Kantorovich operators which preserve some test functions. Journal of Inequalities and Applications, 317. doi:10.1186/s13660-016-1257-z
• Nasiruzzaman, M., & Aljohani, A. F. (2020a). Approximation by Szász-Jakimovski-Leviatan-type operators via aid of Appell polynomials. Journal of Function Spaces, 9657489. doi:10.1155/2020/9657489
• Nasiruzzaman, M., & Aljohani, A. F. (2020b). Approximation by parametric extension of Szász-Mirakjan-Kantorovich operators involving the Appell polynomials. Journal of Function Spaces, 8863664. doi:10.1155/2020/8863664
• Rosenblum, M. (1994). Generalized Hermite polynomials and the Bose-like oscillator calculus. In: A. Feintuch & I. Gohberg (Eds.), Nonselfadjoint Operators and Related Topics (pp. 369-396). Birkhäuser, Basel. doi:10.1007/978-3-0348-8522-5_15
• Sucu, S. (2014). Dunkl analogue of Szász operators. Applied Mathematics and Computation, 244, 42-48. doi:10.1016/j.amc.2014.06.088
• Sucu, S. (2020). Approximation by sequence of operators including Dunkl-Appell polynomials. Bulletin of the Malaysian Mathematical Sciences Society, 43(3), 2455-2464. doi:10.1007/s40840-019-00813-w
• Sucu, S., İçöz, G., & Varma, S. (2012). On some extensions of Szász operators including Boas-Buck-type polynomials. Abstract and Applied Analysis, 680340. doi:10.1155/2012/680340
• Szász, O. (1950). Generalization of S. Bernstein's polynomials to the infinite interval. Journal of Research of the National Bureau of Standards, 45(3), 239-245.
• Yazici, S., Yeşildal, F. T., & Çekim, B. (2022). On a generalization of Szász-Mirakyan operators including Dunkl-Appell polynomials. Turkish Journal of Mathematics, 46(6), 2353-2365. doi:10.55730/1300-0098.3273