Convergence estimates for some composition operators

Abstract

There are different methods available in literature to construct a new operator. One of the methods to construct an operator is the composition method. It is known that Baskakov operators can be achieved by composition of Post Widder $P_n$ and Sz\'asz-Mirakjan $S_n$ operators in that order, which is a discretely defined operator. But when we consider different order composition namely $S_n\circ P_n$, we get another different operator. Here we study such and we establish some convergence estimates for the composition operators $S_n\circ P_n$, along with difference with other operators. Finally we found the difference between two compositions by considering numeric values.

Keywords

• Szasz operators
• Post-Widder operators
• Moment generating function
• Convergence.

References

1. U. Abel, V. Gupta: On Composition of integral-type operators and discrete operators, Math Pannonica, (2024), DOI: https://doi.org/10.1556/314.2024.00001
2. T. Acar, A. Aral and I. Ra¸sa: Positive linear operators preserving τ and τ2, Constr. Math. Anal., 2 (3) (2019), 98–102.
3. T. Acar, V. Gupta and A. Aral: Rate of convergence for generalized Szász operators, Bull. Math. Sci., 1 (2011), 99–113.
4. A. M. Acu, T. Acar and V. A. Radu: Approximation by modified Uρn operators, Rev. R. Acad. Cienc. Exactas Fìs. Nat. Ser. A Mat. RACSAM, 113 (2019), 2715–2729 .
5. J. A. Adell, J. de la Cal: Preservation of moduli of continuity for Bernstein-type operators, In Proceedings of the International Conference on Approximation, Probability and Related Fields, Santa Barbara, (Edited by G. A. Anastassiou and S. T. Rachev), pp. 1-18, Plenum, (1994).
6. A. Aral: On a new approach in the space of measurable functions, Constr. Math. Anal., 6 (4) (2023), 237–248.
7. A. Aral, V. Gupta: On the q analogue of Stancu-Beta operators, Applied Mathematics Letters, 25 (1) (2012), 67–71.
8. J. Bustamante: Directs estimates and a Voronovskaja-type formula for Mihesan operators, Constr. Math. Anal., 5 (4) (2022), 202–213.

9. J. Bustamante: Weighted approximation by generalized Baskakov operators reproducing affine functions, Modern Math. Methods, 1 (1) (2023), 30–42.
10. R. A. DeVore, G. G. Lorentz: Constructive Approximation, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, New York-London (1933).
11. O. Dogru, V. Gupta: Monotonicity and the asymptotic estimate of Bleimann Butzer and Hahn operators based on q-integers, Georgian Mathematical J., 12 (3) (2005), 415–422.
12. Z. Finta: King operators which preserve xj , Constr. Math. Anal., 6 (2) (2023), 90–101.
13. A. D. Gadjiev: The convergence problem for a sequence of positive linear operators on unbounded sets, and theorems analogous to that of P. P. Korovkin, Dokl. Akad. Nauk SSSR, V. 218, N. 5, 1974, pp. 1001–1004.
14. N. K. Govil, V. Gupta and D. Soyba¸s: Certain new classes of Durrmeyer type operators, Appl. Math. Comput., 225 (2013), 195–203.
15. V. Gupta: Convergence of operators based on some special functions, Rev. R. Acad. Cienc. Exactas Fìs. Nat. Ser. A Mat. RACSAM, 118 (2024), 99.
16. V. Gupta: New operators based on Laguerre polynomials, Rev. R. Acad. Cienc. Exactas Fìs. Nat. Ser. A Mat. RACSAM, 118 (2024), 19.
17. V. Gupta: Convergence estimates for gamma operator, Bull. Malays. Math. Sci. Soc., 43 (3) (2020), 2065–2075.
18. V. Gupta: A form of Gamma operators due to Rathore, Rev. R. Acad. Cienc. Exactas Fìs. Nat. Ser. A Mat. RACSAM, 117 (2023), 81.
19. V. Gupta, N. Malik: Direct estimations of new generalized Baskakov-Szász operators, Publ. Math. Inst. (Beograd), 99 (113) (2016), 265–279.
20. V. Gupta, G. Tachev: Approximation with Positive Linear Operators and Linear Combinations, Series: Developments in Mathematics, 50, Springer, Cham (2017).
21. V. Gupta, G. Tachev: General form of Voronovskaja’s theorem in terms of weighted modulus of continuity, Results Math., 69 (3-4) (2016), 419–430.
22. V. Gupta, G. Tachev and A. M. Acu: Modified Kantorovich operators with better approximation properties, Numer Algor., 81 (2019), 125–149.
23. N. Ispir: On modified Baskakov operators on weighted spaces, Turkish J. Math., 25 (2001), 355–365.
24. R. Pˇaltˇanea: Estimates of approximation in terms of a weighted modulus of continuity, Bull. Transilvania Univ. Brasov, 4 (53) (2011), 67–74.