Convergence estimates for some composition operators

Year 2024, , 69 - 76, 15.06.2024

Vijay Gupta , Ruchi Gupta

https://doi.org/10.33205/cma.1474535

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

• 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
• T. Acar, A. Aral and I. Ra¸sa: Positive linear operators preserving τ and τ2, Constr. Math. Anal., 2 (3) (2019), 98–102.
• T. Acar, V. Gupta and A. Aral: Rate of convergence for generalized Szász operators, Bull. Math. Sci., 1 (2011), 99–113.
• 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 .
• 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).
• A. Aral: On a new approach in the space of measurable functions, Constr. Math. Anal., 6 (4) (2023), 237–248.
• A. Aral, V. Gupta: On the q analogue of Stancu-Beta operators, Applied Mathematics Letters, 25 (1) (2012), 67–71.
• J. Bustamante: Directs estimates and a Voronovskaja-type formula for Mihesan operators, Constr. Math. Anal., 5 (4) (2022), 202–213.
• J. Bustamante: Weighted approximation by generalized Baskakov operators reproducing affine functions, Modern Math. Methods, 1 (1) (2023), 30–42.
• R. A. DeVore, G. G. Lorentz: Constructive Approximation, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, New York-London (1933).
• 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.
• Z. Finta: King operators which preserve xj , Constr. Math. Anal., 6 (2) (2023), 90–101.
• 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.
• N. K. Govil, V. Gupta and D. Soyba¸s: Certain new classes of Durrmeyer type operators, Appl. Math. Comput., 225 (2013), 195–203.
• V. Gupta: Convergence of operators based on some special functions, Rev. R. Acad. Cienc. Exactas Fìs. Nat. Ser. A Mat. RACSAM, 118 (2024), 99.
• V. Gupta: New operators based on Laguerre polynomials, Rev. R. Acad. Cienc. Exactas Fìs. Nat. Ser. A Mat. RACSAM, 118 (2024), 19.
• V. Gupta: Convergence estimates for gamma operator, Bull. Malays. Math. Sci. Soc., 43 (3) (2020), 2065–2075.
• V. Gupta: A form of Gamma operators due to Rathore, Rev. R. Acad. Cienc. Exactas Fìs. Nat. Ser. A Mat. RACSAM, 117 (2023), 81.
• V. Gupta, N. Malik: Direct estimations of new generalized Baskakov-Szász operators, Publ. Math. Inst. (Beograd), 99 (113) (2016), 265–279.
• V. Gupta, G. Tachev: Approximation with Positive Linear Operators and Linear Combinations, Series: Developments in Mathematics, 50, Springer, Cham (2017).
• 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.
• V. Gupta, G. Tachev and A. M. Acu: Modified Kantorovich operators with better approximation properties, Numer Algor., 81 (2019), 125–149.
• N. Ispir: On modified Baskakov operators on weighted spaces, Turkish J. Math., 25 (2001), 355–365.
• R. Pˇaltˇanea: Estimates of approximation in terms of a weighted modulus of continuity, Bull. Transilvania Univ. Brasov, 4 (53) (2011), 67–74.