Approximation and Estimation Errors of New Kind of Laugerre and Rathore Operators

Year 2025, Volume: 8 Issue: 4, 212 - 224, 30.12.2025

Jaspreet Kaur , Meenu Goyal , Khursheed Ansari

Abstract

In the present article, we study the approximation properties of new discrete operators based on Laguerre polynomials constructed by Gupta [28]. We study the convergence and rate of approximation for these operators on compact interval. Also, we prove some quantitative Voronovskaja type results as well as Grüss type asymptotic formulae with appropriate modulus of continuity defined in weighted spaces. At last, we present the convergence by using certain examples.

Keywords

Positive linear operators , Weighted approximation , Voronovskaja type asymptotic theorem , Modulus of continuity.

References

• [1] D Aydın Arı and G. Uğur Yılmaz, A note on Kantorovich type operators which preserve affine functions, Fundam. J. Math. Appl., 7(1) (2024), 53–58. $ \href{https://doi.org/10.33401/fujma.1424382}{\mbox{[CrossRef]}} $
• [2] S.N. Bernstein, Demonstration du theoreme de Weierstrass fond´ee sur le calcul des probabilites, Commun. Soc. Math. Kharkov, 13 (1913), 1–2. $ \href{https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=khmo&paperid=107&option_lang=rus}{\mbox{[Web]}} $
• [3] P.L. Butzer, Linear combinations of Bernstein polynomials, Canad. J. Math., 5(2) (1953), 559–567. $ \href{https://doi.org/10.4153/CJM-1953-063-7}{\mbox{[CrossRef]}} $
• [4] Q. Chai, B. Lian and G. Zhou, Approximation properties of l-Bernstein, J. Inequal. Appl., 61 (2018), 1–11. $ \href{https://doi.org/10.1186/s13660-018-1653-7}{\mbox{[CrossRef]}} $
• [5] H. Çiçek and A. İzgi, Approximation by modified bivariate Bernstein-Durrmeyer and GBS bivariate Bernstein Durrmeyer operators on a triangular region, Fundam. J. Math. Appl., 5(2) (2022), 135–145. $ \href{https://doi.org/https://doi.org/10.33401/fujma.1009058}{\mbox{[CrossRef]}} $
• [6] Z. Ditzian and V. Totik, Moduli of Smoothness, Springer, New York, (1987). $ \href{https://books.google.com.tr/books?hl=tr&lr=&id=ctXTBwAAQBAJ&oi=fnd&pg=PA2&dq=Moduli+of+Smoothness&ots=J0Kz1TXuMH&sig=gB0chi0EqeumP98zTI1yCqFQCZo&redir_esc=y#v=onepage&q=Moduli%20of%20Smoothness&f=false}{\mbox{[Web]}} $
• [7] A.D. Gadjiev and A.M. Ghorbanalizadeh, Approximation properties of a new type Bernstein–Stancu polynomials of one and two variables, Appl. Math. Comput., 216(3) (2010), 890–901. $ \href{https://doi.org/10.1016/j.amc.2010.01.099}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/77949487975?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000275739200020}{\mbox{[Web of Science]}} $
• [8] V. Gupta and R.P. Agarwal, Convergence Estimates in Approximation Theory, Springer, New York, (2014). $ \href{https://www.scopus.com/pages/publications/84930521181?origin=resultslist}{\mbox{[Scopus]}} $
• [9] D.D. Stancu, Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pure Appl., 13 (1968), 1173–1194. $\href{https://cir.nii.ac.jp/crid/1370565164571100049}{\mbox{[Web]}} $
• [10] L.V. Kantorovich, Sur certains developpements suivant les polynomes de la forme de S. Bernstein, C. R. Acad. Sci. USSR, 20, (1930), 563–568.
• [11] J.L. Durrmeyer, Une formule d’inversion de la transformee de Laplace: applications a la theorie des moments, These de 3e cycle, Paris, (1967). $ \href{https://search.worldcat.org/title/Une-formule-d'inversion-de-la-transformee-de-Laplace-:-Applications-a-la-theorie-des-moments/oclc/493213510}{\mbox{[Web]}} $
• [12] R. Aslan, Some approximation properties of Riemann-Liouville type fractional Bernstein-Stancu-Kantorovich operators with order of a, Iran. J. Sci., 49 (2025), 481–494. $ \href{https://doi.org/10.1007/s40995-024-01754-1}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/105001070638?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001382366200001}{\mbox{[Web of Science]}} $
• [13] N. Deo and R. Pratap, a-Bernstein–Kantorovich operators, Afr. Mat., 31 (2020), 609–618. $ \href{https://doi.org/10.1007/s13370-019-00746-4}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85075692720?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000499619200002}{\mbox{[Web of Science]}} $
• [14] M.M. Derriennic, Sur l’approximation de fonctions integrables sur [0,1] par des polynomes de Bernstein modifies, J. Approx. Theory, 31 (1981), 323–343. $ \href{https://doi.org/10.1016/0021-9045(81)90101-5}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/0010994129?origin=resultslist}{\mbox{[Scopus]}} $
• [15] V. Gupta, A large family of linear positive operators, Rend. Circ. Mat. Palermo II Ser., 69(3) (2019), 1–9. $\href{https://doi.org/10.1007/s12215-019-00430-3}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85068168064?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000574481000002}{\mbox{[Web of Science]}} $
• [16] V. Gupta, A note on the general family of operators preserving linear functions, RACSAM, 113 (2019), 3717–3725. $ \href{https://doi.org/10.1007/s13398-019-00727-z}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85069641032?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000483725900048}{\mbox{[Web of Science]}} $
• [17] K.J. Ansari F. Özger and Z. Ödemiş Özger, Numerical and theoretical approximation results for Schurer Stancu operators with shape parameter l, Comput. Appl. Math., 41 (2022), 181. $ \href{https://doi.org/10.1007/s40314-022-01877-4}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85130260118?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000796970600001}{\mbox{[Web of Science]}} $
• [18] K.J. Ansari and F. Usta, A generalization of Szasz-Mirakyan operators based on a non-negative parameter, Symmetry, 14 (2022), 1596. $\href{https://doi.org/10.3390/sym14081596}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85137361549?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000845543000001}{\mbox{[Web of Science]}} $
• [19] R. Aslan, Some approximation results on l-Szasz-Mirakjan-Kantorovich operators, Fundam. J. Math. Appl., 4(3) (2021), 150–158. $ \href{https://doi.org/https://doi.org/10.33401/fujma.903140}{\mbox{[CrossRef]}} $
• [20] E. Baytunç, H. Aktuğlu and N.I. Mahmudov, A new generalization of Szasz-Mirakjan Kantorovich operators for better error estimation, Fundam. J. Math. Appl., 6(4) (2023), 194–210. $ \href{https://doi.org/https://doi.org/10.33401/fujma.1355254}{\mbox{[CrossRef]}} $
• [21] A. Kajla, P. Sehrawat and K.J. Ansari, Higher order Sz´asz–Kantorovich operators linking Charlier polynomials, Math. Found. Comput., 10 (2025), 165–176. $ \href{https://doi.org/https://doi.org/10.3934/mfc.2025030}{\mbox{[CrossRef]}} $
• [22] M.A. Mursaleen, Quadratic function preserving wavelet type Baskakov operators for enhanced function approximation, Comput. Appl. Math., 44(8) (2025), Article 395. $ \href{https://doi.org/10.1007/s40314-025-03357-x}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/105012481797?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001543422700001}{\mbox{[Web of Science]}} $
• [23] N. Rao, M. Farid and M. Raiz, On the approximations and symmetric properties of Frobenius-Euler-Şimşek polynomials connecting Szasz operators, Symmetry, 17(5) (2025), 648. $ \href{https://doi.org/10.3390/sym17050648}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/105006855372?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001496915100001}{\mbox{[Web of Science]}} $
• [24] R.K.S. Rathore, Linear combinations of linear positive operators and generating relations of special functions, Ph.D. Thesis, Delhi, (1973).
• [25] U. Abel and V. Gupta, The rate of convergence of a generalization of Post-Widder operators and Rathore operators, Adv. Oper. Theory, 8(3) (2023), 43. $ \href{https://doi.org/10.1007/s43036-023-00272-y}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85160247426?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000995348300001}{\mbox{[Web of Science]}} $
• [26] V. Gupta, A form of Gamma operator due to Rathore, Rev. Real Acad. Cienc. Exactas Fıs. Nat. Ser. A-Mat., 117 (2023), 81. $ \href{https://doi.org/10.1007/s13398-023-01413-x}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85149960525?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000943359200001}{\mbox{[Web of Science]}} $
• [27] S. Sucu, G. Icox and S. Varma, On some extensions of Szasz operators introducing Boas–Buck-type polynomials, Abstr. Appl. Anal., (2012), 680340. $ \href{https://doi.org/10.1155/2012/680340}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/84867024615?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000308491300001}{\mbox{[Web of Science]}} $
• [28] V. Gupta, New operators based on Laguerre polynomials, Rev. Real Acad. Cienc. Exactas Fıs. Nat. Ser. A-Mat., 118 (2024), 19. $ \href{https://doi.org/10.1007/s13398-023-01521-8}{\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85175709478?origin=resultslist}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001096267000002}{\mbox{[Web of Science]}} $
• [29] J. Peetre, A theory of interpolation of normed spaces, Notas de Mathematica, Inst. Mat. Pura e Aplicada, Rio de Janeiro, 39 (1968).
• [30] P.L. Butzer, and H. Berens, Semi-Groups of Operators and Approximation, Springer, New York, (1967). $ \href{https://books.google.com.tr/books?hl=tr&lr=&id=PgbpCAAAQBAJ&oi=fnd&pg=PA1&dq=.L.+Butzer,+H.+Berens,+Semi-Groups+of+Operators+and+Approximation,+Springer,+New+York,+(1967).&ots=3ga8J0P_Li&sig=vjis0LGDj1G7EWQ8bKOcwnnlKPU&redir_esc=y#v=onepage&q&f=false}{\mbox{[Web]}} $
• [31] P.P. Korovkin, On convergence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk SSSR, 90 (1959), 961–964. $ \href{https://cir.nii.ac.jp/crid/1370565164571100042}{\mbox{[Web]}} $