Some Approximation Properties of $\alpha$-Stancu-Chlodowsky Operators

Year 2026, Volume: 9 Issue: 1 , 50 - 62 , 30.03.2026

Reşat Aslan

https://doi.org/10.33401/fujma.1857068

https://izlik.org/JA27WF47XB

Abstract

In the present paper, we construct Stancu variant of $\alpha$-Chlodowsky operators based on a parameter $0\leq\alpha\leq1$. We calculate some needed moment estimates. Next, we investigate direct results of the proposed operators. Also, we derive the order of convergence in terms of the weighted modulus of continuity and in order to check the asymptotic behavior of proposed operators we present Voronovskaya's type approximation theorem. Finally, we provide various illustrations and numerical example to demonstrate the convergence performance, accuracy and significance of the constructed operators.

Keywords

Stancu-Chlodowsky operators , Shape parameter $\alpha$ , Weighted convergence , Voronovskaya's type theorem

References

• [1] S.N. Bernstein, Demonstration du theoreme de Weierstrass fondee sur le calcul des probabilites, Comp. Comm. Soc. Mat. Charkow Ser., 13 (1912), 1–2. $ \href{https://www.mathnet.ru/links/9fc3c50b832c6d3a29e3fd0fbd398e4f/khmo107.pdf}{\,\mbox{[Web]}} $
• [2] I. Chlodowsky, Sur le developpment des fonctions definies dans un intervalle infini en series de polynomes de S. N. Bernstein, Compositio Math., 4 (1937), 380–392. $ \href{https://www.semanticscholar.org/paper/Sur-le-developpement-des-fonctions-definies-dans-un-Chlodovsky/ef04e9d5430586729c5bd9f07c5e8d1b89bb6bab}{\,\mbox{[CrossRef]}} $
• [3] D.D. Stancu, Asupra unei generalizari a polinoamelor lui Bernstein, Studia Univ. Babes-Bolyai Ser. Math.-Phys., 41 (1969), 31–45.
• [4] J.L. Durrmeyer, Une formule d’inversion de la transformee de Laplace: Applications a la theorie des moments, these de 3e cycle, Faculte des Sciences de l’Universite de Paris, 1971.
• [5] E. İbikli, On Stancu type generalization of Bernstein-Chlodowsky polynomials, Mathematica, 42(65) (2000), 37–43.
• [6] M. Sofyalıoğlu, K. Kanat and B. Çekim, Parametric generalization of the modified Bernstein operators, Filomat, 36(5) (2022), 1699–1709. $ \href{https://doi.org/10.2298/FIL2205699S}{\,\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85133701181?origin=resultslist}{\,\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000834761900021}{\,\mbox{[Web of Science]}} $
• [7] İ. Büyükyazıcı, Approximation by Stancu-Chlodowsky polynomials, Comput. Math. Appl., 59 (1)(2010), 274–282. $ \href{https://doi.org/10.1016/j.camwa.2009.07.054}{\,\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/73449092800?origin=resultslist}{\,\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000274004500028}{\,\mbox{[Web of Science]}} $
• [8] A. Aral and T. Acar, Weighted approximation by new Bernstein-Chlodowsky-Gadjiev operators, Filomat, 27(2) (2013), 371–380. $ \href{https://doi.org/10.2298/FIL1302371A}{\,\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/84880098673?origin=resultslist}{\,\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000322027700017}{\,\mbox{[Web of Science]}} $
• [9] X. Chen, J. Tan, Z. Liu and J. Xie, Approximation of functions by a new family of generalized Bernstein operators, J. Math. Anal. Appl., 450(1) (2017), 244–261. $ \href{https://doi.org/10.1016/j.jmaa.2016.12.075}{\,\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85009754923?origin=resultslist}{\,\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000394404800015}{\,\mbox{[Web of Science]}} $
• [10] S.A. Mohiuddine, T. Acar and A. Alotaibi, Construction of a new family of Bernstein-Kantorovich operators, Math. Meth. Appl. Sci., 40(18) (2017), 7749–7759. $\href{https://doi.org/10.1002/mma.4559}{\,\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85040983645?origin=resultslist}{\,\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000419218900100}{\,\mbox{[Web of Science]}} $
• [11] A. Aral and H. Erbay, Parametric generalization of Baskakov operators, Math. Commun., 24(1) (2019), 119–131. $ \href{https://www.scopus.com/pages/publications/85059053748?origin=resultslist}{\,\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000482798000009}{\,\mbox{[Web of Science]}} $
• [12] P. Agrawal, D. Kumar and B. Baxhaku, On the rate of convergence of modified a-Bernstein operators based on q-integers, J. Numer. Anal. Approx. Theory, 51(1) (2022), 3–36. $ \href{https://doi.org/10.33993/jnaat511-1244}{\,\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85138221763?origin=resultslist}{\,\mbox{[Scopus]}} $
• [13] F. Özger, H.M. Srivastava and S.A. Mohiuddine, Approximation of functions by a new class of generalized Bernstein-Schurer operators, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math., 114(4) (2020), 173. $ \href{https://doi.org/10.1007/s13398-020-00903-6}{\,\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85088294504?origin=resultslist}{\,\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000553515500002}{\,\mbox{[Web of Science]}} $
• [14] S. Berwal, S.A. Mohiuddine, A. Kajla and A. Alotaibi, Approximation by Riemann-Liouville type fractional a-Bernstein-Kantorovich operators, Math. Meth. Appl. Sci., 47(11) (2024), 8275–8288. $ \href{https://doi.org/10.1002/mma.10014}{\,\mbox{[CrossRef]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001206269700001}{\,\mbox{[Web of Science]}} $
• [15] N. Rao, M. Farid and M. Raiz, Symmetric properties of l-Szasz operators coupled with generalized beta functions and approximation theory, Symmetry, 16(12) (2024), 1703. $ \href{https://doi.org/10.3390/sym16121703}{\,\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85213227389?origin=resultslist}{\,\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001386589900001}{\,\mbox{[Web of Science]}}$
• [16] P. Patel, Some approximation properties of new families of positive linear operators, Filomat, 33(17) (2019), 5477–5488. $ \href{https://doi.org/10.2298/FIL1917477P}{\,\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85077874616?origin=resultslist}{\,\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000503870100006}{\,\mbox{[Web of Science]}} $
• [17] S.A. Mohiuddine and F. Özger, Approximation of functions by Stancu variant of Bernstein-Kantorovich operators based on shape parameter a, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math., 114(2) (2020), 70. $ \href{https://doi.org/10.1007/s13398-020-00802-w}{\,\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85078635513?origin=resultslist}{\,\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000514593700002}{\,\mbox{[Web of Science]}} $
• [18] N. Turhan, F. Özger and M. Mursaleen, Kantorovich-Stancu type (a, l, s)-Bernstein operators and their approximation properties, Math. Comput. Model. Dyn. Syst., 30(1) (2024), 228–265. $ \href{https://doi.org/10.1080/13873954.2024.2335382}{\,\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85190362917?origin=resultslist}{\,\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001202221600001}{\,\mbox{[Web of Science]}} $
• [19] J. Yadav, S.A. Mohiuddine, A. Kajla and A. Alotaibi, a-Bernstein integral type operators, Bull. Iran. Math. Soc., 49(5) (2023), 59. $ \href{https://doi.org/10.1007/s41980-023-00806-3}{\,\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85169078532?origin=resultslist}{\,\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001055252000001}{\,\mbox{[Web of Science]}} $
• [20] R. Aslan and M. Mursaleen, Some approximation results on a class of new type l-Bernstein polynomials, J. Math. Inequal., 16(2) (2022), 445–462. $ \href{https://doi.org/10.7153/jmi-2022-16-32}{\,\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85133519055?origin=resultslist}{\,\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000822722600001}{\,\mbox{[Web of Science]}} $
• [21] K.J. Ansari and F. Usta, A generalization of Szasz-Mirakyan operators based on a non-negative parameter, Symmetry, 14(8) (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]}} $
• [22] 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(4) (2022), 1–18. $ \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]}} $
• [23] M. Ayman-Mursaleen, Quadratic function preserving wavelet type Baskakov operators for enhanced function approximation, Comput. Appl. Math., 44(8) (2025), 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]}} $
• [24] J. Kaur, M. Goyal and K.J. Ansari, A generalization of modified a-Bernstein operators and its related estimations and errors, Arab. J. Math., 13(3) (2024), 521–531. $ \href{https://doi.org/10.1007/s40065-024-00482-z}{\,\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85210183084?origin=resultslist}{\,\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001362580100001}{\,\mbox{[Web of Science]}} $
• [25] Md. Nasiruzzaman and A.F. Aljohani, Approximation by a-Bernstein–Schurer operators and shape preserving properties via q-analogue, Math. Meth. Appl. Sci., 46(2) (2023), 2354–2372. $ \href{https://doi.org/10.1002/mma.8649}{\,\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85135890922?origin=resultslist}{\,\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000842356000001}{\,\mbox{[Web of Science]}} $
• [26] Z. Ödemiş Özger, S. Bansal, R. Aslan and N. Rao, On a novel class of (l;m)-Bernstein-Stancu operators: Approximation results on Bögel spaces and associated graphical error estimates, Comput. Appl. Math., 45(3) (2026), 98. $\href{https://doi.org/10.1007/s40314-025-03553-9}{\,\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/105024850938?origin=resultslist}{\,\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001640210600029}{\,\mbox{[Web of Science]}} $
• [27] M. Raiz, R.S. Rajawat and V.N. Mishra, a-Schurer Durrmeyer operators and their approximation properties, An. Univ. Craiova Ser. Mat. Inform., 50(1) (2023), 189–204. $ \href{https://doi.org/https://www.scopus.com/pages/publications/85166752737?origin=resultslist}{\,\mbox{[CrossRef]}} \href{10.52846/ami.v50i1.1663}{\,\mbox{[Scopus]}} $
• [28] N. Rao, M. Shahzad and N.K. Jha, Study of two dimensional a-modified Bernstein bi-variate operators, Filomat, 39(5) (2025), 1509–1522. $ \href{https://doi.org/10.2298/FIL2505509R}{\,\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/105000101681?origin=resultslist}{\,\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001469415800007}{\,\mbox{[Web of Science]}} $
• [29] R. Aslan, Some approximation properties of Riemann-Liouville type fractional Bernstein-Stancu-Kantorovich operators with order of a, Iranian J. Sci., 49(2) (2025), 481–494. $ \href{https://doi.org/10.1007/s40995-024-01754-1}{\,\mbox{[CrossRef]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001382366200001}{\,\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001382366200001}{\,\mbox{[Web of Science]}}$
• [30] N.F. Odabaşi and İ. Yüksel, Parametric extension of a certain family of summation-integral type operators, Cumhuriyet Sci. J., 44(2) (2023), 315–327. $ \href{https://doi.org/10.17776/csj.1173496}{\,\mbox{[CrossRef]}} $
• [31] N.L. Braha, T. Mansour and H.M. Srivastava, A parametric generalization of the Baskakov-Schurer-Szasz-Stancu approximation operators, Symmetry, 13(6) (2021), 1–24. $ \href{https://doi.org/10.3390/sym13060980}{\,\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85108008108?origin=resultslist}{\,\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000666292400001}{\,\mbox{[Web of Science]}} $
• [32] M. Ayman-Mursaleen, Md. Nasiruzzaman, N. Rao, M. Dilshad and K.S. Nisar, Approximation by the modified l-Bernstein polynomial in terms of basis function, AIMS Math., 9(2) (2024), 4409–4426. $ \href{https://doi.org/10.3934/math.2024217}{\,\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/85182641046?origin=resultslist}{\,\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001156350000001}{\,\mbox{[Web of Science]}} $
• [33] J.J. Quan, R. Aslan, İ. Yüksel, N.F. Odabaşı and Q.B. Cai, On approximation properties of a-Baskakov-Schurer-Stancu operators: graphical investigations, J. Inequal. Appl., 2025(1) (2025), 1–16. $ \href{https://doi.org/10.1186/s13660-025-03377-5}{\,\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/105018880331?origin=resultslist}{\,\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001594103300002}{\,\mbox{[Web of Science]}} $
• [34] M. Smuc, On a Chlodovsky variant of a-Bernstein operators, Bull. Transilv. Univ. Brasov, Ser. III, Math. Inform. Phys., 59 (2017), 165–178. $ \href{https://webbut.unitbv.ro/index.php/Series_III/article/view/1795/1540}{\,\mbox{[Web]}} $
• [35] R.A. DeVore and G.G. Lorentz, Constructive Approximation,Springer, Heidelberg, (1993).
• [36] A.D. Gadzhiev, Theorems of the type P.P. Korovkin theorems, Math. Zametki; English transl. in Math. Notes, 20 (1976), 781–786.
• [37] A.D. Gadzhiev, The convergence problem for a sequence of linear operators on unbounded sets and theorem analogous to that of P.P. Korovkin, Soviet Math. Dokl., 15 (1974), 1433–1436. $\href{https://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=dan&paperid=38607&option_lang=eng}{\,\mbox{[Web]}} $
• [38] A.D. Gadzhiev, R.O. Efendiev and E. İbikli, Generalized Bernstein-Chlodowsky polynomials, Rocky Mountain J. Math., 28(4) (1998), 1267– 1277. $ \href{https://doi.org/10.1216/rmjm/1181071716}{\,\mbox{[CrossRef]}} \href{https://www.scopus.com/pages/publications/0032223487?origin=resultslist}{\,\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000079804000005}{\,\mbox{[Web of Science]}} $