Research Article
BibTex RIS Cite

Parametric Extension of a Certain Family of Summation-Integral Type Operators

Year 2023, , 315 - 327, 30.06.2023
https://doi.org/10.17776/csj.1173496

Abstract

In this paper, we introduce a parametric extension of a certain family of summation-integral type operators on the interval [0,∞). Firstly, we obtain test functions and central moments. Secondly, we investigate weighted approximation properties for these operators and estimate the rate of convergence. Then, we give a pointwise approximation for the Peetre K-functional and functions of the Lipschitz class. Moreover, we demonstrate Voronovskaja type theorem for the operators. Finally, the convergence properties of operators to some functions are illustrated by graphics.

References

  • [1] Weierstrass K., Uber die analytische Darstellbarkeit sogenannter willkurlicher Functionen einer reellen Veranderlichen Sitzungsberichteder, Koniglich Preussischen Akedemie der Wissenschcaften zu Berlin, (1885) 633-639, 789-805.
  • [2] Baskakov V.A., An instance of sequance of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk. SSSR (N.S.), 113 (1957) 249-251.
  • [3] Durrmeyer J.L., Une Formule de d’inversion de la transform’ee de Laplace: applications`a la th’eorie des moments, These de 3e cycle, Facult’e des Sciences de I’Universit’e de Paris, (1967).
  • [4] May C.P., On Phillips operator, J. Approximation Theory, 20(4) (1977) 315–332.
  • [5] Srivastava H.M., Gupta V., A certain family of summation-integral type operators, Math. Comput. Modelling, 37(12-13) (2003) 1307-1315.
  • [6] İspir N., Yüksel İ., On the Bezier variant of Srivastava-Gupta operators, Appl. Math. E-Notes, 5 (2005) 129-137.
  • [7] Yüksel İ., İspir N., Weighted approximation by a certain family of summation integral-type operators, Comput. Math. Appl., 52(10-11) (2006) 1463-1470.
  • [8] Chen X., Tan J., Lıu Z., J.Xie J., Approximation of functions by a new family of generalized Bernstein operators. J.Math. Anal. Appl. 450 (2017), 244-261.
  • [9] Cai Q.-B., Lian B.-Y., Zhou G., Approximation properties of λ-Bernstein operators, J. Inequal. Appl., 61 (2018) 1-11.
  • [10] Aral A., Erbay H., Parametric generalization of Baskakov operators, Math. Commun., 24(1) (2019) 119–131.
  • [11] Yüksel İ., Dinlemez Kantar Ü., Altın B., On approximation of Baskakov-Durrmeyer type operators of two variables, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 78(1) (2016) 123-134.
  • [12] İnce İlarslan H.G., Erbay H., Aral A., Kantorovich-type generalization of parametric Baskakov operators,Math. Methods Appl. Sci., 42(18) (2019) 6580-6587.
  • [13] Nasiruzzaman Md., Rao N., Wazir S., Kumar R., Approximation on parametric extension of Baskakov-Durrmeyer operators on weighted spaces, J. Inequal. Appl., 103 (2019).
  • [14] Cai Q.-B., Dinlemez Kantar Ü., Çekim B., Approximation properties for the genuine modified Bernstein-Durrmeyer-Stancu operators, Appl. Math. J. Chinese Univ. Ser. B, 35(4) (2020) 468-478.
  • [15] Mohiuddine S.A., Özger F., Approximation of functions by Stancu variant of Bernstein-Kantorovich operators based on shape parameter α, Rev. R. Acad. Clenc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 114(70) (2020).
  • [16] Çetin N., Approximation by α-Bernstein-Schurer operator, Hacet. J. Math. Stat., 50(3) (2021) 732-743.
  • [17] Cai Q.-B., Torun G., Dinlemez Kantar Ü., Approximation properties of generalized λ-Bernstein-Stancu-Type operators, J. Math., (2021) 1-17.
  • [18] Sofyalıoğlu M., Kanat K., Çekim B., Parametric generalizaiton of the Meyer-König-Zeller operators, Chaos Solitions Fractals, 152 (2021).
  • [19] Cai Q.-B., Sofyalıoğlu M., Kanat K., Çekim B., Some approximation results for the new modification of Bernstein-Beta operators, AIMS Math., 7(2) (2022) 1831-1844.
  • [20] Torun G., Mercan Boyraz M., Dinlemez Kantar Ü., Investigation of the asymptotic behavior of generalized Baskakov-Durrmeyer-Stancu type operators, Cumhuriyet Sci. J., 43(1) (2022) 98-104.
  • [21] Mishra V.N., Patel P., Approximation properties of q-Baskakov-Durrmeyer-Stancu Operators, Math. Sci., 7 (2013) 38.
  • [22] Gupta V., Aral A., Some approximation properties of q-Baskakov-Durrmeyer operators, Appl. Math. Comput., 218(3) (2011) 783-788.
  • [23] Erdoğan S., Olgun A., Approximation properties of modified Jain-Gamma operators, Carpathian Math. Publ., 13(3) (2021) 651-665.
  • [24] Mishra V.N., Khatri K., Mishra L.N., Deepmala, Inverse result in simultaneous approximation by Baskakov-Durrmeyer-Stancu operators, J. Inequal. Appl, (2013) 586.
  • [25] Mishra V.N., Khatri K., Mishra L.N., Statistical approximation by Kantorovich type discrete q-beta operators, Adv. Difference Equ., (2013) 345.
  • [26] Gandhi R.B., Deepmala, Mishra V.N., Local and global results for modified Szász-Mirakjan operators, Math. Methods Appl. Sci., 40(7) (2017) 2491–2504.
  • [27] Mishra L.N., Pandey S., Mishra V.N., King type generalization of Baskakov operators based on (p,q) calculus with better approximation properties, Analysis 40(4) (2020) 163–173.
  • [28] Mishra L.N., Mishra V.N., Approximation by Jakimovski-Leviatan-Paltanea operators involving Boas-Buck-type polynomials, Mathematics in Engineering, Science and Aerospace(MESA), 12(4) (2021) 1153-1165.
  • [29] Sofyalıoğlu M., Kanat K., Approximation by the new modification of Bernstein-Stancu operators, Cumhuriyet Sci. J., 42(4) (2021) 862-872.
  • [30] Raiz M., Kumar A., Mishra V.N., Rao N., Dunkl analogue of Szász-Schurer-Beta operators and their approximation behaviour, Math. Found. Comput., 5 (2022) 315-330.
  • [31] Khatri K., Mishra V.N., Generalized Szász-Mirakyan operators involving Brenke type polynomials, Appl. Math. Comput., 324 (2018) 228–238.
  • [32] Gadzhiev A.D., The convergence problem for a sequence of positive linear operators on unbounded sets, and theorems analogous to that of P.P. Korovkin, (in Russian), Dokl. Akad. Nauk. SSSR, 218(5) (1974) 1001-1004; (in English), Sov. Math. Dokl., 15(5) (1974) 1433-1436.
  • [33] Gadzhiev A., Theorems of the type P.P. Korovkin type theorems, Math. Zametki, 20(5) (1976) 781-786; English Translation, Math. Notes, 20(5/6) (1976) 996-998.
  • [34] İspir N., On modified Baskakov operators on weighted spaces, Turkish J. Math., 25(3) (2001) 355-365.
  • [35] Shisha O., Mond B., The degree of convergence of sequences of linear positive operators, Proc. Nat. Acad. Sci. U.S.A., 60 (1968) 1196-1200.
  • [36] Peetre J., A theory of interpolation of normed spaces, notas de matematica 39, Instituto de Matematica Pura e Aplicada, Conselho Nacional de Pesquisas, Rio de Janeiro, (1968).
Year 2023, , 315 - 327, 30.06.2023
https://doi.org/10.17776/csj.1173496

Abstract

References

  • [1] Weierstrass K., Uber die analytische Darstellbarkeit sogenannter willkurlicher Functionen einer reellen Veranderlichen Sitzungsberichteder, Koniglich Preussischen Akedemie der Wissenschcaften zu Berlin, (1885) 633-639, 789-805.
  • [2] Baskakov V.A., An instance of sequance of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk. SSSR (N.S.), 113 (1957) 249-251.
  • [3] Durrmeyer J.L., Une Formule de d’inversion de la transform’ee de Laplace: applications`a la th’eorie des moments, These de 3e cycle, Facult’e des Sciences de I’Universit’e de Paris, (1967).
  • [4] May C.P., On Phillips operator, J. Approximation Theory, 20(4) (1977) 315–332.
  • [5] Srivastava H.M., Gupta V., A certain family of summation-integral type operators, Math. Comput. Modelling, 37(12-13) (2003) 1307-1315.
  • [6] İspir N., Yüksel İ., On the Bezier variant of Srivastava-Gupta operators, Appl. Math. E-Notes, 5 (2005) 129-137.
  • [7] Yüksel İ., İspir N., Weighted approximation by a certain family of summation integral-type operators, Comput. Math. Appl., 52(10-11) (2006) 1463-1470.
  • [8] Chen X., Tan J., Lıu Z., J.Xie J., Approximation of functions by a new family of generalized Bernstein operators. J.Math. Anal. Appl. 450 (2017), 244-261.
  • [9] Cai Q.-B., Lian B.-Y., Zhou G., Approximation properties of λ-Bernstein operators, J. Inequal. Appl., 61 (2018) 1-11.
  • [10] Aral A., Erbay H., Parametric generalization of Baskakov operators, Math. Commun., 24(1) (2019) 119–131.
  • [11] Yüksel İ., Dinlemez Kantar Ü., Altın B., On approximation of Baskakov-Durrmeyer type operators of two variables, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 78(1) (2016) 123-134.
  • [12] İnce İlarslan H.G., Erbay H., Aral A., Kantorovich-type generalization of parametric Baskakov operators,Math. Methods Appl. Sci., 42(18) (2019) 6580-6587.
  • [13] Nasiruzzaman Md., Rao N., Wazir S., Kumar R., Approximation on parametric extension of Baskakov-Durrmeyer operators on weighted spaces, J. Inequal. Appl., 103 (2019).
  • [14] Cai Q.-B., Dinlemez Kantar Ü., Çekim B., Approximation properties for the genuine modified Bernstein-Durrmeyer-Stancu operators, Appl. Math. J. Chinese Univ. Ser. B, 35(4) (2020) 468-478.
  • [15] Mohiuddine S.A., Özger F., Approximation of functions by Stancu variant of Bernstein-Kantorovich operators based on shape parameter α, Rev. R. Acad. Clenc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 114(70) (2020).
  • [16] Çetin N., Approximation by α-Bernstein-Schurer operator, Hacet. J. Math. Stat., 50(3) (2021) 732-743.
  • [17] Cai Q.-B., Torun G., Dinlemez Kantar Ü., Approximation properties of generalized λ-Bernstein-Stancu-Type operators, J. Math., (2021) 1-17.
  • [18] Sofyalıoğlu M., Kanat K., Çekim B., Parametric generalizaiton of the Meyer-König-Zeller operators, Chaos Solitions Fractals, 152 (2021).
  • [19] Cai Q.-B., Sofyalıoğlu M., Kanat K., Çekim B., Some approximation results for the new modification of Bernstein-Beta operators, AIMS Math., 7(2) (2022) 1831-1844.
  • [20] Torun G., Mercan Boyraz M., Dinlemez Kantar Ü., Investigation of the asymptotic behavior of generalized Baskakov-Durrmeyer-Stancu type operators, Cumhuriyet Sci. J., 43(1) (2022) 98-104.
  • [21] Mishra V.N., Patel P., Approximation properties of q-Baskakov-Durrmeyer-Stancu Operators, Math. Sci., 7 (2013) 38.
  • [22] Gupta V., Aral A., Some approximation properties of q-Baskakov-Durrmeyer operators, Appl. Math. Comput., 218(3) (2011) 783-788.
  • [23] Erdoğan S., Olgun A., Approximation properties of modified Jain-Gamma operators, Carpathian Math. Publ., 13(3) (2021) 651-665.
  • [24] Mishra V.N., Khatri K., Mishra L.N., Deepmala, Inverse result in simultaneous approximation by Baskakov-Durrmeyer-Stancu operators, J. Inequal. Appl, (2013) 586.
  • [25] Mishra V.N., Khatri K., Mishra L.N., Statistical approximation by Kantorovich type discrete q-beta operators, Adv. Difference Equ., (2013) 345.
  • [26] Gandhi R.B., Deepmala, Mishra V.N., Local and global results for modified Szász-Mirakjan operators, Math. Methods Appl. Sci., 40(7) (2017) 2491–2504.
  • [27] Mishra L.N., Pandey S., Mishra V.N., King type generalization of Baskakov operators based on (p,q) calculus with better approximation properties, Analysis 40(4) (2020) 163–173.
  • [28] Mishra L.N., Mishra V.N., Approximation by Jakimovski-Leviatan-Paltanea operators involving Boas-Buck-type polynomials, Mathematics in Engineering, Science and Aerospace(MESA), 12(4) (2021) 1153-1165.
  • [29] Sofyalıoğlu M., Kanat K., Approximation by the new modification of Bernstein-Stancu operators, Cumhuriyet Sci. J., 42(4) (2021) 862-872.
  • [30] Raiz M., Kumar A., Mishra V.N., Rao N., Dunkl analogue of Szász-Schurer-Beta operators and their approximation behaviour, Math. Found. Comput., 5 (2022) 315-330.
  • [31] Khatri K., Mishra V.N., Generalized Szász-Mirakyan operators involving Brenke type polynomials, Appl. Math. Comput., 324 (2018) 228–238.
  • [32] Gadzhiev A.D., The convergence problem for a sequence of positive linear operators on unbounded sets, and theorems analogous to that of P.P. Korovkin, (in Russian), Dokl. Akad. Nauk. SSSR, 218(5) (1974) 1001-1004; (in English), Sov. Math. Dokl., 15(5) (1974) 1433-1436.
  • [33] Gadzhiev A., Theorems of the type P.P. Korovkin type theorems, Math. Zametki, 20(5) (1976) 781-786; English Translation, Math. Notes, 20(5/6) (1976) 996-998.
  • [34] İspir N., On modified Baskakov operators on weighted spaces, Turkish J. Math., 25(3) (2001) 355-365.
  • [35] Shisha O., Mond B., The degree of convergence of sequences of linear positive operators, Proc. Nat. Acad. Sci. U.S.A., 60 (1968) 1196-1200.
  • [36] Peetre J., A theory of interpolation of normed spaces, notas de matematica 39, Instituto de Matematica Pura e Aplicada, Conselho Nacional de Pesquisas, Rio de Janeiro, (1968).
There are 36 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Natural Sciences
Authors

Nadire Fulda Odabaşı 0000-0001-5825-1177

İsmet Yüksel 0000-0002-2631-2382

Publication Date June 30, 2023
Submission Date September 10, 2022
Acceptance Date February 24, 2023
Published in Issue Year 2023

Cite

APA Odabaşı, N. F., & Yüksel, İ. (2023). Parametric Extension of a Certain Family of Summation-Integral Type Operators. Cumhuriyet Science Journal, 44(2), 315-327. https://doi.org/10.17776/csj.1173496