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

Year 2023, Volume: 44 Issue: 2, 315 - 327, 30.06.2023

Nadire Fulda Odabaşı , İsmet Yüksel

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.

Keywords

rate of convergence weighted spaces weighted modulus of continuity

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, Volume: 44 Issue: 2, 315 - 327, 30.06.2023

Nadire Fulda Odabaşı , İsmet Yüksel

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 2023Volume: 44 Issue: 2

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