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Investigation of the Asymptotic Behavior of Generalized Baskakov-Durrmeyer-Stancu Type Operators

Year 2022, Volume: 43 Issue: 1, 98 - 104, 30.03.2022

Gülten Torun , Meliha Mercan Boyraz Ülkü Dinlemez Kantar

Abstract

In this manuscript, we firstly find the Korovkin test functions for the Baskakov operators, secondly, we find the generalized Baskakov-Durrmeyer-Stancu type operators. Thirdly, we give the modulus of continuity for the generalized Baskakov-Durrmeyer-Stancu type operators. Then, the asymptotic approach of these operators has been studied by using the Voronovskaja-type theorem. Finally, it is demonstrated that the generalized Baskakov-Durrmeyer-Stancu type operators converge to the considered function by plotting the graphs. Moreover, the convergence of the generalized Baskakov-Durrmeyer-Stancu type operators is compared with that of some other operators to the same function.

Keywords

Voronovskaja type theorem Voronovskaja type theorem Baskakov operators

References

  • [1] Weierstrass K., Uber die analytische Darstellbarkeit sogenannter willkurlicher Functionen einer reellen Veranderlichen Sitzungsberichteder, Koniglich Preussischen Akademie der Wissenschcaften zu Berlin, (1885) 633-639, 789-805.
  • [2] Bernstein S. N., Démonstration du théoréme deWeierstrass fondée sur la calcul des probabilitiés, Commun. Soc.Math. Charkow Sér. 13(2) (1912) 1-2 .
  • [3] Stancu D.D., Approximation of functions by a new class of linear polynomial operators, Rev. Roum. Math. Pures Appl. 13 (1968) 1173-1194.
  • [4] Baskakov V. A., An instance of a sequence of positive linear operators in the space of continuous functions, Doklady Akademii Nauk SSSR, 113:2 (1957) 249–251,
  • [5] Mihesan V., Uniform approximation with positive linear operators generated by generalized Baskakov method, Automat. Comput. Appl. Math. 7 (1998).34-37.
  • [6] Wafi A., Khatoon S., On the order of approximation of functions by generalized Baskakov operators, Indian J. Pure Appl. Math. (2004) 35347-358.
  • [7] Erencin A., Bascanbaz-Tunca G., Approximation properties of a class of linear positive operators in weighted spaces. C. R. Acad. Bulg. Sci. 63(10), (2010)1397–1404.
  • [8] Durrmeyer J. L., Une formule d’inversion de la transform´ee de Laplace: applications`a la th´eorie des moments, These de 3e cycle, Facult´e des Sciences de l’Universit´e de Paris, (1967).
  • [9] Lupas A., Die Folge der Betaoperatoren, Dissertation, Universit¨at Stuttgart, (1972).
  • [10] Erencin A., Durrmeyer type modification of generalized Baskakov operators, Appl. Math. Comput., 218 (2011) 4384-4390.
  • [11] Agrawal P. N., Gupta V., Kumar A. S., Generalized Baskakov-Durrmeyer type operators, Rend. Circ. Mat. Palermo, 63 (2014) 193–209.
  • [12] Kumar A. S., Finta Z., Agrawal P. N., On generalized Baskakov-Durrmeyer-Stancu type operators, Demonstr. Math., 50 (2017) 144–155.
  • [13] Gadhziev A.D., Theorems of the type of P.P. Korovkin type theorems, Matematicheskie Zametki, 20(5) (1976) 781-786.
  • [14] Erençin A., Olgun A., Taşdelen F., Generalized Baskakov type operators, Mathematica Slovaca, 67(5) (2017) 1269-1277.
  • [15] Yesildal FT., Bodur M., Bivariate Baskakov type operators, Revısta De La Real Academıa De Cıencıas Exactas Fısıcas Y Naturales Serıe A-Matematıcas, 113(4) (2019) 3269-3281.
  • [16] Aktaş R., Söylemez D., Taşdelen F., Stancu type generalization of Szász-Durrmeyer operators involving Brenke-type polynomials, Filomat, 33(3) (2019) 855-868.
  • [17] Bodur M., Yılmaz O. G., Aral A., Approximation by Baskakov-Szász-Stancu Operators Preserving Exponential Functions, Constr. Math. Anal., 1(1) (2018) 1-8.

Year 2022, Volume: 43 Issue: 1, 98 - 104, 30.03.2022

Gülten Torun , Meliha Mercan Boyraz Ülkü Dinlemez Kantar

Abstract

References

  • [1] Weierstrass K., Uber die analytische Darstellbarkeit sogenannter willkurlicher Functionen einer reellen Veranderlichen Sitzungsberichteder, Koniglich Preussischen Akademie der Wissenschcaften zu Berlin, (1885) 633-639, 789-805.
  • [2] Bernstein S. N., Démonstration du théoréme deWeierstrass fondée sur la calcul des probabilitiés, Commun. Soc.Math. Charkow Sér. 13(2) (1912) 1-2 .
  • [3] Stancu D.D., Approximation of functions by a new class of linear polynomial operators, Rev. Roum. Math. Pures Appl. 13 (1968) 1173-1194.
  • [4] Baskakov V. A., An instance of a sequence of positive linear operators in the space of continuous functions, Doklady Akademii Nauk SSSR, 113:2 (1957) 249–251,
  • [5] Mihesan V., Uniform approximation with positive linear operators generated by generalized Baskakov method, Automat. Comput. Appl. Math. 7 (1998).34-37.
  • [6] Wafi A., Khatoon S., On the order of approximation of functions by generalized Baskakov operators, Indian J. Pure Appl. Math. (2004) 35347-358.
  • [7] Erencin A., Bascanbaz-Tunca G., Approximation properties of a class of linear positive operators in weighted spaces. C. R. Acad. Bulg. Sci. 63(10), (2010)1397–1404.
  • [8] Durrmeyer J. L., Une formule d’inversion de la transform´ee de Laplace: applications`a la th´eorie des moments, These de 3e cycle, Facult´e des Sciences de l’Universit´e de Paris, (1967).
  • [9] Lupas A., Die Folge der Betaoperatoren, Dissertation, Universit¨at Stuttgart, (1972).
  • [10] Erencin A., Durrmeyer type modification of generalized Baskakov operators, Appl. Math. Comput., 218 (2011) 4384-4390.
  • [11] Agrawal P. N., Gupta V., Kumar A. S., Generalized Baskakov-Durrmeyer type operators, Rend. Circ. Mat. Palermo, 63 (2014) 193–209.
  • [12] Kumar A. S., Finta Z., Agrawal P. N., On generalized Baskakov-Durrmeyer-Stancu type operators, Demonstr. Math., 50 (2017) 144–155.
  • [13] Gadhziev A.D., Theorems of the type of P.P. Korovkin type theorems, Matematicheskie Zametki, 20(5) (1976) 781-786.
  • [14] Erençin A., Olgun A., Taşdelen F., Generalized Baskakov type operators, Mathematica Slovaca, 67(5) (2017) 1269-1277.
  • [15] Yesildal FT., Bodur M., Bivariate Baskakov type operators, Revısta De La Real Academıa De Cıencıas Exactas Fısıcas Y Naturales Serıe A-Matematıcas, 113(4) (2019) 3269-3281.
  • [16] Aktaş R., Söylemez D., Taşdelen F., Stancu type generalization of Szász-Durrmeyer operators involving Brenke-type polynomials, Filomat, 33(3) (2019) 855-868.
  • [17] Bodur M., Yılmaz O. G., Aral A., Approximation by Baskakov-Szász-Stancu Operators Preserving Exponential Functions, Constr. Math. Anal., 1(1) (2018) 1-8.
There are 17 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Natural Sciences
Authors

Gülten Torun 0000-0002-1897-0174

Meliha Mercan Boyraz 0000-0002-8288-7949

Ülkü Dinlemez Kantar 0000-0002-5656-3924

Publication Date March 30, 2022
Submission Date November 30, 2021
Acceptance Date February 26, 2022
Published in Issue Year 2022Volume: 43 Issue: 1

Cite

APA Torun, G., Mercan Boyraz, M., & Dinlemez Kantar, Ü. (2022). Investigation of the Asymptotic Behavior of Generalized Baskakov-Durrmeyer-Stancu Type Operators. Cumhuriyet Science Journal, 43(1), 98-104.
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