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On Approximation Properties of Stancu Type Post-Widder Operators Preserving Exponential Functions

Year 2022, Volume: 9 Issue: 2, 173 - 186, 30.06.2022
https://doi.org/10.54287/gujsa.1113567

Abstract

In this article, Stancu type Post-Widder operators are introduced, which are a modification of the Post-Widder operators that preserve the functions constant and e^2ax for fixed a>0. The uniform convergence of these modified operators for the function f on [0,∞) is examined and then the convergence rate is investigated with the help of the continuity module. The Voronovskaja type asymptotic formula is obtained to examine the asymptotic behavior of these operators. Finally, numerical examples and graphs are given to show the convergence of Stancu type Post- Widder operators and compared with Post Widder operators.

References

  • Acar, T., Aral, A., & Gonska, H. (2017). On Szász-Mirakyan operators preserving e^2ax,a>0. Mediterranean Journal of Mathematics, 14(6), 1-14. doi:10.1007/s00009-016-0804-7
  • Aral, A., Inoan, D., & Raşa, I. (2019). Approximation properties of Szász-Mirakyan operators preserving exponential functions. Positivity, 23, 233-246. doi:10.1007/s11117-018-0604-3
  • Bodur, M., Gürel-Yılmaz, Ö., & Aral, A. (2018). Approximation by Baskakov-Szász-Stancu operators preserving exponential functions. Constructive Mathematical Analysis, 1(1), 1-8. doi:10.33205/cma.450708
  • Boyanov, B. D., & Veselinov, V. M. (1970). A note on the approximation of functions in an infinite interval by linear positive operators. Bull Math Soc Sci Math Roum, 14(62), 9-13.
  • Cai, Q.-B., Dinlemez Kantar, Ü. & Çekim, B. (2020). Approximation properties for the genuine modified Bernstein-Durrmeyer-Stancu operators. Applied Mathematics-A Journal of Chinese Universities, 35(4), 468-478. doi:10.1007/s11766-020-3918-y
  • Cai, Q.-B., Sofyalıoğlu, M., Kanat, K. & Çekim, B. (2021a). Some approximation results for the new modification of Bernstein-Beta operators. AIMS Mathematics, 7(2): 1831-1844. doi:10.3934/math.2022105
  • Cai, Q.-B., Torun, G., & Dinlemez Kantar, Ü. (2021b). Approximation properties of generalized λ–Bernstein–Stancu-type operators. Journal of Mathematics, 2021, 5590439. doi:10.1155/2021/5590439
  • DeVore, R. A., & Lorentz, G. G. (1993). Constructive Approximation. (Grundlehren der mathematischen Wissenschaften, Vol. 303) Springer, Berlin.
  • Dinlemez Kantar, Ü. & Ergelen, G. (2019). A voronovskaja-type theorem for a kind of Durrmeyer-Bernstein-Stancu operators. Gazi University Journal of Science, 32(4), 1228-1236. doi:10.35378/gujs.513478
  • Dinlemez Kantar, Ü. & Yüksel, İ. (2022). Investigating (p, q)-hybrid Durrmeyer-type operators in terms of their approximation properties. Gazi University Journal of Science Part A: Engineering and Innovation, 9(1), 1-11. doi:10.54287/gujsa.1029633
  • Gupta, V., & Agrawal, D. (2019). Convergence by modified Post-Widder operators. RACSAM, 113(2), 1475-1486. doi:10.1007/s13398-018-0562-4
  • Gupta, V., & Maheshwari, P. (2019). Approximation with certain Post-Widder operators. Publications de L'Institut Mathématique, 105(119), 131-136. doi:10.2298/PIM1919131G
  • Gupta, V., & Tachev, G. (2022). Some results on Post-Widder operators preserving test function x^r. Kragujevac Journal of Mathematics, 46(1), 149-165. doi:10.46793/KgJMat2201.149G
  • Gürel-Yılmaz, Ö., Gupta, V., & Aral, A. (2017). On Baskakov operators preserving the exponential function. Journal of Numerical Analysis and Approximation Theory, 46(2), 150-161.
  • Gürel-Yılmaz, Ö., Bodur, M., & Aral, A. (2018). On approximation properties of Baskakov-Schurer-Szász operators preserving exponential functions. Filomat, 32(15), 5433-5440. doi:10.2298/FIL1815433Y
  • Hacısalihoğlu, H., & Hacıyev, A. (1995). Lineer Pozitif Operatör Dizilerin Yakınsaklığı. Ankara University F.F., Ankara.
  • Holhoş, A. (2010). The rate of approximation of functions in an infinite interval by positive linear operators. Stud Univ Babes-Bolyai Math., 55(2). 133-142.
  • Kanat, K., & Sofyalıoğlu, M. (2021). On Stancu type Szász-Mirakyan-Durrmeyer operators preserving e^2ax,a>0. Gazi University Journal of Science, 34(1), 196-209. doi:10.35378/gujs.691419
  • May, C. P. (1976). Saturation and inverse theorems for combinations of a class of exponential type operators. Canadian Journal of Mathematics, 28(6), 1224-1250. doi:10.4153/CJM-1976-123-8
  • Rathore, R. K. S., & Singh, O. P. (1980). On convergence of derivatives of Post-Widder operators. Indian J. Pure Appl. Math., 11(5), 547-561.
  • Rempulska, L., & Skorupka, M. (2009). On approximation by Post-Widder and Stancu operators preserving x². Kyungpook Mathematical Journal, 49(1), 57-65. doi:10.5666/KMJ.2009.49.1.057
  • Sofyalıoğlu, M., & Kanat, K. (2019). Approximation properties of generalized Baskakov–Schurer–Szasz–Stancu operators preserving e^2ax,a>0. Journal of Inequalities and Applications and Applications, 2019(1), 112. doi:10.1186/s13660-019-2062-2
  • Sofyalıoğlu, M., & Kanat, K. (2020). Approximation properties of the Post-Widder operators preserving e^2ax,a>0. Mathematical Methods in the Applied Sciences, 43(7), 4272-4285. doi:10.1002/mma.6192
  • Sofyalıoğlu, M., Kanat, K., & Çekim, B. (2021). Parametric generalization of the Meyer-König-Zeller operators. Chaos, Solitons & Fractals, 152, 111417. doi:10.1016/j.chaos.2021.111417
  • 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. doi:10.17776/csj.1030637
  • Widder, D. V. (1941). The Laplace Transform. (Princeton Mathematical Series, 6) Princeton University Press, Princeton, New Jersey.
Year 2022, Volume: 9 Issue: 2, 173 - 186, 30.06.2022
https://doi.org/10.54287/gujsa.1113567

Abstract

References

  • Acar, T., Aral, A., & Gonska, H. (2017). On Szász-Mirakyan operators preserving e^2ax,a>0. Mediterranean Journal of Mathematics, 14(6), 1-14. doi:10.1007/s00009-016-0804-7
  • Aral, A., Inoan, D., & Raşa, I. (2019). Approximation properties of Szász-Mirakyan operators preserving exponential functions. Positivity, 23, 233-246. doi:10.1007/s11117-018-0604-3
  • Bodur, M., Gürel-Yılmaz, Ö., & Aral, A. (2018). Approximation by Baskakov-Szász-Stancu operators preserving exponential functions. Constructive Mathematical Analysis, 1(1), 1-8. doi:10.33205/cma.450708
  • Boyanov, B. D., & Veselinov, V. M. (1970). A note on the approximation of functions in an infinite interval by linear positive operators. Bull Math Soc Sci Math Roum, 14(62), 9-13.
  • Cai, Q.-B., Dinlemez Kantar, Ü. & Çekim, B. (2020). Approximation properties for the genuine modified Bernstein-Durrmeyer-Stancu operators. Applied Mathematics-A Journal of Chinese Universities, 35(4), 468-478. doi:10.1007/s11766-020-3918-y
  • Cai, Q.-B., Sofyalıoğlu, M., Kanat, K. & Çekim, B. (2021a). Some approximation results for the new modification of Bernstein-Beta operators. AIMS Mathematics, 7(2): 1831-1844. doi:10.3934/math.2022105
  • Cai, Q.-B., Torun, G., & Dinlemez Kantar, Ü. (2021b). Approximation properties of generalized λ–Bernstein–Stancu-type operators. Journal of Mathematics, 2021, 5590439. doi:10.1155/2021/5590439
  • DeVore, R. A., & Lorentz, G. G. (1993). Constructive Approximation. (Grundlehren der mathematischen Wissenschaften, Vol. 303) Springer, Berlin.
  • Dinlemez Kantar, Ü. & Ergelen, G. (2019). A voronovskaja-type theorem for a kind of Durrmeyer-Bernstein-Stancu operators. Gazi University Journal of Science, 32(4), 1228-1236. doi:10.35378/gujs.513478
  • Dinlemez Kantar, Ü. & Yüksel, İ. (2022). Investigating (p, q)-hybrid Durrmeyer-type operators in terms of their approximation properties. Gazi University Journal of Science Part A: Engineering and Innovation, 9(1), 1-11. doi:10.54287/gujsa.1029633
  • Gupta, V., & Agrawal, D. (2019). Convergence by modified Post-Widder operators. RACSAM, 113(2), 1475-1486. doi:10.1007/s13398-018-0562-4
  • Gupta, V., & Maheshwari, P. (2019). Approximation with certain Post-Widder operators. Publications de L'Institut Mathématique, 105(119), 131-136. doi:10.2298/PIM1919131G
  • Gupta, V., & Tachev, G. (2022). Some results on Post-Widder operators preserving test function x^r. Kragujevac Journal of Mathematics, 46(1), 149-165. doi:10.46793/KgJMat2201.149G
  • Gürel-Yılmaz, Ö., Gupta, V., & Aral, A. (2017). On Baskakov operators preserving the exponential function. Journal of Numerical Analysis and Approximation Theory, 46(2), 150-161.
  • Gürel-Yılmaz, Ö., Bodur, M., & Aral, A. (2018). On approximation properties of Baskakov-Schurer-Szász operators preserving exponential functions. Filomat, 32(15), 5433-5440. doi:10.2298/FIL1815433Y
  • Hacısalihoğlu, H., & Hacıyev, A. (1995). Lineer Pozitif Operatör Dizilerin Yakınsaklığı. Ankara University F.F., Ankara.
  • Holhoş, A. (2010). The rate of approximation of functions in an infinite interval by positive linear operators. Stud Univ Babes-Bolyai Math., 55(2). 133-142.
  • Kanat, K., & Sofyalıoğlu, M. (2021). On Stancu type Szász-Mirakyan-Durrmeyer operators preserving e^2ax,a>0. Gazi University Journal of Science, 34(1), 196-209. doi:10.35378/gujs.691419
  • May, C. P. (1976). Saturation and inverse theorems for combinations of a class of exponential type operators. Canadian Journal of Mathematics, 28(6), 1224-1250. doi:10.4153/CJM-1976-123-8
  • Rathore, R. K. S., & Singh, O. P. (1980). On convergence of derivatives of Post-Widder operators. Indian J. Pure Appl. Math., 11(5), 547-561.
  • Rempulska, L., & Skorupka, M. (2009). On approximation by Post-Widder and Stancu operators preserving x². Kyungpook Mathematical Journal, 49(1), 57-65. doi:10.5666/KMJ.2009.49.1.057
  • Sofyalıoğlu, M., & Kanat, K. (2019). Approximation properties of generalized Baskakov–Schurer–Szasz–Stancu operators preserving e^2ax,a>0. Journal of Inequalities and Applications and Applications, 2019(1), 112. doi:10.1186/s13660-019-2062-2
  • Sofyalıoğlu, M., & Kanat, K. (2020). Approximation properties of the Post-Widder operators preserving e^2ax,a>0. Mathematical Methods in the Applied Sciences, 43(7), 4272-4285. doi:10.1002/mma.6192
  • Sofyalıoğlu, M., Kanat, K., & Çekim, B. (2021). Parametric generalization of the Meyer-König-Zeller operators. Chaos, Solitons & Fractals, 152, 111417. doi:10.1016/j.chaos.2021.111417
  • 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. doi:10.17776/csj.1030637
  • Widder, D. V. (1941). The Laplace Transform. (Princeton Mathematical Series, 6) Princeton University Press, Princeton, New Jersey.
There are 26 citations in total.

Details

Primary Language English
Journal Section Mathematics
Authors

Gülten Torun 0000-0002-1897-0174

Publication Date June 30, 2022
Submission Date May 7, 2022
Published in Issue Year 2022 Volume: 9 Issue: 2

Cite

APA Torun, G. (2022). On Approximation Properties of Stancu Type Post-Widder Operators Preserving Exponential Functions. Gazi University Journal of Science Part A: Engineering and Innovation, 9(2), 173-186. https://doi.org/10.54287/gujsa.1113567