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Investigating (p,q)-hybrid Durrmeyer-type Operators in terms of Their Approximation Properties

Year 2022, Volume 9, Issue 1, 1 - 11, 30.03.2022
https://doi.org/10.54287/gujsa.1029633

Abstract

This study introduces (p,q)-hybrid Durrmeyer-Stancu type linear positive operators, which are generalized forms of q-hybrid Durrmeyer-Stancu-type linear positive operators and examines their approximation properties. The first modulus of continuity on a finite interval is introduced using Peetre’s K-functional. Then, the weighted approximation theorem in a weighted space is provided using Gadzhiev’s weighted Korovkin-type theorem. Finally, these operators’ rates of convergence are obtained for the continuous functions.

References

  • Acar, T., Aral, A., & Mohiuddine, S. A. (2016). On Kantorovich modication of (p, q)-Baskakov operators. Journal of Inequalities and Applications, 2016, 98. doi:10.1186/s13660-016-1045-9
  • Acar, T., Aral A., & Mohiuddine, S. A. (2018). Approximation by Bivariate (p, q)-Bernstein-Kantorovich operators. Iranian Journal of Science and Technology, Transactions A: Science, 42(2), 655-662. doi:10.1007/s40995-016-0045-4
  • Aral, A., Gupta, V., & Agarwal, R. P. (2013). Applications of q-calculus in operator theory. Springer, New York. doi:10.1007/978-1-4614-6946-9
  • Cai, Q-B., Sofyalıoglu, M., Kanat, K., & Çekim, B. (2021). Some approximation results for the new modification of Bernstein-Beta operators AIMS Mathematics, 7(2), 1831-1844. doi:10.3934/math.2022105
  • De Sole, A., & Kac, V. G. (2005). On integral representations of q-gamma and q-beta functions. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 16(1), 11-29.
  • Dinlemez, Ü., Yüksel, İ., & Altın, B. (2014). A note on the approximation by the q-hybrid summation integral type operators. Taiwanese Journal of Mathematics, 18(3), 781-792.
  • Doğru, O., & Gupta, V. (2005). Monotonicity and the asymptotic estimate of Bleimann Butzer and Hahn operators based on q -integers. Georgian Mathematical Journal, 12(3), 415-422.
  • Doğru, O., & Gupta, V. (2006). Korovkin-type approximation properties of bivariate q-Meyer-König and Zeller operators. Calcolo, 43(1), 51-63. doi:10.1007/s10092-006-0114-8
  • Gadzhiev, A. D. (1976). Theorems of the type of P. P. Korovkin type theorems. Matematicheskie Zametki, 20(5), 781-786. English Translation: Math. Notes, 20(5/6), 996-998, (1976).
  • Gupta, V., & Heping, W. (2008). The rate of convergence of q-Durrmeyer operators for 0<q<1. Mathematical Methods in the Applied Sciences, 31(16), 1946-1955. doi:10.1002/mma.1012
  • Gupta, V., & Aral, A. (2010). Convergence of the q-analogue of Szász-beta operators. Applied Mathematics and Computation, 216(2), 374-380. doi:10.1016/j.amc.2010.01.018
  • Gupta, V., & Karsli, H. (2012). Some approximation properties by q-Szász-Mirakyan-Baskakov-Stancu operators. Lobachevskii Journal of Mathematics, 33(2), 175-182. doi:10.1134/S1995080212020138
  • Gupta, V. (2018). (p, q)-Szász-Mirakyan-Baskakov Operators. Complex Analysis and Operator Theory, 12, 17-25. doi:10.1007/s11785-015-0521-4
  • Jackson, F. H. (1910). On q-Definite Integrals. The Quarterly Journal of Pure and Applied Mathematics, 41(15), 193-203.
  • Kac, V. G., & Cheung, P. (2002). Quantum Calculus. Part of the Universitext book series, Springer-Verlag, New York. doi:10.1007/978-1-4613-0071-7
  • Kanat, K., & Sofyalıoğlu, M. (2018). Approximation by (p, q)-Lupaş–Schurer–Kantorovich operators. Journal of Inequalities and Applications, 2018, 263. doi:10.1186/s13660-018-1858-9
  • Kanat, K., & Sofyalıoğlu, M. (2021). On Stancu type Szász-Mirakyan-Durrmeyer Operators Preserving exp(2ax), a > 0. Gazi University Journal of Science, 34(1), 196-209. doi:10.35378/gujs.691419
  • Koelink, H. T., & Koornwinder, T. H. (1990). q-special functions, a tutorial. In: M. Gerstenhaber, & J. Stasheff (Eds.) Deformation Theory and Quantum Groups with Applications to Mathematical Physics (Proceedings of an AMS-IMS-SIAM 1990), Contemporary Mathematics, 134, 141-142.
  • Lupaş, A. (1987). A q-analogue of the Bernstein operator. In: Seminar on numerical and statistical calculus, University of Cluj-Napoca, 9, 85-92.
  • Mursaleen, M., Ansari, K. J., & Khan, A. (2015a). On (p, q)-analogue of Bernstein operators. Applied Mathematics and Computation, 266, 874-882. doi:10.1016/j.amc.2015.04.090
  • Mursaleen, M., Ansari, K. J., & Khan, A. (2015b). Some Approximation Results by (p, q)-analogue of Bernstein-Stancu operators. Applied Mathematics and Computation, 264, 392-402. doi:10.1016/j.amc.2015.03.135
  • Mursaleen, M., Nasiuzzaman, Md., & Nurgali, A. (2015c). Some approximation results on Bernstein-Schurer operators dened by (p, q)-integers. Journal of Inequalities and Applications, 2015, 249. doi:10.1186/s13660-015-0767-4
  • Phillips, G. M. (1997). Bernstein polynomials based on the q-integers. Annals of Numerical Mathematics, 4(1-4), 511-518.
  • Sahai, V., & Yadav, S. (2007). Representations of two parameter quantum algebras and (p, q)-special functions. Journal of Mathematical Analysis and Applications, 335(1), 268-279. doi:10.1016/j.jmaa.2007.01.072
  • 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
  • Yüksel, İ. (2013). Direct results on the q-mixed summation integral type operators. J. Applied Functional Analysis, 8(2), 235-245.

Year 2022, Volume 9, Issue 1, 1 - 11, 30.03.2022
https://doi.org/10.54287/gujsa.1029633

Abstract

References

  • Acar, T., Aral, A., & Mohiuddine, S. A. (2016). On Kantorovich modication of (p, q)-Baskakov operators. Journal of Inequalities and Applications, 2016, 98. doi:10.1186/s13660-016-1045-9
  • Acar, T., Aral A., & Mohiuddine, S. A. (2018). Approximation by Bivariate (p, q)-Bernstein-Kantorovich operators. Iranian Journal of Science and Technology, Transactions A: Science, 42(2), 655-662. doi:10.1007/s40995-016-0045-4
  • Aral, A., Gupta, V., & Agarwal, R. P. (2013). Applications of q-calculus in operator theory. Springer, New York. doi:10.1007/978-1-4614-6946-9
  • Cai, Q-B., Sofyalıoglu, M., Kanat, K., & Çekim, B. (2021). Some approximation results for the new modification of Bernstein-Beta operators AIMS Mathematics, 7(2), 1831-1844. doi:10.3934/math.2022105
  • De Sole, A., & Kac, V. G. (2005). On integral representations of q-gamma and q-beta functions. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 16(1), 11-29.
  • Dinlemez, Ü., Yüksel, İ., & Altın, B. (2014). A note on the approximation by the q-hybrid summation integral type operators. Taiwanese Journal of Mathematics, 18(3), 781-792.
  • Doğru, O., & Gupta, V. (2005). Monotonicity and the asymptotic estimate of Bleimann Butzer and Hahn operators based on q -integers. Georgian Mathematical Journal, 12(3), 415-422.
  • Doğru, O., & Gupta, V. (2006). Korovkin-type approximation properties of bivariate q-Meyer-König and Zeller operators. Calcolo, 43(1), 51-63. doi:10.1007/s10092-006-0114-8
  • Gadzhiev, A. D. (1976). Theorems of the type of P. P. Korovkin type theorems. Matematicheskie Zametki, 20(5), 781-786. English Translation: Math. Notes, 20(5/6), 996-998, (1976).
  • Gupta, V., & Heping, W. (2008). The rate of convergence of q-Durrmeyer operators for 0<q<1. Mathematical Methods in the Applied Sciences, 31(16), 1946-1955. doi:10.1002/mma.1012
  • Gupta, V., & Aral, A. (2010). Convergence of the q-analogue of Szász-beta operators. Applied Mathematics and Computation, 216(2), 374-380. doi:10.1016/j.amc.2010.01.018
  • Gupta, V., & Karsli, H. (2012). Some approximation properties by q-Szász-Mirakyan-Baskakov-Stancu operators. Lobachevskii Journal of Mathematics, 33(2), 175-182. doi:10.1134/S1995080212020138
  • Gupta, V. (2018). (p, q)-Szász-Mirakyan-Baskakov Operators. Complex Analysis and Operator Theory, 12, 17-25. doi:10.1007/s11785-015-0521-4
  • Jackson, F. H. (1910). On q-Definite Integrals. The Quarterly Journal of Pure and Applied Mathematics, 41(15), 193-203.
  • Kac, V. G., & Cheung, P. (2002). Quantum Calculus. Part of the Universitext book series, Springer-Verlag, New York. doi:10.1007/978-1-4613-0071-7
  • Kanat, K., & Sofyalıoğlu, M. (2018). Approximation by (p, q)-Lupaş–Schurer–Kantorovich operators. Journal of Inequalities and Applications, 2018, 263. doi:10.1186/s13660-018-1858-9
  • Kanat, K., & Sofyalıoğlu, M. (2021). On Stancu type Szász-Mirakyan-Durrmeyer Operators Preserving exp(2ax), a > 0. Gazi University Journal of Science, 34(1), 196-209. doi:10.35378/gujs.691419
  • Koelink, H. T., & Koornwinder, T. H. (1990). q-special functions, a tutorial. In: M. Gerstenhaber, & J. Stasheff (Eds.) Deformation Theory and Quantum Groups with Applications to Mathematical Physics (Proceedings of an AMS-IMS-SIAM 1990), Contemporary Mathematics, 134, 141-142.
  • Lupaş, A. (1987). A q-analogue of the Bernstein operator. In: Seminar on numerical and statistical calculus, University of Cluj-Napoca, 9, 85-92.
  • Mursaleen, M., Ansari, K. J., & Khan, A. (2015a). On (p, q)-analogue of Bernstein operators. Applied Mathematics and Computation, 266, 874-882. doi:10.1016/j.amc.2015.04.090
  • Mursaleen, M., Ansari, K. J., & Khan, A. (2015b). Some Approximation Results by (p, q)-analogue of Bernstein-Stancu operators. Applied Mathematics and Computation, 264, 392-402. doi:10.1016/j.amc.2015.03.135
  • Mursaleen, M., Nasiuzzaman, Md., & Nurgali, A. (2015c). Some approximation results on Bernstein-Schurer operators dened by (p, q)-integers. Journal of Inequalities and Applications, 2015, 249. doi:10.1186/s13660-015-0767-4
  • Phillips, G. M. (1997). Bernstein polynomials based on the q-integers. Annals of Numerical Mathematics, 4(1-4), 511-518.
  • Sahai, V., & Yadav, S. (2007). Representations of two parameter quantum algebras and (p, q)-special functions. Journal of Mathematical Analysis and Applications, 335(1), 268-279. doi:10.1016/j.jmaa.2007.01.072
  • 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
  • Yüksel, İ. (2013). Direct results on the q-mixed summation integral type operators. J. Applied Functional Analysis, 8(2), 235-245.

Details

Primary Language English
Subjects Engineering, Multidisciplinary
Journal Section Mathematics
Authors

Ülkü DİNLEMEZ KANTAR> (Primary Author)
GAZİ ÜNİVERSİTESİ
0000-0002-5656-3924
Türkiye


İsmet YÜKSEL>
GAZİ ÜNİVERSİTESİ
0000-0002-2631-2382
Türkiye

Publication Date March 30, 2022
Published in Issue Year 2022, Volume 9, Issue 1

Cite

APA 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

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