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Approximation Properties of Stancu-Type (p,q)-Baskakov Operators

Year 2019, Volume: 8 Issue: 3, 889 - 902, 30.09.2019
https://doi.org/10.17798/bitlisfen.582238

Abstract

This paper deals with constructing the Stancu type
modication of the Baskakov operators using (p,q) - integers. We study the rate of convergence by using
Peetre's K-functional and modulus of continuity. In addition, we examine the
pointwise estimation of the new operators for continuous and bounded functions
in a Lipschitz space and a Lipschitz-type maximal function. Finally, we
illustrate the convergence of the constructed operators to some functions with
the help of MATLAB.

References

  • Reference 1. Aral, A., Gupta, V. 2011. Generalized q -Baskakov operators, Mathematica Slovaca, 61 (4): 619-634.
  • Reference 2. Büyükyazıcı, İ., Atakut, Ç. 2010. On Stancu type generalization of q -Baskakov operators, Mathematical and Computer Modelling, 52: 752-759.
  • Reference 3. Mishra, V. N., Sharma, P., Kılıçman, A., Jain, D. 2016. Statistical Approximation Properties of Stancu Type q - Baskakov-Kantorovich Operators, Filomat, 30 (7): 1853-1868.
  • Reference 4. Mursaleen, M., Ansari, K. J., Khan, A. 2015. On (p,q) -analogue of Bernstein Operators, Appl. Math. and Comput., 266: 874-882. (Erratum: Appl. Math. Comput., 266: 874-882, (2015)).
  • Reference 5. Acar, T., Mursaleen, M., Mohiuddine, S. A. 2018. Stancu Type (p,q) -Szasz-Mirakyan-Baskakov operators, Commun. Fac. Sci. Univ. Ank. Series A1, 67 (1): 116-128,
  • Reference 6. Acar, T., Aral, A., Mohiuddine, S. A. 2016. On Kantorovich modification of (p,q) -Baskakov operators, J. Inequal. Appl., 2016: 98.
  • Reference 7. Aral, A., Ulusoy, G., Deniz, E. 2018. A new construction of Szasz-Mirakyan operators, Numerical Algorithms, 77 (2): 313-326.
  • Reference 8. Kanat, K., Sofyalıoğlu, M. 2018. Some approximation results for Stancu type Lupaş-Schurer operators based on (p,q) - integers, Appl. Math. and Comp., 317: 129-142.
  • Reference 9. Mishra, V. N., Pandey, S. 2017. On (p,q) Baskakov–Durrmeyer–Stancu Operators, Adv. Appl. Clifford Algebras, 27: 1633-1646.
  • Reference 10. Yüksel, İ., Dinlemez Kantar, U., Altın, B. 2017. On the (p,q)-Stancu Generalization of a Genuine Baskakov-Durrmeyer Type Operators, Int. J. of Analysis and App., 15 (2): 138-145.
  • Reference 11. Aral, A., Gupta, V. 2016. (p,q) -type beta functions of second kind, Adv. Oper. Theory, 1 (1): 134-146.
  • Reference 12. Hounkonnou, M. N., Kyemba, JDB. 2013. R (p,q) -calculus: differentiation and integration, SUT J Math., 49 (2): 145–167.
  • Reference 13. Sadjang, P. N. 2013. On the fundamental theorem of (p,q) -calculus and some (p,q) -Taylor formulas, arXiv:1309.3934v1 [math.QA].
  • Reference 14. Gadzhiev, A. D. 1976. Theorems of the type of P. P. Korovkin type theorems, Math. Zametki, 20 (5): 781–786, (1976). English Translation, Math. Notes 20 (5–6): 996–998. https://doi.org/10.1007/BF01146928.
  • Reference 15. DeVore, R. A., Lorentz, G. G. 1993. Constructive Approximation, Berlin: Springer, 177.
  • Reference 16. Lenze, B. 1988. On Lipschitz type maximal functions and their smoothness spaces, Ned. Akad. Indag. Math., 50: 53-63.

Stancu Tipli (p,q)-Baskakov Operatörlerinin Yaklaşım Özellikleri

Year 2019, Volume: 8 Issue: 3, 889 - 902, 30.09.2019
https://doi.org/10.17798/bitlisfen.582238

Abstract

Bu makale Baskakov
operatörlerinin Stancu tipli modifikasyonlarının (p,q)
-tamsayılar kullanılarak inşaa edilmesiyle
ilgilenmektedir. Yaklaşım derecesi Peetre-K fonksiyonelleri ve süreklilik
modülü kullanılarak çalışılmıştır. Buna ek olarak yeni operatörlerin noktasal
yaklaşımı bir Lipschitz uzayındaki sürekli ve sınırlı fonksiyonlar ve bir Lipschitz-tipli
maksimal fonksiyon için incelenmiştir. Sonuç olarak, üretilen operatörlerin
bazı fonksiyonlara yakınsaklığı MATLAB yardımıyla  elde edilen grafiklerle gösterilmiştir.

References

  • Reference 1. Aral, A., Gupta, V. 2011. Generalized q -Baskakov operators, Mathematica Slovaca, 61 (4): 619-634.
  • Reference 2. Büyükyazıcı, İ., Atakut, Ç. 2010. On Stancu type generalization of q -Baskakov operators, Mathematical and Computer Modelling, 52: 752-759.
  • Reference 3. Mishra, V. N., Sharma, P., Kılıçman, A., Jain, D. 2016. Statistical Approximation Properties of Stancu Type q - Baskakov-Kantorovich Operators, Filomat, 30 (7): 1853-1868.
  • Reference 4. Mursaleen, M., Ansari, K. J., Khan, A. 2015. On (p,q) -analogue of Bernstein Operators, Appl. Math. and Comput., 266: 874-882. (Erratum: Appl. Math. Comput., 266: 874-882, (2015)).
  • Reference 5. Acar, T., Mursaleen, M., Mohiuddine, S. A. 2018. Stancu Type (p,q) -Szasz-Mirakyan-Baskakov operators, Commun. Fac. Sci. Univ. Ank. Series A1, 67 (1): 116-128,
  • Reference 6. Acar, T., Aral, A., Mohiuddine, S. A. 2016. On Kantorovich modification of (p,q) -Baskakov operators, J. Inequal. Appl., 2016: 98.
  • Reference 7. Aral, A., Ulusoy, G., Deniz, E. 2018. A new construction of Szasz-Mirakyan operators, Numerical Algorithms, 77 (2): 313-326.
  • Reference 8. Kanat, K., Sofyalıoğlu, M. 2018. Some approximation results for Stancu type Lupaş-Schurer operators based on (p,q) - integers, Appl. Math. and Comp., 317: 129-142.
  • Reference 9. Mishra, V. N., Pandey, S. 2017. On (p,q) Baskakov–Durrmeyer–Stancu Operators, Adv. Appl. Clifford Algebras, 27: 1633-1646.
  • Reference 10. Yüksel, İ., Dinlemez Kantar, U., Altın, B. 2017. On the (p,q)-Stancu Generalization of a Genuine Baskakov-Durrmeyer Type Operators, Int. J. of Analysis and App., 15 (2): 138-145.
  • Reference 11. Aral, A., Gupta, V. 2016. (p,q) -type beta functions of second kind, Adv. Oper. Theory, 1 (1): 134-146.
  • Reference 12. Hounkonnou, M. N., Kyemba, JDB. 2013. R (p,q) -calculus: differentiation and integration, SUT J Math., 49 (2): 145–167.
  • Reference 13. Sadjang, P. N. 2013. On the fundamental theorem of (p,q) -calculus and some (p,q) -Taylor formulas, arXiv:1309.3934v1 [math.QA].
  • Reference 14. Gadzhiev, A. D. 1976. Theorems of the type of P. P. Korovkin type theorems, Math. Zametki, 20 (5): 781–786, (1976). English Translation, Math. Notes 20 (5–6): 996–998. https://doi.org/10.1007/BF01146928.
  • Reference 15. DeVore, R. A., Lorentz, G. G. 1993. Constructive Approximation, Berlin: Springer, 177.
  • Reference 16. Lenze, B. 1988. On Lipschitz type maximal functions and their smoothness spaces, Ned. Akad. Indag. Math., 50: 53-63.
There are 16 citations in total.

Details

Primary Language English
Journal Section Araştırma Makalesi
Authors

Kadir Kanat 0000-0002-7738-903X

Melek Sofyalıoğlu 0000-0001-7837-2785

Publication Date September 30, 2019
Submission Date June 25, 2019
Acceptance Date July 17, 2019
Published in Issue Year 2019 Volume: 8 Issue: 3

Cite

IEEE K. Kanat and M. Sofyalıoğlu, “Approximation Properties of Stancu-Type (p,q)-Baskakov Operators”, Bitlis Eren Üniversitesi Fen Bilimleri Dergisi, vol. 8, no. 3, pp. 889–902, 2019, doi: 10.17798/bitlisfen.582238.

Bitlis Eren University
Journal of Science Editor
Bitlis Eren University Graduate Institute
Bes Minare Mah. Ahmet Eren Bulvari, Merkez Kampus, 13000 BITLIS