On a New Type of q-Baskakov Operators

Ersin Şimşek

doi:10.19113/sdufbed.29379

On a New Type of q-Baskakov Operators

Abstract

In this work, we have introduced a new type of $q$-analogous of Baskakov Operators. Their respective formulae for central moments are thereby obtained. The approximation properties and the approximation rapid of the sequences of the operators which are defined have been established in terms of the modulus of smoothness.

Keywords

q-Analysis,Korovkin's theorem; Baskakov operators

References

[1] Jackson, F. H. 1908. On q-functions and a certain difference operator. Transactions Royal Society Edinburgh, 46(1908), 253-281.
[2] Aral, A., Gupta and V., Agarwal, R. P. 2013. Applications of q-Calculus in Operator Theory. Springer-Verlag New York, 262s.
[3] Kac, V., Cheung, P. 2002. Quantum Calculus. Universitext Springer-Verlag New York, 112s.
[4] Ernst, T. 2000. The History of q-Calculus and a New Method. U.U.D.M. Report Uppsala, Department of Mathematics, Uppsala University, 230s.
[5] Lupas, A. 1987. A q-analogue of the Bernstein operator. University of Cluj-Napoca Seminar on numerical and statistical calculus, 9(1987), 85-92.
[6] Phillips, G. M. 1997. Bernstein polynomials based on the q-integers. Annals of Numerical Mathematics, 4(1997), 511-518.
[7] Heping, W. 2008. Properties of convergence for w;q-Bernstein polynomials. Journal of Mathematical Analysis and Applications, 340(2)(2008), 1096-1108.
[8] Heping, W., Meng, F. 2005. The rate of convergence of q-Bernstein polynomials for 0 < q < 1. Journal of Approximation Theory, 136(2005), 151-158.

[9] II’inski, A., Ostrovska, S. 2002. Convergence of generalized Bernstein polynomials. Journal of Approximation Theory, 116(2002), 100-112.
[10] Ostrovska, S. 2003. q-Bernstein polynomials and their iterates. Journal of Approximation Theory, 123(2003), 232-255.
[11] Bustamante, J. 2017. Bernstein operators and their properties. Birkhäuser Basel, 420s.
[12] Kajla, A., Ispir, N., Agrawal, P.N., Goyal, M. 2016. q-Bernstein-Schurer-Durrmeyer type operators for functions of one and two variables. Applied Mathematics and Computation, 275(2016), 372-385.
[13] Agrawal, P.N., Goyal, M., Kajla, A. 2015. q-Bernstein-Schurer-Durrmeyer type operators for functions of one and two variables. Bollettino dell’Unione Matematica Italiana, 8(2015) , 169–180.
[14] Baskakov, V. A. 1957. An example of sequence of linear positive operators in the space of continuous functions. Doklady Akademii Nauk SSSR, 113(1957), 259-251.
[15] Aral, A., Gupta, V. 2009. On q-Baskakov type operators. Demonstratio Mathematica, 42(1)(2009), 109-122.
[16] Aral, A., Gupta, V. 2011. Generalized q-Baskakov operators. Mathematica Slovaca, 61(4)(2011), 619-634.
[17] Radu, C. 2009. On statistical approximation of a general class of positive linear operators extended in q-calculus. Applied Mathematics and Computation, 215(6)(2009), 2317-2325.
[18] Korovkin, P. P. 1960. Linear Operators and Approximation Theory. Hindustan Pub. Corp., 222s.
[19] Simsek, E., Tunc, T. 2017. On the Construction of q- Analogues for some Positive Linear Operators. Filomat, 31:13 (2017), 4287-4295.
[20] Simsek E., Tunc, T. 2018. On Approximation Properties of some Class Positive Linear Operators in q-Analysis. Journal of Mathematical Inequalities, Accepted (2018).
[21] Rajkovic, P. M., Stankovic, M. S., Marinkovic, S. D. 2002. Mean value theorems in q-calculus. Applied Mathematics and Computation, 54(2002), 171-178.
[22] Carlitz, L. 1948. q-Bernoulli numbers and polynomials. Duke Mathematical Journal, 63(1948), 987-1000.

Details

Primary Language

Turkish

Subjects

-

Journal Section

-

Authors

Ersin Şimşek

Publication Date

March 29, 2018

Submission Date

September 7, 2017

Acceptance Date

-

Published in Issue

Year 2018 Volume: 22 Number: 1

DOI

https://doi.org/10.19113/sdufbed.29379

IZ

https://izlik.org/JA68EA65ZU

Cite

RIS / Bibtex

APA

Şimşek, E. (2018). On a New Type of q-Baskakov Operators. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 22(1), 121-125. https://doi.org/10.19113/sdufbed.29379

AMA

1.Şimşek E. On a New Type of q-Baskakov Operators. J. Nat. Appl. Sci. 2018;22(1):121-125. doi:10.19113/sdufbed.29379

Chicago

Şimşek, Ersin. 2018. “On a New Type of Q-Baskakov Operators”. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi 22 (1): 121-25. https://doi.org/10.19113/sdufbed.29379.

EndNote

Şimşek E (April 1, 2018) On a New Type of q-Baskakov Operators. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi 22 1 121–125.

IEEE

[1]E. Şimşek, “On a New Type of q-Baskakov Operators”, J. Nat. Appl. Sci., vol. 22, no. 1, pp. 121–125, Apr. 2018, doi: 10.19113/sdufbed.29379.

ISNAD

Şimşek, Ersin. “On a New Type of Q-Baskakov Operators”. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi 22/1 (April 1, 2018): 121-125. https://doi.org/10.19113/sdufbed.29379.

JAMA

1.Şimşek E. On a New Type of q-Baskakov Operators. J. Nat. Appl. Sci. 2018;22:121–125.

MLA

Şimşek, Ersin. “On a New Type of Q-Baskakov Operators”. Süleyman Demirel Üniversitesi Fen Bilimleri Enstitüsü Dergisi, vol. 22, no. 1, Apr. 2018, pp. 121-5, doi:10.19113/sdufbed.29379.

Vancouver

1.Ersin Şimşek. On a New Type of q-Baskakov Operators. J. Nat. Appl. Sci. 2018 Apr. 1;22(1):121-5. doi:10.19113/sdufbed.29379