q-Baskakov Operatörlerinin Yeni bir Tipi Üzerine
Bu çalı¸smada, q-Baskakov operatörlerinin yeni bir türü tanıtılmı¸stır. Merkezi momentler için formüller elde edildilmi¸s. Tanımlanan q-Baskakov operatörlerinin dizilerinin yakla¸sım özellikleri ve yakınsama oranı, süreklilik modülü yardımıyla belirlenmi¸stir.
On a New Type of q-Baskakov Operators
In this work, we have introduced a new type of q-analogous of BaskakovOperators. Their respective formulae for central moments are thereby obtained. Theapproximation properties and the approximation rapid of the sequences of the operatorswhich are defined have been established in terms of the modulus of smoothness.
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- [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.