Korovkin type theorem in the space $C_{b}[0,\infty)$
Keywords:
Korovkin theorem, modulus of continuity, $K$-functional, $q$-integers, $q$-BaskakovAbstract
A Korovkin type theorem is established in the space $C_{b}[0,\infty)$ of all continuous and bounded functions on $[0,\infty)$ for a sequence of positive linear operators, the approximation error being estimated with the aid of the usual modulus of continuity. As applications we obtain quantitative results for $q$-Baskakov operators.References
bibitem{Andrews} Andrews, G. E., Askey, R., Roy, R., emph{Special Functions}, Cambridge Univ. Press, Cambridge, 1999.
bibitem{Aral1} Aral, A., Gupta, V., emph{On $q$-Baskakov type operators}, Demonstratio Math., textbf{42}(2009), no. 1, 107-120.
bibitem{Aral2} Aral, A., Gupta, V., emph{Generalized $q$-Baskakov operators}, Math. Slovaca, textbf{61}(2011), no. 4, 619-634.
bibitem{Baskakov} Baskakov, V. A., emph{An example of a sequence of linear positive operators in the space of continuous functions}, Dokl. Akad. Nauk SSSR, textbf{113}(1957), 249-251 (in Russian).
bibitem{DeVore} DeVore, R. A., Lorentz, G. G., emph{Constructive Approximation}, Springer, Berlin, 1993.
bibitem{Finta1} Finta, Z., emph{Note on a Korovkin-type theorem}, J. Math. Anal. Appl., textbf{415}(2014), 750-759.
bibitem{Finta2} Finta, Z., emph{Korovkin type theorem for sequences of operators depending on a parameter}, Demonstratio Math., textbf{48}(2015), no. 3, 381-403.
bibitem{Gadjiev1} Gadjiev, A. D., emph{A problem on the convergence of a sequence of positive linear operators on unbounded sets, and theorems that are analogous to P.P. Korovkin's theorem}, Dokl. Akad. Nauk SSSR, textbf{218}(1974), 1001-1004 (in Russian); English translation: Soviet Math. Dokl., textbf{15} (1974), no. 5, 1433-1436.
bibitem{Gadjiev2} Gadjiev, A. D., emph{Theorems of the type of P.P. Korovkin's theorems}, Mat. Zametki, textbf{20}(1976), no. 5, 781-786 (in Russian); English translation: Soviet Math. Dokl., textbf{20} (1976), no. 5-6, 995-998.
bibitem{Ilinskii} Il'inskii, A., Ostrovska, S., emph{Convergence of generalized Bernstein polynomials}, J. Approx. Theory, textbf{116}(2002), 100-112.
bibitem{Kac} Kac, V., Cheung, P., emph{Quantum Calculus}, Springer, New York, 2002.
bibitem{Phillips} Phillips, G. M., emph{Bernstein polynomials based on the q-integers}, Ann. Numer. Math., textbf{4}(1997), 511-518.
bibitem{Radu} Radu, C., emph{On statistical approximation of a general class of positive linear operators extended in $q$-calculus}, Appl. Math. Comput., textbf{215}(2009), 2317-2325.
bibitem{Wang} Wang, H., emph{Korovkin-type theorem and application}, J. Approx. Theory, textbf{132}(2005), 258-264.
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