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Statistical summability and approximation by de la Vallée-Poussin mean. (English) Zbl 1216.40003
The authors introduce two concepts of statistical $\lambda$-convergence and strongly ${\lambda }_{q}$-convergence $\left(0 and give some relations between $\lambda$-statistical convergence and these newly defined concepts. They also prove a Korovkin type approximation theorem by using the newly defined summability method.

##### MSC:
 40A35 Ideal and statistical convergence 41A36 Approximation by positive operators 47A58 Operator approximation theory
##### References:
 [1] Fast, H.: Sur la convergence statistique, Colloq. math. 2, 241-244 (1951) · Zbl 0044.33605 [2] Steinhaus, H.: Sur la convergence ordinaire et la convergence asymptotique, Colloq. math. 2, 73-84 (1951) [3] Mursaleen, M.: $\lambda$-statistical convergence, Math. slovaca 50, 111-115 (2000) · Zbl 0953.40002 [4] Leindler, L.: Über die de la vallée–pousinsche summierbarkeit allgemeiner orthogonalreihen, Acta math. Acad. sci. Hungar. 16, 375-387 (1965) · Zbl 0138.28802 · doi:10.1007/BF01904844 [5] Dirik, F.; Demirci, K.: Korovkin type approximation theorem for functions of two variables in statistical sense, Turkish J. Math. 33, 1-11 (2009) [6] Doğru, O.; Örkcü, M.: Statistical approximation by a modification of q-Meyer–könig and zeller operators, Appl. math. Lett. 23, 261-266 (2010) · Zbl 1183.41013 · doi:10.1016/j.aml.2009.09.018 [7] Gadziev, A. D.; Orhan, C.: Some approximation theorems via statistical convergence, Rocky mountain J. Math. 32, 129-138 (2002) · Zbl 1039.41018 · doi:10.1216/rmjm/1030539612 · doi:http://math.la.asu.edu/~rmmc/rmj/vol32-1/CONT32-1/CONT32-1.html [8] Gadziev, A. D.: The convergence problems for a sequence of positive linear operators on unbounded sets, and theorems analogous to that of P.P. Korovkin, Sov. math. Dokl. 15, 1433-1436 (1974) · Zbl 0312.41013 [9] P.P. Korovkin, Linear operators and the theory of approximation, India, Delhi, 1960.