A general Korovkin result under generalized convergence. (English) Zbl 1463.41051
Summary: In this paper the classic result of Korovkin about the convergence of sequences of functions defined from sequences of linear operators is reformulated in terms of generalized convergence. This convergence extends some others given in the literature. The operator of the sequence fulfill a shape preserving property more general than the positivity. This property is related with certain extension of the notion of derivative. This extended derivative is precisely the object of the approximation process. The study is completed by analysing the conditions for the existence of an asymptotic formula, from which some interesting consequences are derived as a local version of the shape preserving property. Finally, as applications of the previous results, the author uses the following notion of generalized convergence, an extension of Nörlund-Cesaro summability given by V. Loku and N. L. Braha [Armen. J. Math. 9, No. 1, 35–42 (2017; Zbl 1379.40006)]. A way to transfer a notion of generalized convergence to approximation theory by means of linear operators is shown.
MSC:
| 41A36 | Approximation by positive operators |
| 40D05 | General theorems on summability |
| 41A28 | Simultaneous approximation |
Citations:
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