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Korovkin-type convergence results for non-positive operators

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Nowak O.

Open Mathematics

2010

tom 8

nr 5

890-907

Korovkin-type approximation theory usually deals with convergence analysis for sequences of positive operators. In this work we present qualitative Korovkin-type convergence results for a class of sequences of non-positive operators, more precisely regular operators with vanishing negative parts under a limiting process. Sequences of that type are called sequences of almost positive linear operators and have not been studied before in the context of Korovkin-type approximation theory. As an example we show that operators related to the multivariate scattered data interpolation technique moving least squares interpolation originally due to Lancaster and Šalkauskas [Surfaces generated by moving least squares methods, Math. Comp., 1981, 37, 141–158] give rise to such sequences. This work also generalizes Korovkin-type results regarding Shepard interpolation [Korovkin-type convergence results for multivariate Shepard formulae, Rev. Anal. Numér. Théor. Approx., 2009, 38, 170–176] due to the author. Moreover, this work establishes connections and differences between the concepts of sequences of almost positive linear operators and sequences of quasi-positive or convexity-monotone linear operators introduced and studied by Campiti in [Convexity-monotone operators in Korovkin theory, Rend. Circ. Mat. Palermo (2) Suppl., 1993, 33, 229–238].

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1 Open Mathematics

1 Nowak O.

1 2010

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Korovkin-type convergence results for non-positive operators