Korovkin-type convergence results for non-positive operators
Korovkin-type approximation theory usually deals with convergence analysis for sequences of positive operators. The author presents 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, the author shows that operators related to the multivariate scattered data interpolation technique moving least squares interpolation (originally due to \textit{P. Lancaster} and \textit{K. Salkauskas} [``Surfaces generated by moving least squares methods'', Math. Comput. 37, 141--158 (1981; Zbl 0469.41005)]) give rise to such sequences. The present paper also generalizes Korovkin-type results regarding Shepard interpolation due to the author [``Korovkin-type convergence results for multivariate Shepard formulae'', Rev. Anal. Numér. Théor. Approx. 38, No. 2, 170--176 (2009; Zbl 1224.41009)]. 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 \textit{M. Campiti} in [``Convexity-monotone operators in Korovkin theory'', Suppl. Rend. Circ. Mat. Palermo, II. Ser. 33, 229--238 (1994; Zbl 0929.41022)].
multivariate scattered data interpolation
scientific article; zbMATH DE number 5900620
scientific article