# Korovkin-type convergence results for non-positive operators

Open Mathematics (2010)

- Volume: 8, Issue: 5, page 890-907
- ISSN: 2391-5455

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topOliver Nowak. "Korovkin-type convergence results for non-positive operators." Open Mathematics 8.5 (2010): 890-907. <http://eudml.org/doc/269305>.

@article{OliverNowak2010,

abstract = {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].},

author = {Oliver Nowak},

journal = {Open Mathematics},

keywords = {Korovkin-type approximation theory; Positive operator; Non-positive operator; Regular operator; Moving least squares interpolation; Multivariate scattered data interpolation; non-positive operator; regular operator; moving least squares interpolation; multivariate scattered data interpolation},

language = {eng},

number = {5},

pages = {890-907},

title = {Korovkin-type convergence results for non-positive operators},

url = {http://eudml.org/doc/269305},

volume = {8},

year = {2010},

}

TY - JOUR

AU - Oliver Nowak

TI - Korovkin-type convergence results for non-positive operators

JO - Open Mathematics

PY - 2010

VL - 8

IS - 5

SP - 890

EP - 907

AB - 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].

LA - eng

KW - Korovkin-type approximation theory; Positive operator; Non-positive operator; Regular operator; Moving least squares interpolation; Multivariate scattered data interpolation; non-positive operator; regular operator; moving least squares interpolation; multivariate scattered data interpolation

UR - http://eudml.org/doc/269305

ER -

## References

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