Korovkin-type convergence results for non-positive operators

• Autorzy: Oliver Nowak
• Języki publikacji: EN

Abstrakty

EN

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].

Słowa kluczowe

EN

Korovkin-type approximation theory; Positive operator; Non-positive operator; Regular operator; Moving least squares interpolation; Multivariate scattered data interpolation

Kategorie tematyczne

• 41A36: Approximation by positive operators
• 41A63: Multidimensional problems (should also be assigned at least one other classification number in this section)
• 41A05: Interpolation
• 41A35: Approximation by operators (in particular, by integral operators)

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2010
• Tom: 8
• Numer: 5
• Strony: 890-907

Daty

wydano

2010-10-01

online

2010-09-28

Twórcy

autor

Oliver Nowak

• Swiss Federal Institute of Technology

Bibliografia

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• [17] Nowak O., Korovkin-type convergence results for multivariate Shepard formulae, Rev. Anal. Numér. Théor. Approx., 2009, 38(2), 170–176
• [18] Nowak O., High-order convergence of Moving Least Squares Interpolation under regular data distributions, preprint available at http://public.me.com/oliver.nowak/highorder.pdf
• [19] Nowak O., Sonar T., Upwind-biased finite difference approximations from interpolating moving least squares, preprint available at http://public.me.com/oliver.nowak/upwind.pdf
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• [23] Sonar T., Difference operators from interpolating moving least squares and their deviation from optimality, ESAIM, Math. Model. Numer. Anal., 2005, 39(5), 883–908 http://dx.doi.org/10.1051/m2an:2005039
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Identyfikatory

DOI

10.2478/s11533-010-0058-8