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Korovkin-type convergence results for non-positive operators. (English) Zbl 1220.41004

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 P. Lancaster and 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 M. Campiti in [“Convexity-monotone operators in Korovkin theory”, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 33, 229–238 (1994; Zbl 0929.41022)].

MSC:

41A35 Approximation by operators (in particular, by integral operators)
41A36 Approximation by positive operators
41A05 Interpolation in approximation theory
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[1] Agratini O., On approximation of functions by positive linear operators, Stud. Cercet. Ştiinţ. Ser. Mat. Univ. Bacąau, 2006, 16 Suppl., 17-28; · Zbl 1174.41334
[2] Altomare F., Campiti M., Korovkin-type Approximation Theory and its Applications, de Gruyter Stud. Math., 17, de Gruyter, Berlin-New York, 1994; · Zbl 0924.41001
[3] Bohman H., On approximation of continuous and of analytic functions, Ark. Mat., 1952, 2(1), 43-56 http://dx.doi.org/10.1007/BF02591381; · Zbl 0048.29901
[4] Campiti M., Convexity-monotone operators in Korovkin theory, In: Proceedings of the 2nd International Conference in Functional Analysis and Approximation Theory, Acquafredda di Maratea (Potenza), September 14-19, 1992, Rend. Circ. Mat. Palermo (2) Suppl., 1994, 33, 229-238; · Zbl 0929.41022
[5] DeVore R.A., The Approximation of Continuous Functions by Positive Linear Operators, Lecture Notes in Math., 293, Springer, Berlin-Heidelberg-New York, 1972; · Zbl 0276.41011
[6] Farwig R., Multivariate interpolation of scattered data by moving least squares methods, In: Algorithms for Approximation, Proc. IMA Conf., Shrivenham, July 1985, The Institute of Mathematics and its Applications Conference Series, New Series, 10, Clarendon Press, Oxford, 1987, 193-211;
[7] Farwig R., Rate of convergence for moving least squares interpolation methods: the univariate case, In: Progress in Approximation Theory, Academic Press, 1991, 313-327;
[8] Korovkin P.P., On convergence of linear positive operators in the space of continuous functions, Dokl. Akad. Nauk SSSR, 1953, 90, 961-964, (in Russian);
[9] Korovkin P.P., Linear Operators and Approximation Theory, International Monographs on advanced Mathematics & Physics, Hindustan Publishing Corp., Delhi, 1960;
[10] Lancaster P., Šalkauskas K., Surfaces generated by moving least squares methods, Math. Comp., 1981, 37(155), 141-158; · Zbl 0469.41005
[11] Levin D., The approximation power of moving least-squares, Math. Comp., 1998, 67(224), 1517-1531 http://dx.doi.org/10.1090/S0025-5718-98-00974-0; · Zbl 0911.41016
[12] Lorentz G.G., Approximation of Functions, 2nd ed., Chelsea Publishing Company, New York, 1986; · Zbl 0643.41001
[13] Muñoz-Delgado F.J., Ramírez-González V., Cárdenas-Morales D., Qualitative Korovkin-type results on conservative approximation, J. Approx. Theory, 1998, 94(1), 144-159 http://dx.doi.org/10.1006/jath.1998.3182; · Zbl 0911.41015
[14] Netuzhylov H., Sonar T., Yomsatieankul W., Finite difference operators from moving least squares interpolation, ESAIM, Math. Model. Numer. Anal., 2007, 41(5), 959-974 http://dx.doi.org/10.1051/m2an:2007042; · Zbl 1153.65356
[15] Nishishiraho T., Convergence of quasi-positive linear operators, Atti Semin. Mat. Fis. Univ. Modena, 1992, 40(2), 519-526; · Zbl 0799.47016
[16] Nowak O., Exakte kleinste Quadrate Interpolierende: Konvergenzresultate vom Korovkin-Typ und Anwendungen im Kontext der numerischen Approximation von Erhaltungsgleichungen, Ph.D. thesis, TU Braunschweig, 2009; · Zbl 1197.65017
[17] Nowak O., Korovkin-type convergence results for multivariate Shepard formulae, Rev. Anal. Numér. Théor. Approx., 2009, 38(2), 170-176; · Zbl 1224.41009
[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;
[20] Popoviciu T., Asupra demonstratiei teoremei lui Weierstrass cu ajutorul polinoamelor de interpolare, Editura Academiei Republicii Populare Române, 1951, 1664-1667;
[21] Popoviciu T., On the proof of Weierstrass’ theorem using interpolation polynomials (translated by Daniela Kasco), East J. Approx., 1998, 4(1), 107-110; · Zbl 0914.41001
[22] Shepard D., A two-dimensional interpolation function for irregularly-spaced data, In: Proceedings of the 23rd ACM National Conference, 1968, 517-524;
[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; · Zbl 1085.39018
[24] Wendland H., Scattered Data Approximation, Cambridge Monogr. Appl. Comput. Math., 17, Cambridge University Press, Cambridge, 2005; · Zbl 1075.65021
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