In this paper, considering A-statistical convergence instead of Pringsheim’s sense for double sequences, we prove a Korovkin-type approximation theorem for sequences of positive linear operators defined on the space of all real valued Bögel-type continuous and periodic functions on the whole real two-dimensional space. A strong application is also presented. Furthermore, we obtain some rates of A-statistical convergence in our approximation.
@article{bwmeta1.element.doi-10_2478_s11533-009-0025-4,
author = {Fadime Dirik and Oktay Duman and Kamil Demirci},
title = {Statistical approximation to B\"ogel-type continuous and periodic functions},
journal = {Open Mathematics},
volume = {7},
year = {2009},
pages = {539-549},
zbl = {1179.41011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0025-4}
}
Fadime Dirik; Oktay Duman; Kamil Demirci. Statistical approximation to Bögel-type continuous and periodic functions. Open Mathematics, Tome 7 (2009) pp. 539-549. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0025-4/
[1] Anastassiou G.A., Duman O., A Baskakov type generalization of statistical Korovkin theory, J. Math. Anal. Appl., 2008, 340, 476–486 http://dx.doi.org/10.1016/j.jmaa.2007.08.040 | Zbl 1133.41004
[2] Anastassiou G.A., Gal S.G., Approximation theory: Moduli of continuity and global smoothness preservation, Birkhäuser, Boston, 2000 | Zbl 0937.41001
[3] Badea I., Modulus of continuity in Bögel sense and some applications for approximation by a Bernstein-type operator, Stud. Univ. Babeş-Bolyai Math., 1973, 18, 69–78 (in Romanian)
[4] Badea C., Badea I., Cottin C., A Korovkin-type theorem for generalizations of Boolean sum operators and approximation by trigonometric pseudopolynomials, Anal. Numér. Théor. Approx., 1988, 17, 7–17 | Zbl 0671.42003
[5] Badea C., Badea I., Gonska H.H., A test function and approximation by pseudopolynomials, Bull. Austral. Math. Soc., 1986, 34, 53–64 http://dx.doi.org/10.1017/S0004972700004494 | Zbl 0595.41017
[6] Badea C., Cottin C., Korovkin-type theorems for generalized Boolean sum operators, Approximation theory (Kecskemét, 1990), 51–68, Colloq. Math. Soc. János Bolyai, North-Holland, Amsterdam, 1991, 58 | Zbl 0765.41020
[7] Bojanic R., Shisha O., Approximation of continuous, periodic functions by discrete positive linear operators, J. Approx. Theory, 1974, 11, 231–235 http://dx.doi.org/10.1016/0021-9045(74)90015-X | Zbl 0298.41014
[8] Bögel K., Mehrdimensionale differentiation von funktionen mehrerer veränderlicher, J. Reine Angew. Math., 1934, 170, 197–217 | Zbl 60.0215.02
[9] Bögel K., Über mehrdimensionale differentiation, integration und beschränkte variation, J. Reine Angew. Math., 1935, 173, 5–29 | Zbl 0011.05903
[10] Bögel K., Über die mehrdimensionale differentiation, Jahresber. Deutsch. Math.-Verein., 1962, 65, 45–71 | Zbl 0111.05901
[11] Cottin C., Approximation by bounded pseudo-polynomials, In: Musielak J. et al (Eds.), Function Spaces, Teubner-Texte zur Mathematik, 1991, 120, 152–160 | Zbl 0741.41004
[12] Duman O., Statistical approximation for periodic functions, Demonstratio Math., 2003, 36, 873–878 | Zbl 1065.41041
[13] Duman O., Erkuş E., Gupta V., Statistical rates on the multivariate approximation theory, Math. Comput. Modelling, 2006, 44, 763–770 http://dx.doi.org/10.1016/j.mcm.2006.02.009 | Zbl 1132.41330
[14] Erkuş E., Duman O., Srivastava H.M., Statistical approximation of certain positive linear operators constructed by means of the Chan-Chyan-Srivastava polynomials, Appl. Math. Comput., 2006, 182, 213–222 http://dx.doi.org/10.1016/j.amc.2006.01.090 | Zbl 1103.41024
[15] Hamilton H.J., Transformations of multiple sequences, Duke Math. J., 1936, 2, 29–60 http://dx.doi.org/10.1215/S0012-7094-36-00204-1 | Zbl 0013.30301
[16] Hardy G.H., Divergent Series, Oxford Univ. Press, London, 1949 | Zbl 0032.05801
[17] Karakuş S., Demirci K., Duman O., Equi-statistical convergence of positive linear operators, J. Math. Anal. Appl., 2008, 339, 1065–1072 http://dx.doi.org/10.1016/j.jmaa.2007.07.050 | Zbl 1131.41008
[18] Moricz F., Statistical convergence of multiple sequences, Arch. Math. (Basel), 2004, 81, 82–89 | Zbl 1041.40001
[19] Mursaleen, Edely O.H.H., Statistical convergence of double sequences, J. Math. Anal. Appl., 2003, 288, 223–231 http://dx.doi.org/10.1016/j.jmaa.2003.08.004 | Zbl 1032.40001
[20] Pringsheim A., Zur theorie der zweifach unendlichen zahlenfolgen, Math. Ann., 1900, 53, 289–321 http://dx.doi.org/10.1007/BF01448977
[21] Robison G.M., Divergent double sequences and series, Amer. Math. Soc. Transl., 1926, 28, 50–73 http://dx.doi.org/10.2307/1989172 | Zbl 52.0223.01
[22] Schumaker L.L., Spline Functions: Basic Theory, John Wiley & Sons, New York, 1981 | Zbl 0449.41004