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BY-NC-ND 3.0 license Open Access Published by De Gruyter February 29, 2012

A-summation process and Korovkin-type approximation theorem for double sequences of positive linear operators

  • Sevda Karakuş EMAIL logo and Kami̇l Demi̇rci̇
From the journal Mathematica Slovaca

Abstract

The aim of this paper is to present a Korovkin-type approximation theorem on the space of all continuous real valued functions on any compact subset of the real two-dimensional space by using a A-summation process. We also study the rates of convergence of positive linear operators with the help of the modulus of continuity.

[1] ATLIHAN, Ö. G.— ORHAN, C.: Matrix summability and positive linear operators, Positivity 11 (2007), 387–389. http://dx.doi.org/10.1007/s11117-007-2049-y10.1007/s11117-007-2049-ySearch in Google Scholar

[2] ATLIHAN, Ö. G.— ORHAN, C.: Summation process of positive linear operators, Comput. Math. Appl. 56 (2008), 1188–1195. http://dx.doi.org/10.1016/j.camwa.2008.02.02010.1016/j.camwa.2008.02.020Search in Google Scholar

[3] DEMİRCİ, K.— DİRİK, F.— KARAKUŞ, S.: A-summability and Korovkin-type approximation theorem for double sequences of positive linear operators (Submitted). Search in Google Scholar

[4] DİRİK, F.— DEMİRCİ, K.: Korovkin type approximation theorem for functions of two variables in statistical sense, Turkish J. Math. 34 (2010), 73–83. Search in Google Scholar

[5] HARDY, G. H.: Divergent Series, Oxford Univ. Press, London, 1949. Search in Google Scholar

[6] HAMILTON, H. J.: Transformations of multiple sequences, Duke Math. J. 2 (1936), 29–60. http://dx.doi.org/10.1215/S0012-7094-36-00204-110.1215/S0012-7094-36-00204-1Search in Google Scholar

[7] MÓRICZ, F.: Statistical convergence of multiple sequences, Arch. Math. (Basel) 81 (2004), 82–89. http://dx.doi.org/10.1007/s00013-003-0506-910.1007/s00013-003-0506-9Search in Google Scholar

[8] MÓRICZ, F.— RHOADES, B. E.: Almost convergence of double sequences and strong regularity of summability matrices, Math. Proc. Cambridge Philos. Soc. 104 (1988), 283–294. http://dx.doi.org/10.1017/S030500410006546410.1017/S0305004100065464Search in Google Scholar

[9] MURSALEEN-EDELY, O. H. H.: Statistical convergence of double sequences, J.Math. Anal. Appl. 288 (2003), 223–231. http://dx.doi.org/10.1016/j.jmaa.2003.08.00410.1016/j.jmaa.2003.08.004Search in Google Scholar

[10] MURSALEEN-SAVAŞ, E.: Almost regular matrices for double sequences, Studia Sci. Math. Hungar. 40 (2003), 205–212. Search in Google Scholar

[11] PATTERSON, R. F.— SAVAŞ, E.: Uniformly summable double sequences, Studia Sci. Math. Hungar. 44 (2007), 147–158. Search in Google Scholar

[12] PRINGSHEIM, A.: Zur theorie der zweifach unendlichen zahlenfolgen, Math. Ann. 53 (1900), 289–321. http://dx.doi.org/10.1007/BF0144897710.1007/BF01448977Search in Google Scholar

[13] ROBISON, G. M.: Divergent double sequences and series. In: Amer. Math. Soc. Transl. Ser. 2, Vol. 28, Amer. Math. Soc., Providence, RI, 1926, pp. 50–73. Search in Google Scholar

[14] RADU, C.: A-summability and approximation of continuous periodic functions, Studia Univ. Babeş-Bolyai Math. 52 (2007), 155–161. Search in Google Scholar

[15] SAVAŞ, E.— RHOADES, B. E.: Double summability factor theorems and applications, Math. Inequal. Appl. 10 (2007), 125–149. Search in Google Scholar

[16] STANCU, D. D.: A method for obtaining polynomials of Bernstein type of two variables, Amer. Math. Monthly 70 (1963), 260–264. http://dx.doi.org/10.2307/231312110.2307/2313121Search in Google Scholar

[17] VOLKOV, V. I.: On the convergence of sequences of positive linear operators in the space of two variables, Dokl. Akad. Nauk. SSSR (N.S.) 115 (1957), 17–19. Search in Google Scholar

Published Online: 2012-2-29
Published in Print: 2012-4-1

© 2012 Mathematical Institute, Slovak Academy of Sciences

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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