Abstract and Applied Analysis

Korovkin Second Theorem via B -Statistical A -Summability

M. Mursaleen and A. Kiliçman

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Abstract

Korovkin type approximation theorems are useful tools to check whether a given sequence ( L n ) n 1 of positive linear operators on C [ 0,1 ] of all continuous functions on the real interval [ 0,1 ] is an approximation process. That is, these theorems exhibit a variety of test functions which assure that the approximation property holds on the whole space if it holds for them. Such a property was discovered by Korovkin in 1953 for the functions 1, x , and x 2 in the space C [ 0,1 ] as well as for the functions 1, cos, and sin in the space of all continuous 2 π -periodic functions on the real line. In this paper, we use the notion of B -statistical A -summability to prove the Korovkin second approximation theorem. We also study the rate of B -statistical A -summability of a sequence of positive linear operators defined from C 2 π ( ) into C 2 π ( ) .

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 598963, 6 pages.

Dates
First available in Project Euclid: 18 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1366306746

Digital Object Identifier
doi:10.1155/2013/598963

Mathematical Reviews number (MathSciNet)
MR3034945

Zentralblatt MATH identifier
06161356

Citation

Mursaleen, M.; Kiliçman, A. Korovkin Second Theorem via $B$ -Statistical $A$ -Summability. Abstr. Appl. Anal. 2013 (2013), Article ID 598963, 6 pages. doi:10.1155/2013/598963. https://projecteuclid.org/euclid.aaa/1366306746


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References

  • H. Fast, “Sur la convergence statistique,” Colloquium Mathematicum, vol. 2, pp. 241–244, 1951.
  • Mursaleen and O. H. H. Edely, “Statistical convergence of double sequences,” Journal of Mathematical Analysis and Applications, vol. 288, no. 1, pp. 223–231, 2003.
  • S. A. Mohiuddine, A. Alotaibi, and M. Mursaleen, “Statistical convergence of double sequences in locally solid Riesz spaces,” Abstract and Applied Analysis, vol. 2012, Article ID 719729, 9 pages, 2012.
  • A. R. Freedman and J. J. Sember, “Densities and summability,” Pacific Journal of Mathematics, vol. 95, no. 2, pp. 293–305, 1981.
  • J. Connor, “On strong matrix summability with respect to a modulus and statistical convergence,” Canadian Mathematical Bulletin. Bulletin Canadien de Mathématiques, vol. 32, no. 2, pp. 194–198, 1989.
  • E. Kolk, “Matrix summability of statistically convergent sequences,” Analysis, vol. 13, no. 1-2, pp. 77–83, 1993.
  • O. H. H. Edely and M. Mursaleen, “On statistical $A$-summability,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 672–680, 2009.
  • M. Mursaleen and O. H. H. Edely, “Generalized statistical convergence,” Information Sciences, vol. 162, no. 3-4, pp. 287–294, 2004.
  • O. H. H. Edely, “$B$-statistically $A$-summability,” Thai Journal of Mathematics. In press.
  • J. A. Fridy and C. Orhan, “Lacunary statistical convergence,” Pacific Journal of Mathematics, vol. 160, no. 1, pp. 43–51, 1993.
  • Mursaleen, “$\lambda $-statistical convergence,” Mathematica Slovaca, vol. 50, no. 1, pp. 111–115, 2000.
  • F. Móricz, “Tauberian conditions, under which statistical convergence follows from statistical summability $(C,1)$,” Journal of Mathematical Analysis and Applications, vol. 275, no. 1, pp. 277–287, 2002.
  • F. Móricz, “Theorems relating to statistical harmonic summability and ordinary convergence of slowly decreasing or oscillating sequences,” Analysis, vol. 24, no. 2, pp. 127–145, 2004.
  • F. Móricz and C. Orhan, “Tauberian conditions under which statistical convergence follows from statistical summability by weighted means,” Studia Scientiarum Mathematicarum Hungarica, vol. 41, no. 4, pp. 391–403, 2004.
  • P. P. Korovkin, “On convergence of linear positive operators in the space of continuous functions,” Doklady Akademii Nauk, vol. 90, pp. 961–964, 1953.
  • P. P. Korovkin, Linear Operators and Approximation Theory, Hindustan, Delhi, India, 1960.
  • O. Duman, “Statistical approximation for periodic functions,” Demonstratio Mathematica, vol. 36, no. 4, pp. 873–878, 2003.
  • S. Karakuş and K. Demirci, “Approximation for periodic functions via statistical \emphA-summability,” Acta Mathematica Universitatis Comenianae, vol. 81, no. 2, pp. 159–169, 2012.
  • F. Altomare, “Korovkin-type theorems and approximation by positive linear operators,” Surveys in Approximation Theory, vol. 5, pp. 92–164, 2010.
  • K. Demirci and F. Dirik, “Approximation for periodic functions via statistical $\sigma $-convergence,” Mathematical Communications, vol. 16, no. 1, pp. 77–84, 2011.
  • O. H. H. Edely, S. A. Mohiuddine, and A. K. Noman, “Korovkin type approximation theorems obtained through generalized statistical convergence,” Applied Mathematics Letters, vol. 23, no. 11, pp. 1382–1387, 2010.
  • S. A. Mohiuddine, A. Alotaibi, and M. Mursaleen, “Statistical summability $(C,1)$ and a Korovkin type approximation theorem,” Journal of Inequalities and Applications, vol. 2012, 172 pages, 2012.
  • M. Mursaleen and R. Ahmad, “Korovkin type approximation theorem through statistical lacunary summability,” Iranian Journal of Science and Technology-Science. In press.
  • M. Mursaleen and A. Alotaibi, “Statistical summability and approximation by de la Vallée-Poussin mean,” Erratum: Applied Mathematics Letters, vol. 25, p. 665, 2012.
  • M. Mursaleen and A. Alotaibi, “Statistical lacunary summability and a Korovkin type approximation theorem,” Annali dell'Universitá di Ferrara, vol. 57, no. 2, pp. 373–381, 2011.
  • M. Mursaleen, V. Karakaya, M. Ertürk, and F. Gürsoy, “Weighted statistical convergence and its application to Korovkin type approximation theorem,” Applied Mathematics and Computation, vol. 218, no. 18, pp. 9132–9137, 2012.
  • C. Radu, “$A$-summability and approximation of continuous periodic functions,” Studia. Universitatis Babeş-Bolyai. Mathematica, vol. 52, no. 4, pp. 155–161, 2007.
  • H. M. Srivastava, M. Mursaleen, and A. Khan, “Generalized equi-statistical convergence of positive linear operators and associated approximation theorems,” Mathematical and Computer Modelling, vol. 55, no. 9-10, pp. 2040–2051, 2012.
  • G. A. Anastassiou, M. Mursaleen, and S. A. Mohiuddine, “Some approximation theorems for functions of two variables through almost convergence of double sequences,” Journal of Computational Analysis and Applications, vol. 13, no. 1, pp. 37–46, 2011.
  • C. Belen, M. Mursaleen, and M. Yildirim, “Statistical $A$-summability of double sequences and a Korovkin type approximation theorem,” Bulletin of the Korean Mathematical Society, vol. 49, no. 4, pp. 851–861, 2012.
  • K. Demirci and F. Dirik, “Four-dimensional matrix transformation and rate of $A$-statistical convergence of periodic functions,” Mathematical and Computer Modelling, vol. 52, no. 9-10, pp. 1858–1866, 2010.
  • M. Mursaleen and A. Alotaibi, “Korovkin type approximation theorem for functions of two variables through statistical $A$-summability,” Advances in Difference Equations, vol. 2012, 65 pages, 2012.
  • S. A. Mohiuddine, “An application of almost convergence in approximation theorems,” Applied Mathematics Letters, vol. 24, no. 11, pp. 1856–1860, 2011.