Abstract and Applied Analysis

Asymptotic Formulae via a Korovkin-Type Result

Daniel Cárdenas-Morales, Pedro Garrancho, and Ioan Raşa

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Abstract

We present a sort of Korovkin-type result that provides a tool to obtain asymptotic formulae for sequences of linear positive operators.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 217464, 12 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1355495695

Digital Object Identifier
doi:10.1155/2012/217464

Mathematical Reviews number (MathSciNet)
MR2922908

Zentralblatt MATH identifier
1241.41012

Citation

Cárdenas-Morales, Daniel; Garrancho, Pedro; Raşa, Ioan. Asymptotic Formulae via a Korovkin-Type Result. Abstr. Appl. Anal. 2012 (2012), Article ID 217464, 12 pages. doi:10.1155/2012/217464. https://projecteuclid.org/euclid.aaa/1355495695


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References

  • E. Voronovskaja, “Détermination de la forme asymptotique d'approximation des fonctions par les polynômes de S. Bernstein,” Doklady Akademii Nauk SSSR, pp. 79–85, 1932.
  • R. A. DeVore, The Approximation of Continuous Functions by Positive Linear Operators, vol. 293 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1972.
  • F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and Its Applications, vol. 17, Walter de Gruyter, Berlin, Germany, 1994.
  • J. M. Aldaz, O. Kounchev, and H. Render, “Shape preserving properties of generalized Bernstein operators on extended Chebyshev spaces,” Numerische Mathematik, vol. 114, no. 1, pp. 1–25, 2009.
  • M. A. Özarslan and O. Duman, “MKZ type operators providing a better estimation on (1/2, 3),” Canadian Mathematical Bulletin. Bulletin Canadien de Mathématiques, vol. 50, no. 3, pp. 434–439, 2007.
  • E. W. Cheney and A. Sharma, “Bernstein power series,” Canadian Journal of Mathematics. Journal Canadien de Mathématiques, vol. 16, pp. 241–252, 1964.
  • J. P. King, “Positive linear operators which preserve ${x}^{2}$,” Acta Mathematica Hungarica, vol. 99, no. 3, pp. 203–208, 2003.
  • H. Gonska and P. Piţul, “Remarks on an article of J. P. King,” Commentationes Mathematicae Universitatis Carolinae, vol. 46, no. 4, pp. 645–652, 2005.
  • D. Cárdenas-Morales, P. Garrancho, and F. J. Muñoz-Delgado, “Shape preserving approximation by Bernstein-type operators which fix polynomials,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1615–1622, 2006.
  • O. Agratini, “Linear operators that preserve some test functions,” International Journal of Mathematics and Mathematical Sciences, Article ID 94136, 11 pages, 2006.
  • O. Duman and M. A. Özarslan, “Szász-Mirakjan type operators providing a better error estimation,” Applied Mathematics Letters. An International Journal of Rapid Publication, vol. 20, no. 12, pp. 1184–1188, 2007.
  • H. Gonska, P. Piţul, and I. Raşa, “General King-type operators,” Results in Mathematics, vol. 53, no. 3-4, pp. 279–286, 2009.
  • D. Cárdenas-Morales, P. Garrancho, and I. Raşa, “Bernstein-type operators which preserve polynomials,” Computers & Mathematics with Applications. An International Journal, vol. 62, no. 1, pp. 158–163, 2011.
  • Z. Finta, “Estimates for Bernstein type operators,” Mathematical Inequalities & Applications, vol. 15, no. 1, pp. 127–135, 2012.
  • O. Shisha and B. Mond, “The degree of convergence of sequences of linear positive operators,” Proceedings of the National Academy of Sciences of the United States of America, vol. 60, pp. 1196–1200, 1968.