Abstract and Applied Analysis

On Some Extensions of Szasz Operators Including Boas-Buck-Type Polynomials

Sezgin Sucu, Gürhan İçöz, and Serhan Varma

Full-text: Open access

Abstract

This paper is concerned with a new sequence of linear positive operators which generalize Szasz operators including Boas-Buck-type polynomials. We establish a convergence theorem for these operators and give the quantitative estimation of the approximation process by using a classical approach and the second modulus of continuity. Some explicit examples of our operators involving Laguerre polynomials, Charlier polynomials, and Gould-Hopper polynomials are given. Moreover, a Voronovskaya-type result is obtained for the operators containing Gould-Hopper polynomials.

Article information

Source
Abstr. Appl. Anal., Volume 2012 (2012), Article ID 680340, 15 pages.

Dates
First available in Project Euclid: 14 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1355495833

Digital Object Identifier
doi:10.1155/2012/680340

Mathematical Reviews number (MathSciNet)
MR2965458

Zentralblatt MATH identifier
1250.41014

Citation

Sucu, Sezgin; İçöz, Gürhan; Varma, Serhan. On Some Extensions of Szasz Operators Including Boas-Buck-Type Polynomials. Abstr. Appl. Anal. 2012 (2012), Article ID 680340, 15 pages. doi:10.1155/2012/680340. https://projecteuclid.org/euclid.aaa/1355495833


Export citation

References

  • P. P. Korovkin, “On convergence of linear positive operators in the space of continuous functions,” Doklady Akademii Nauk SSSR, vol. 90, pp. 961–964, 1953 (Russian).
  • O. Szasz, “Generalization of S. Bernstein's polynomials to the infinite interval,” Journal of Research of the National Bureau of Standards, vol. 45, pp. 239–245, 1950.
  • A. Jakimovski and D. Leviatan, “Generalized Szász operators for the approximation in the infinite interval,” Mathematica, vol. 11, pp. 97–103, 1969.
  • M. E. H. Ismail, “On a generalization of Szász operators,” Mathematica, vol. 39, no. 2, pp. 259–267, 1974.
  • S. Varma, S. Sucu, and G. \.Içöz, “Generalization of Szasz operators involving Brenke type polynomials,” Computers & Mathematics with Applications, vol. 64, no. 2, pp. 121–127, 2012.
  • T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, NY, USA, 1978.
  • M. E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, Cambridge, UK, 2005.
  • F. Altomare and M. Campiti, Korovkin-Type Approximation Theory and Its Applications, vol. 17 of de Gruyter Studies in Mathematics, Walter de Gruyter, Berlin, Germany, 1994.
  • I. Gavrea and I. Raşa, “Remarks on some quantitative Korovkin-type results,” Revue d'Analyse Numérique et de Théorie de l'Approximation, vol. 22, no. 2, pp. 173–176, 1993.
  • V. V. Zhuk, “Functions of the Lip1 class and S. N. Bernstein's polynomials,” Vestnik Leningradskogo Universiteta. Matematika, Mekhanika, Astronomiya, vol. 1, pp. 25–30, 1989.
  • S. Varma and F. Taşdelen, “Szász type operators involving Charlier polynomials,” Mathematical and Computer Modelling, vol. 56, no. 5-6, pp. 118–122, 2012.
  • H. W. Gould and A. T. Hopper, “Operational formulas connected with two generalizations of Hermite polynomials,” Duke Mathematical Journal, vol. 29, pp. 51–63, 1962.
  • K. Douak, “The relation of the $d$-orthogonal polynomials to the Appell polynomials,” Journal of Computational and Applied Mathematics, vol. 70, no. 2, pp. 279–295, 1996.
  • J. Van Iseghem, “Vector orthogonal relations. Vector QD-algorithm,” Journal of Computational and Applied Mathematics, vol. 19, no. 1, pp. 141–150, 1987.
  • P. Maroni, “L'orthogonalité et les récurrences de polynômes d'ordre supérieur à deux,” Annales Faculté des Sciences de Toulouse Série 5, vol. 10, no. 1, pp. 105–139, 1989.
  • H. Gonska, “On the degree of approximation in Voronovskaja's theorem,” Studia. Universitatis Babeş-Bolyai. Mathematica, vol. 52, no. 3, pp. 103–115, 2007.