Asymptotic approximation with Stancu Beta operators
Keywords:
Stancu beta operators, asymptotic approximation, asymptotic expansion, Stirling numbers of the first and second kind, reciprocal factorialsAbstract
The concern of this paper is a beta type operator \(L_n\) recently introduced by D.D. Stancu.
We present the complete asymptotic expansion for \(L_n\) as \(n\) tends to infinity.
All coefficients of \(n^{-k} \ (k=1,2, \ldots)\) are calculated explicitly in terms of Stirling numbers of the first and second kind.
Moreover, we give an asymptotic expansion for \(L_n\) into a series of reciprocal factorials.
Downloads
References
U. Abel, The moments for the Meyer-König and Zeller operators, J. Approx. Theory, 82 (1995), pp. 352-361, https://doi.org/10.1006/jath.1995.1084
U. Abel, On the asymptotic approximation with operators of Bleimann, Butzer and Hahn, Indag. Math., New Ser., 7 (1996), pp. 1-9, https://doi.org/10.1016/0019-3577(96)88653-8
U. Abel, The complete asymptotic expansion for the Meyer-König and Zeller operators, J. Math. Anal. Appl., 208 (1997), pp. 109-119,https://doi.org/10.1006/jmaa.1997.5295
U. Abel, The asymptotic expansion for the Stancu operators, submitted.
U. Abel, Asymptotic approximation with Kantorovich polynomials, submitted.
J. A. Adell and J. de la Cal, On a Bernstein-type operator associated with the inverse Polya-Eggenberger distribution, Rend. Circolo Matem. Palermo, Ser. II, Nr. 33 (1993), pp. 143-154.
F. Altomare and M. Campiti, "Korovkin-type approximation theory and its applications," Walter de Gruyter, Berlin, New York, 1994, https://doi.org/10.1515/9783110884586
S. N. Bernstein, Complément à l'article de E. Voronowskaja, Dokl. Akad. Nauk USSR, 4 (1932), pp. 86-92.
R. A. DeVore and G. G. Lorentz, "Constructive approximation", Springer, Berlin, Heidelberg 1993, https://doi.org/10.1007/978-3-662-02888-9
C. Jordan, "Calculus of finite differences," Chelsea, New York, 1965.
M. K. Khan, Approximation properties of beta operators, Progress in Approx. Theory, Academic Press, New York, 1991, pp. 483-495.
G. G. Lorentz, "Bernstein polynomials," University of Toronto Press, Toronto, 1953.
A. Lupaş, "Die Folge der Beta-Operatoren," Dissertation, Universität Stuttgart, 1972.
G. Mühlbach, Verallgemeinerungen der Bernstein-und der Lagrange-Polynome. Bemerkungen zu einer Klasse linearer Polynomoperatoren von D. D. Stancu, Rev. Roumaine Math. Pures Appl., 15 (1970), pp. 1235-1252.
P. C. Sikkema, On some linear positive operators, Indag. Math., 32 (1970), pp. 327-337, https://doi.org/10.1016/s1385-7258(70)80037-3
P. C. Sikkema, On the asymptotic approximation with operators of Meyer-König and Zeller, Indag. Math., 32 (1970), pp. 428-440, https://doi.org/10.1016/s1385-7258(70)80047-6
D. D. Stancu, On the beta approximating operators of second kind, Rev. Anal. Numér. Théor. Approx., 24 (1995), pp. 231-239,https://ictp.acad.ro/jnaat/journal/article/view/1995-vol24-nos1-2-art26
R. Upreti, Approximation properties of Beta operators, J. Approx. Theory, 45 (1985), pp. 85-89, https://doi.org/10.1016/0021-9045(85)90036-x
E. V. Voronovskaja, Détermination de la forme asymptotique de l' approximation des fonctions par les polynômes de S. Bernstein, Dokl. Akad. Nauk. SSSR, A (1932), pp. 79-85.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2015 Journal of Numerical Analysis and Approximation Theory
This work is licensed under a Creative Commons Attribution 4.0 International License.
Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
