Abstract
In the present note we prove convergence results for over-iterates of certain (generalized) Bernstein-Stancu operators. Similar assertions were obtained in [11]. However, our approach is different in the sense that it uses the spectrum of the operators involved. It is therefore possible to make global statements on [0, 1].
Keywords Bernstein-Stancu operators, eigenvalues, eigenfunctions, iterates.
Mathematics Subject Classsification (2000): 41A36, 47A75
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
1. Călugăreanu, G.: On operators of S.N. Bernstein. Spectra of operators. (Romanian) Gaz. Mat. Ser. A 71, 448–451 (1966)
2. Cooper, S., Waldron, S.: The eigenstructure of the Bernstein operator. J. Approx. Theory 105, 133–165 (2000)
3. Gonska, H.: Quantitative Aussagen zur Approximation durch positive lineare Operatoren. Dissertation. Gesamthochschule Duisburg (1979)
4. Gonska, H.H., Meier, J.: Quantitative theorems on approximation by Bernstein-Stancu operators. Calcolo 21, 317–335 (1984)
5. Gonska, H., Kacsó, D., Piţul, P.: The degree of convergence of over-iterated positive linear operators. J. Appl. Funct. Anal. 1, 403–423 (2006)
6. Kelisky, R.P., Rivlin, T.J.: Iterates of Bernstein polynomials. Pacific J. Math. 21, 511–520 (1967)
7. Kemeny, J.G., Snell, J.L.: Finite Markov chains. New York: Springer 1976
8. Lupaş, A.: Die Folge der Betaoperatoren. Dissertation. Uni. Stuttgart 1972
9. Ostrovska, S.: q-Bernstein polynomials and their iterates. J. Approx. Theory 123, 232–255 (2003)
10. Raşa, I., Vladislav, T.: Some properties of Bernstein and Stancu operators. In: Stancu, D.D. et al. (eds.): Approximation and optimization. Vol. 1. Transilvania Press: Cluj-Napoca 1997, pp. 345–350
11. Rus, I.A.: Iterates of Stancu operators, via contraction principle. Studia Univ. Babeş-Bolyai. Math. 47 (4), 101–104 (2002)
12. Rus, I.A.: Iterates of Bernstein operators, via contraction principle. J. Math. Anal. Appl. 292, 259–261 (2004)
13. Stancu, D.D.: Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pures Appl. 13, 1173–1194 (1968)
14. Stancu, D.D.: Approximation of functions by means of some new classes of positive linear operators. In: Collatz, L., Meinardus, G. (eds.): Numerische Methoden der Approximationstheorie. Band 1. Basel: Birkhäuser 1972 pp. 187–203
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gonska, H., Piţul, P. & Raşa, I. Over-iterates of Bernstein-Stancu operators. Calcolo 44, 117–125 (2007). https://doi.org/10.1007/s10092-007-0131-2
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10092-007-0131-2