×

On the iterates of positive linear operators preserving the affine functions. (English) Zbl 1196.41014

Summary: We study the limit behavior of the iterates of a large class of positive linear operators preserving the affine functions and, as a byproduct of our result, we obtain the limit of the iterates of Meyer-König and Zeller operators.

MSC:

41A36 Approximation by positive operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adell, J. A.; Badía, F. G.; de la Cal, J., On the iterates of some Bernstein-type operators, J. Math. Anal. Appl., 209, 529-541 (1997) · Zbl 0872.41009
[2] Agratini, O., On the iterates of a class of summation-type linear positive operators, Comput. Math. Appl., 55, 1178-1180 (2008) · Zbl 1151.41013
[3] Agratini, O.; Rus, I. A., Iterates of a class of discrete linear operators via contraction principle, Comment. Math. Univ. Carolin., 44, 555-563 (2003) · Zbl 1096.41015
[4] Altomare, F.; Campiti, M., Korovkin-Type Approximation Theory and Its Applications, de Gruyter Stud. Math., vol. 17 (1994), Walter de Gruyter & Co.: Walter de Gruyter & Co. Berlin, Appendix A by Michael Pannenberg and Appendix B by Ferdinand Beckhoff · Zbl 0924.41001
[5] Becker, M.; Nessel, R. J., A global approximation theorem for Meyer-König and Zeller operators, Math. Z., 160, 195-206 (1978) · Zbl 0376.41007
[6] Cheney, E. W.; Sharma, A., Bernstein power series, Canad. J. Math., 16, 241-252 (1964) · Zbl 0128.29001
[7] Cooper, S.; Waldron, S., The eigenstructure of the Bernstein operator, J. Approx. Theory, 105, 133-165 (2000) · Zbl 0963.41006
[8] Gonska, H.; Kacsó, D.; Piţul, P., The degree of convergence of over-iterated positive linear operators, J. Appl. Funct. Anal., 1, 403-423 (2006) · Zbl 1099.41011
[9] Gonska, H. H.; Zhou, X. L., Approximation theorems for the iterated Boolean sums of Bernstein operators, J. Comput. Appl. Math., 53, 21-31 (1994) · Zbl 0816.41020
[10] Jessen, B., Bemærkninger om konvekse Funktioner og Uligheder imellem Middelværdier, I, Mat. Tidsskrift B, 17-28 (1931)
[11] Karlin, S.; Ziegler, Z., Iteration of positive approximation operators, J. Approx. Theory, 3, 310-339 (1970) · Zbl 0199.44702
[12] Kelisky, R. P.; Rivlin, T. J., Iterates of Bernstein polynomials, Pacific J. Math., 21, 511-520 (1967) · Zbl 0177.31302
[13] May, C. P., Saturation and inverse theorems for combinations of a class of exponential-type operators, Canad. J. Math., 28, 1224-1250 (1976) · Zbl 0342.41018
[14] Meyer-König, W.; Zeller, K., Bernsteinsche Potenzreihen, Studia Math., 19, 89-94 (1960) · Zbl 0091.14506
[15] Rus, I. A., Iterates of Bernstein operators, via contraction principle, J. Math. Anal. Appl., 292, 259-261 (2004) · Zbl 1056.41004
[16] Sikkema, P. C., On the asymptotic approximation with operators of Meyer-König and Zeller, Nederl. Akad. Wetensch. Proc. Ser. A. Nederl. Akad. Wetensch. Proc. Ser. A, Indag. Math., 32, 428-440 (1970) · Zbl 0205.08101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.