The eigenstructure of some positive linear operators
DOI:
https://doi.org/10.33993/jnaat431-994Keywords:
positive linear operators, eigenvalues and eigenpolynomials, iterates and series of positive linear operators, strongly continuous semigroups, asymptotic behaviourAbstract
Of concern is the study of the eigenstructure of some classes of positive linear operators satisfying particular conditions. As a consequence, some results concerning the asymptotic behaviour as \(t\to +\infty\) of particular strongly continuous semigroups \((T(t))_{t\geq 0}\) expressed in terms of iterates of the operators under consideration are obtained as well. All the analysis carried out herein turns out to be quite general and includes some applications to concrete cases of interest, related to the classical Beta, Stancu and Bernstein operators.Downloads
References
F. Altomare and M. Campiti, Korovkin-type Approximation Theory and its Applications, W. de Gruyter, Berlin-New York, 1994. DOI: https://doi.org/10.1515/9783110884586
F. Altomare and I. Raşa, On some classes of diffusion equations and related approximation problems, in: Trends and Applications in Constructive Approximation, M. G. de Bruin, D. H. Mache and J. Szabados (Eds.), ISNM Vol. 151 (2005), 13-26, Birkhäuser-Verlag, Basel. DOI: https://doi.org/10.1007/3-7643-7356-3_2
F. Altomare, V. Leonessa and I. Raşa, On Bernstein-Schabl operators on the unit interval, Zeit. Anal. Anwend., 27 (2008), pp. 353-379. DOI: https://doi.org/10.4171/ZAA/1360
A. Attalienti, Generalized Bernstein-Durrmeyer operators and the associated limit semigroup, J. Approximation Theory, 99 (1999), pp. 289-309, http://ictp.acad.ro/jnaat/journal/article/view/2007-vol36-no1-art5 DOI: https://doi.org/10.1006/jath.1999.3329
A. Attalienti and I. Raşa, Total Positivity: an application to positive linear operators and to their limiting semigroup, Anal. Numér. Théor. Approx., 36 (2007), pp. 51-66, https://ictp.acad.ro/jnaat/journal/article/view/2007-vol36-no1-art5
A. Attalienti and I. Raşa, Asymptotic behaviour of C₀-semigroups, in: Proceedings of the International Conference on Numerical Analysis and Approximation Theory, Cluj-Napoca, Romania, July 5-8, 2006, ISBN 973-686-961-X, 127-130.
A. Attalienti and I. Raşa, Overiterated linear operators and asymptotic behaviour of semigroups, Mediterr. J. Math., 5 (2008), pp. 315-324, https://doi.org/10.1007/s00009-008-0152-3 DOI: https://doi.org/10.1007/s00009-008-0152-3
S. Cooper and S. Waldron, The eigenstructure of the Bernstein operator, J. Approx. Theory, 105 (2000), no. 1, pp. 133-165, https://doi.org/10.1006/jath.2000.3464 DOI: https://doi.org/10.1006/jath.2000.3464
H. Gonska, P. Piţul and I. Raşa, Over-iterates of Bernstein-Stancu operators, Calcolo, 44 (2007), pp. 117-125, https://doi.org/10.1007/s10092-007-0131-2 DOI: https://doi.org/10.1007/s10092-007-0131-2
H. Gonska and I. Raşa, The limiting semigroup of the Bernstein iterates: degree of convergence, Acta Math. Hungar., 111 (2006), pp. 119-130, https://doi.org/10.1007/s10474-006-0038-4 DOI: https://doi.org/10.1007/s10474-006-0038-4
H. Gonska, I. Raşa and E. D. Stănilă, The eigenstructure of operators linking the Bernstein and the genuine Bernstein-Durrmeyer operators, Mediterr. J. Math, 11 (2014), no. 2, pp. 561-576, https://doi.org/10.1007/s00009-013-0347-0 DOI: https://doi.org/10.1007/s00009-013-0347-0
S. Karlin, Total Positivity, Stanf. University Press, Stanford, 1968.
A. Lupaş, Die Folge der Beta Operatoren, Dissertation Universität Stuttgart, 1972.
I. Raşa, Asymptotic behaviour of iterates of positive linear operators, Jaen J. Approx., 1 (2009), pp. 195-204.
I. Raşa, Estimates for the semigroup associated with Bernstein-Schnabl operators, Carpathian J. Math., 28 (2012), no.1, pp. 157-162. DOI: https://doi.org/10.37193/CJM.2012.01.02
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.
