Abstract
We describe a unifying approach for studying the power series of the positive linear operators from a certain class. For the same operators, we give simpler proofs of some known ergodic theorems.
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Abel, U.: Geometric series of Bernstein–Durrmeyer operators. East J. Approx. 15, 439–450 (2009)
Abel, U., Ivan, M., Paltanea, R.: Geometric series of Bernstein operators revisited. J. Math. Anal. Appl. 400, 22–24 (2013)
Abel, U., Ivan, M., Paltanea, R.: Geometric series of positive linear operators and the inverse Voronovskaya theorem on a compact interval. J. Approx. Theory 184, 163–175 (2014)
Acar, T., Aral, A., Rasa, I.: Power series of beta operators. Appl. Math. Comput. 247, 815–823 (2014)
Altomare, F., Campiti, M.: Korovkin Type Approximation Theory and Its Applications. W. de Gruyter, Berlin (1994)
Altomare, F., Cappelletti Montano, M., Leonessa, V., Rasa, I.: Markov operators, positive semigroups and approximation processes. In: De Gruyter studies in mathematics, vol. 61. Walter de Gruyter GmbH, Berlin, Boston (2014)
Altomare, F., Rasa, I.: Lipschitz contractions, unique ergodicity and asymptotics of Markov semigroups. Bollettino UMI 9(5), 1–17 (2012)
Attalienti, A., Rasa, I.: The eigenstructure of some positive linear operators. Anal. Numer. Theor. Approx. 43, 45–58 (2014)
Cooper, S., Waldron, S.: The eigenstructure of Bernstein operators. J. Approx. Theory 105, 133–165 (2000)
Goldstein, J.A.: Semigroups of Linear Operators and Applications. Oxford Univ. Press, New York (1985)
Gonska, H., Rasa, I., Stanila, E.D.: Power series of operators \(\ U_{n}^{\rho }\). Positivity 19, 237–249 (2015)
Heilmann, M., Rasa, I.: Eigenstructure and iterates for uniquely ergodic Kantorovich modifications of operators. Positivity 21, 897–910 (2017)
Heilmann, M., Rasa, I., \(C_{0}\)-semigroups associated with uniquely ergodic Kantorovich modifications of operators, Positivity. https://doi.org/10.1007/s11117-017-0547-0
Nagel, R. (Ed.), One parameter semigroups of positive operators, Lecture Notes Math., vol 1184, Springer-Verlag, (1986)
Paltanea, R.: The power series of Bernstein operators. Autom. Comput. Appl. Math. 15, 247–253 (2006)
Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Springer, New York (1983)
Rasa, I.: Positive operators, Feller semigroups and diffusion equations associated with Altomare projections. Conf. Sem. Math. Univ. Bari 284, 1–26 (2002)
Rasa, I.: Power series of Bernstein operators and approximation of resolvents. Mediterr. J. Math. 9, 635–644 (2012)
Vladislav, T., Rasa, I.: Analiză Numerică. In: Aproximare, problema lui Cauchy abstractă, proiectori Altomare, p. 173. Editura Tehnică, Bucureşti (1999). ISBN 973-31-1336-0. (in Romanian)
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Acar, T., Aral, A. & Raşa, I. Power Series of Positive Linear Operators. Mediterr. J. Math. 16, 43 (2019). https://doi.org/10.1007/s00009-019-1313-2
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DOI: https://doi.org/10.1007/s00009-019-1313-2