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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Rates of convergence for the iterates of Cesàro operators


Authors: José A. Adell and A. Lekuona
Journal: Proc. Amer. Math. Soc. 138 (2010), 1011-1021
MSC (2000): Primary 47B37, 60F05
Published electronically: October 22, 2009
MathSciNet review: 2566567
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Abstract: We obtain sharp rates of convergence in the usual sup-norm for the $ n$th iterates $ D^nf$ and $ C^nf$ of continuous and discrete Cesàro operators, respectively. In both cases the best possible rate of convergence is $ n^{-1/2}$, and such a rate is attained under appropriate integrability conditions on $ f$. Otherwise, the rates of convergence could be extremely poor, depending on the behavior of $ f$ near the boundary. We introduce probabilistic representations of $ D^nf$ and $ C^nf$ involving standardized sums of independent identically distributed random variables and binomial mixtures, respectively, which allow us to use the classical Berry-Esseen theorem.


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Additional Information

José A. Adell
Affiliation: Departamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain
Email: adell@unizar.es

A. Lekuona
Affiliation: Departamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain
Email: lekuona@unizar.es

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10127-2
PII: S 0002-9939(09)10127-2
Keywords: Ces\`{a}ro operator, iterates, rate of convergence, Berry-Esseen theorem, binomial mixture
Received by editor(s): March 2, 2009
Received by editor(s) in revised form: July 16, 2009
Published electronically: October 22, 2009
Additional Notes: This work has been supported by research grants MTM2008-06281-C02-01/MTM and DGA E-64 and by FEDER funds.
Communicated by: Walter Van Assche
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.