Rates of convergence for the iterates of Cesàro operators
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- by José A. Adell and A. Lekuona PDF
- Proc. Amer. Math. Soc. 138 (2010), 1011-1021 Request permission
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.References
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Additional Information
- José A. Adell
- Affiliation: Departamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain
- MR Author ID: 340766
- Email: adell@unizar.es
- A. Lekuona
- Affiliation: Departamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain
- MR Author ID: 663604
- Email: lekuona@unizar.es
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 1011-1021
- MSC (2000): Primary 47B37, 60F05
- DOI: https://doi.org/10.1090/S0002-9939-09-10127-2
- MathSciNet review: 2566567