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Almost Everywhere Convergence of Convolution Measures

Published online by Cambridge University Press:  20 November 2018

Karin Reinhold ,
Anna K. Savvopoulou  and
Christopher M. Wedrychowicz
Karin Reinhold
Affiliation:
Department of Mathematics, University at Albany, SUNY, Albany, NY 12222 USAe-mail: reinhold@albany.edu
Anna K. Savvopoulou
Affiliation:
Department of Mathematical Sciences, Indiana University in South Bend, South Bend, IN, 46545 USAe-mail: annsavvo@iusb.edue-mail: cwedrych@iusb.edu
Christopher M. Wedrychowicz
Affiliation:
Department of Mathematical Sciences, Indiana University in South Bend, South Bend, IN, 46545 USAe-mail: annsavvo@iusb.edue-mail: cwedrych@iusb.edu
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Abstract

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Let $\left( X,\,\mathcal{B},\,m,\,\tau \right)$ be a dynamical system with $\left( X,\mathcal{B},m \right)$ a probability space and $\tau $ an invertible, measure preserving transformation. This paper deals with the almost everywhere convergence in ${{\text{L}}^{1}}\left( X \right)$ of a sequence of operators of weighted averages. Almost everywhere convergence follows once we obtain an appropriate maximal estimate and once we provide a dense class where convergence holds almost everywhere. The weights are given by convolution products of members of a sequence of probability measures $\left\{ {{v}_{i}} \right\}$ defined on $\mathbb{Z}$. We then exhibit cases of such averages where convergence fails.

Keywords

28D
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 55 , Issue 4 , 01 December 2012 , pp. 830 - 841
Copyright
Copyright © Canadian Mathematical Society 2012

References

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