Geodesic
Almost Everywhere Convergence of Convolution Measures
Canadian mathematical bulletin, Tome 55 (2012) no. 4, pp. 830-841

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-2011-124-3
Mots-clés : 28D 
Reinhold, Karin; Savvopoulou, Anna K.; Wedrychowicz, Christopher M. Almost Everywhere Convergence of Convolution Measures. Canadian mathematical bulletin, Tome 55 (2012) no. 4, pp. 830-841. doi: 10.4153/CMB-2011-124-3
@article{10_4153_CMB_2011_124_3,
     author = {Reinhold, Karin and Savvopoulou, Anna K. and Wedrychowicz, Christopher M.},
     title = {Almost {Everywhere} {Convergence} of {Convolution} {Measures}},
     journal = {Canadian mathematical bulletin},
     pages = {830--841},
     year = {2012},
     volume = {55},
     number = {4},
     doi = {10.4153/CMB-2011-124-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-124-3/}
}
TY  - JOUR
AU  - Reinhold, Karin
AU  - Savvopoulou, Anna K.
AU  - Wedrychowicz, Christopher M.
TI  - Almost Everywhere Convergence of Convolution Measures
JO  - Canadian mathematical bulletin
PY  - 2012
SP  - 830
EP  - 841
VL  - 55
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-124-3/
DO  - 10.4153/CMB-2011-124-3
ID  - 10_4153_CMB_2011_124_3
ER  - 
%0 Journal Article
%A Reinhold, Karin
%A Savvopoulou, Anna K.
%A Wedrychowicz, Christopher M.
%T Almost Everywhere Convergence of Convolution Measures
%J Canadian mathematical bulletin
%D 2012
%P 830-841
%V 55
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2011-124-3/
%R 10.4153/CMB-2011-124-3
%F 10_4153_CMB_2011_124_3

[1] [1] Akcoglu, M. A., Alternating sequences with nonpositive operators. Proc. Amer. Math. Soc. 104(1988), no. 4, 1124–1130. Google Scholar | DOI

[2] [2] Bellow, A. and Calderón, A., A weak type inequality for convolution products. In: Harmonic Analysis and Partial Differential Equations. Chicago Lectures in Math. Univ. Chicago Press, Chicago, IL, 1999, pp. 41–48. Google Scholar

[3] [3] Bellow, A., Jones, R. L., and Rosenblatt, J. M., Almost everywhere convergence of convolution products. Ergodic Theory Dynam. Systems 14(1994), no. 3, 415–432. Google Scholar

[4] [4] Bellow, A., Jones, R. L., and Rosenblatt, J. M., Almost everywhere convergence of weighted averages. Math. Ann. 293(1992), no. 3, 399–426. Google Scholar | DOI

[5] [5] Losert, V., The strong sweeping out property for convolution powers. Ergodic Theory Dynam. Systems 21(2001), no. 1, 115–119. Google Scholar

[6] [6] Losert, V., A remark on almost everywhere convegrence of convolution powers. Illinois J. Math. 43(1999), no. 3, 465–479. Google Scholar

[7] [7] Ornstein, D., On the pointwise behavior of iterates of a self-adjoint operator. J. Math. Mech. 18(1968/1969), 473–477. Google Scholar

[8] [8] Petersen, K., Ergodic Theory. Cambridge Studies in AdvancedMathematics 2. Cambridge University Press, Cambridge, 1983. Google Scholar

[9] [9] Petrov, V. V., Sums of Independent Random Variables. Ergebnisse der Mathematik und ihrer Grenzgebiete 82. Springer-Verlag, New York, 1975. Google Scholar

[10] [10] Rosenblatt, J. M., Universally bad sequences in ergodic theory. In: Almost Everywhere Convergence. II. Academic Press, Boston, MA, 1991, pp. 227–245. Google Scholar

[11] [11] Rota, G. C., An “Alternierende Verfahren” for general positive operators. Bull. Amer. Math. Soc. 68(1962), 95–102. Google Scholar | DOI

Cité par Sources :