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The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters

Published online by Cambridge University Press:  19 September 2008

Mustafa Akcoglu ,
Alexandra Bellow ,
Roger L. Jones ,
Viktor Losert ,
Karin Reinhold-Larsson  and
Máté Wierdl
Mustafa Akcoglu
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 1A1, Canada
Alexandra Bellow
Affiliation:
Department of Mathematics, Northwestern University, Evanston, IL 60201, USA
Roger L. Jones
Affiliation:
Department of Mathematics, DePaul University, 2219 N. Kenmore, Chicago, IL 60614, USA
Viktor Losert
Affiliation:
Institute of Mathematics, University of Vienna, A-1090 Vienna, Austria
Karin Reinhold-Larsson
Affiliation:
Department of Mathematics, SUNY at Albany, Albany NY 12222, USA
Máté Wierdl
Affiliation:
Department of Mathematics, Northwestern University, Evanston, IL 60201, USA
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Abstract

In this paper we establish conditions on a sequence of operators which imply divergence. In fact, we give conditions which imply that we can find a set B of measure as close to zero as we like, but such that the operators applied to the characteristic function of this set have a lim sup equal to 1 and a lim inf equal to 0 a.e. (strong sweeping out). The results include the fact that ergodic averages along lacunary sequences, certain convolution powers, and the Riemann sums considered by Rudin are all strong sweeping out. One of the criteria for strong sweeping out involves a condition on the Fourier transform of the sequence of measures, which is often easily checked. The second criterion for strong sweeping out involves showing that a sequence of numbers satisfies a property similar to the conclusion of Kronecker's lemma on sequences linearly independent over the rationals.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 2 , April 1996 , pp. 207 - 253
Copyright
Copyright © Cambridge University Press 1996

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