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Moving Weighted Averages

Published online by Cambridge University Press:  20 November 2018

M. A. Akcoglu  and
Y. Déniel
M. A. Akcoglu
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ontario M5S 1A1
Y. Déniel
Affiliation:
Département de Mathématiques et Informatique, Université de Bretagne occidentale, 6, avenue Victor le Gorgeu, 29287 Brest, France
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Abstract

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Let ℝ denote the real line. Let {Tt}tєℝ be a measure preserving ergodic flow on a non atomic finite measure space (X, ℱ, μ). A nonnegative function φ on ℝ is called a weight function if ∫ℝ φ(t)dt = 1. Consider the weighted ergodic averages

of a function f X —> ℝ, where {θk} is a sequence of weight functions. Some sufficient and some necessary and sufficient conditions are given for the a.e. convergence of Akf, in particular for a special case in which

where φ is a fixed weight function and {(ak, rk)} is a sequence of pairs of real numbers such that rk > 0 for all k. These conditions are obtained by a combination of the methods of Bellow-Jones-Rosenblatt, developed to deal with moving ergodic averages, and the methods of Broise-Déniel-Derriennic, developed to deal with unbounded weight functions.

Keywords

28D9947A35maximal ergodic theoremsrearrangements
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 45 , Issue 3 , 01 June 1993 , pp. 449 - 469
Copyright
Copyright © Canadian Mathematical Society 1993

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