Alexandra Bellow

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

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 Kroneckers lemma on sequences linearly independent over the rationals.

Almost everywhere convergence of convolution powers

Given an ergodic dynamical system ( X , B , m , τ) and a probability measure μ on the integers, define for all f ∈ L 1 ( X ) The almost everywhere convergence of the convolution powers μ n f ( x ) depends on the properties of μ. If μ has finite and then for all f ∈ L p ( X ), 1 p exists for a.e. x . However, if m 2 (μ) is finite and E (μ)≠0, then there exists E ∈ B such that a.e. and a.e. In the case when m 2 (μ) is infinite and E (μ)=0 we give examples for which we have divergence and other examples which show convergence is possible.

An application of number theory to ergodic theory and the construction of uniquely ergodic models

Using a combinatorial result of N. Hindman one can extend Jewett’s method for proving that a weakly mixing measure preserving transformation has a uniquely ergodic model to the general ergodic case. We sketch a proof of this reviewing the main steps in Jewett’s argument.

On restricted weak type (1,1); The discrete case

In this paper we study the relationship between restricted weak type (1,1) and weak type (1,1) for convolution operators on ℝ.

Divergence of averages obtained by sampling a flow

In this paper we consider ergodic averages obtained by sampling at discrete times along a measure preserving ergodic flow. We show, in particular, that if U t is an aperiodic flow, then averages obtained by sampling at times n + t n satisfy the strong sweeping out property for any sequence t n → 0. We also show that there is a flow (which is periodic) and a sequence t n → 0 such that the Cesaro averages of samples at time n + t n do converge a.e. In fact, we show that every uniformly distributed sequence admits a perturbation that makes it a good Lebesgue sequence

The Calderón Brothers, a Happy Mathematical Relation

It is not often that one has the opportunity to observe at close range two remarkable brothers, mathematicians. I had the privilege of being Alberto Calderon’s wife—second wife—which made me Calixto Calderon’s sister-in-law and gave me an unusual vantage point. Allow me then to say a few words about the remarkable Calderon brothers.