Analysis seminar talk
241 Altgeld Hall
Speaker: Terry Adams
Title: Is it reasonable to ask for universally convergent moving averages?
Abstract:
We consider pointwise convergence of moving averages of the form:
\[
M(v_n, L_n)^T f =
\frac{1}{L_n} \sum_{i=v_n+1}^{v_n+L_n} f \circ T^i
\]
where $L_n \geq n$, $T$ is an ergodic transformation preserving a probability measure $\mu$ and $f \in L^p(\mu)$ for some $p \geq 1$. For non-zero $f\in L^1(\mu)$, there is a generic class of ergodic transformations $T$ such that each transformation has an associated moving average which does not converge pointwise. By solving the coboundary equation, we show if $f \in L^2(\mu)$, there always exists an ergodic transformation $T$ with universally convergent moving averages for $L_n \geq n$. However, this does not generalize to $L^p(\mu)$ for $p<2$. We explicitly define functions $f\in \bigcap_{p<2} L^p(\mu)$ such that for each ergodic measure preserving $T$, there exists a moving average which does not converge pointwise. Several of the results are generalized to the case of moving averages where $L_n$ has polynomial growth. This is joint work with Joe Rosenblatt.