Exeter University

Consider a periodic function $f$, such that its restriction to the unit segment lies in the Banach space $L^2=L^2(0,1)$. Denote by $S$ the family of dilations $f(nx)$ for all $n$ positive integer.    The purpose of this talk is to discuss the following question: When does $S$ form a Riesz basis of $L^2$?

In this talk, we will present a new \textit{mutli-term} criteria for determining Riesz basis properties of $S$ in $L^2$. This method was established in [L. Boulton, H. Melkonian; arXiv: 1708.08545 J. (2017), to appear at the Journal of Analysis and its Applications (ZAA)] and it relies on a general framework developed by Hedenmalm, Lindqvist and Seip about 20 years ago, which turns the basis question into one about the localisation of the zeros and poles of a corresponding analytic multiplier. Our results improve upon various criteria formulated previously, which give sufficient conditions for invertibility of the multiplier in terms of sharp estimates on the Fourier coefficients.

First order asymptotic of scalar renewal sequences with infinite mean characterized by regular variation has been classified in the 60's (Garsia and Lamperti). In the recent years, the question of higher order asymptotic for renewal sequences with infinite mean was motivated by obtaining 'mixing rates' for dynamical systems with infinite measure. In this talk I will present the recent results we have obtained on higher order expansion for renewal sequences with infinite mean (not necessarily generated by independent processes) in the regime of slow regular variation (with small exponents).  I will also discuss some consequences of these results for error rates in certain limit theorems (such as arcsine law for null recurrent Markov processes).