Almost sure convergence of weighted series of contractions
The authors investigate the pointwise convergence of randomly weighted series of contractions \(\sum_{k=1}^{\infty}W_k(\omega)T^{p_k}\) . The functions \(W_k\) form a sequence of independent, mean zero, square integrable variables defined on the probability measure space and \(T\) is a linear contraction on \(L^2(\mu)\). The sequence \(p_k\) of non-negative integers is non-decreasing. They provide a sufficient condition that guarantees the ``universal'' a.e. convergence of such randomly generated series. The universality comes from the fact that they can find a set of \(\omega\) of full measure for which the series converges a.e for any contraction \(T\). The method is based on some extension of Salem and Zygmund classical results providing uniform estimates for random polynomials.
scientific article; zbMATH DE number 1839929