On random frequent universality
Extending a recent result by \textit{M. Nikula} [Complex Anal. Oper. Theory 8, No. 7, 1455--1474 (2014; Zbl 1315.47010)], the authors provide probabilistic proofs of known results about frequent hypercyclicity and universality. Namely, by an application of Kolmogorov's Zero-One Law and Birkhoff's Ergodic Theorem, it is shown that for a sequence of complex iid random variables \((X_n)_{n\in\mathbb{N}}\) whose distribution is supported by the whole plane \(\mathbb{C}\) and satisfies a certain decay property at infinity, the random power series \(f(z)=\sum_{n\geq 0}X_n\frac{z^n}{n!}\) is almost surely frequently hypercyclic for the differentiation operator on the Fréchet space of entire functions \(H(\mathbb{C})\) and has almost surely a frequently universal generalized Taylor expansion at the origin. Furthermore, the authors use their arguments to derive some new results about frequent hypercyclicity for certain polynomials of weighted backward shifts on \(\ell_p\).
scientific article; zbMATH DE number 6420114
scientific article