Complex Tauberian theorems for Laplace transforms have been strikingly useful tools in diverse areas of mathematics such as number theory and spectral theory for differential operators. Many results in the area from the last three decades have been motivated by applications in operator theory and semigroups.
In this talk we shall discuss some recent developments on complex Tauberian theory for Laplace transforms and power series. We will focus on two groups of statements, usually labeled as Ingham-Fatou-Riesz theorems and Wiener-Ikehara theorems. Several classical applications will be discussed in order to explain the nature of these Tauberian theorems.
The results we will present considerably improve earlier Tauberians, on the one hand, by relaxing boundary requirements on Laplace transforms to local pseudofunction boundary behavior, with possible exceptional null sets of boundary singularities, and, on the other hand, by simultaneously considering one-sided Tauberian conditions. Using pseudofunctions allows us to take boundary hypotheses to a minimum, producing “if and only if” type results. In the case of power series, we will extend the Katznelson-Tzafriri theorem, one of the cornerstones in the modern asymptotic theory of operator.