Tauberian Theorems for the Laplace-Stieltjes Transform
C. J. K. Batty · Transactions of the American Mathematical Society · 1990
Let $\alpha :[0,\infty ) \to {\mathbf {C}}$ be a function of locally bounded variation, with $\alpha (0) = 0$, whose Laplace-Stieltjes transform $g(z) = \int _0^\infty {{e^{ - zt}}d\alpha (t)}$ is absolutely convergent for $\operatorname {Re} z > 0$. Let $E$ be the singular set of $g$ in $i{\mathbf {R}}$, and suppose that $0 otin E$. Various estimates for $\lim {\sup _{t \to \infty }}|\alpha (t) - g(0)|$ are obtained. In particular, $\alpha (t) \to g(0)$ as $t \to \infty$ if \[ \begin {gathered} ({\text {i)}}\quad E {\text {is null,}} \hfill \\ {\text {(ii)}}\quad \sup \limits _{y \in E} \sup \limits _{t > 0} \left | {\int _0^t {{e^{ - iys}} d\alpha (s)} } \right | < \infty , \hfill \\ ({\text {iii)}}\quad \lim \limits _{\delta \downarrow 0} \lim \sup \limits _{t \to \infty } \sup \limits _{t - \delta \leqslant s \leqslant t} |\alpha (s) - \alpha (t)| = 0. \hfill \\ \end {gathered} \] This result contains Tauberian theorems for Laplace transforms, power series, and Dirichlet series.