Tauberian theorem for the Laplace-Stieltjes integral and Dirichlet series
The author considers the functions f defined by the Laplace-Stieltjes integral \[ (1)\quad f(z)=\int^{\infty}_{0}e^{-\lambda (t)z} d\sigma (t) \] with complex-valued \(\lambda\) and \(\sigma\) satisfying a number of technical conditions. He proves 9 Tauberian theorems about this integral. The typical and the most interesting one is the following. Suppose \(z_ 0=x_ 0+iy_ 0\) belongs to the boundary of the region of convergence of (1) and let \(\eta (t)=\int^{t}_{0}e^{-\lambda (s)z_ 0} d\sigma (s).\) Suppose, moreover, that the limit \(\ell =\lim_{x\to x^+_ 0}f(x+iy_ 0)\) exists. Then \[ \int^{p}_{0}\lambda (t)d\eta (t)=o(\lambda (p)),\quad p\to \infty \] is the necessary and sufficient condition for the convergence of the integral (1) at \(z=z_ 0\) to the value \(\ell\).
scientific article; zbMATH DE number 4171593