Irrational series I Laplace transform in a neighborhood of $-\mathrm{\infty }$

Discrete sums of exponentials $g(w)=\sum a_{\beta }{\mathrm{e}}^{\beta w}$ with positive exponents may converge not normally in neighborhoods $H$ of $-\mathrm{\infty }$ which do not contain half-planes. In order to obtain a decomposition of a holomorphic function $g$ in $H$ as a sum of exponentials we study the Laplace transform in general neighborhoods of $-\mathrm{\infty }$. We adress questions such as continuity of Laplace and inverse Laplace transformations, continuity for the operation of taking partial sums, and resummation formulas.