  We study the holomorphic extension associated with power series, i.e., the
analytic continuation from the unit disk to the cut-plane $\mathbb{C} \setminus
[1,+\infty)$. Analogous results are obtained also in the study of trigonometric
series: we establish conditions on the series coefficients which are sufficient
to guarantee the series to have a KMS analytic structure. In the case of power
series we show the connection between the unique (Carlsonian) interpolation of
the coefficients of the series and the Laplace transform of a probability
distribution. Finally, we outline a procedure which allows us to obtain a
numerical approximation of the jump function across the cut starting from a
finite number of power series coefficients. By using the same methodology, the
thermal Green functions at real time can be numerically approximated from the
knowledge of a finite number of noisy Fourier coefficients in the expansion of
the thermal Green functions along the imaginary axis of the complex time plane.
