Holomorphic extensions associated with series expansions

TL;DR

This paper investigates the holomorphic extension of power and trigonometric series, establishing conditions for analytic continuation, connecting series coefficients to Laplace transforms, and proposing numerical methods for approximating jump functions and Green functions.

Contribution

It introduces new conditions for analytic continuation of series, links coefficient interpolation to Laplace transforms, and develops numerical procedures for approximating jump functions and Green functions from finite data.

Findings

Conditions for holomorphic extension of series are established.

A connection between coefficient interpolation and Laplace transforms is demonstrated.

Numerical methods for approximating jump functions and Green functions are proposed.

Abstract

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.