On Borel summability and analytic functionals

TL;DR

This paper establishes a link between the convergence of formal power series, Borel summability, and exponential type entire functions, providing new characterizations of analytic functionals and ultradistributions.

Contribution

It introduces a novel characterization of entire functions of exponential type via Borel summability of their Taylor series.

Findings

Formal power series with positive radius of convergence are Borel summable over a circle.

Entire functions of exponential type correspond to uniformly Borel summable Taylor series.

Analytic functionals can be represented as Borel sums of their moment Taylor series.

Abstract

We show that a formal power series has positive radius of convergence if and only if it is uniformly Borel summable over a circle with center at the origin. Consequently, we obtain that an entire function $f$ is of exponential type if and only if the formal power series $\sum_{n=0}^{\infty}f^{(n)}(0)z^{n}$ is uniformly Borel summable over a circle centered a the origin. We apply these results to obtain a characterization of those Silva tempered ultradistributions which are analytic functionals. We also use Borel summability to represent analytic functionals as Borel sums of their moment Taylor series over the Borel polygon.