On Borel summability and analytic functionals

Abstract

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 ∞ n=0 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.

The short-time Fourier transform of distributions of exponential type and tauberian theorems for shift-asymptotics

“…[18]). We refer to [7,14,25,30] for some applications of the Silva spaces. Our second goal is to present a new kind of Tauberian theorems.…”

On the non-triviality of certain spaces of analytic functions. Hyperfunctions and ultrahyperfunctions of fast growth

“…More recently, microlocal analysis, edge of the wedge theorems, and Bochner-Schwartz theorems in the context of ultrahyperfunctions have been investigated in [3,13,43]; applications of tempered ultrahyperfunctions can be found e.g. in [10,32]. Also in recent times, ultrahyperfunctions have shown to be quite useful in mathematical physics, particularly as a framework for Wightman-type axiomatic formulations of relativistic quantum field theory with a fundamental length [2,12,31,39].…”