Asymptotics of the Fourier and Laplace transforms in weighted spaces of analytic functions

TL;DR

This paper investigates the asymptotic behavior of Fourier and Laplace transforms within weighted spaces of analytic functions, providing insights into zero distributions and extremal majorants in complex analysis.

Contribution

It offers new asymptotic formulas for transforms in weighted Hardy and entire function spaces, addressing open problems related to zero depth and majorant functions.

Findings

Asymptotics of Fourier transform near zero in weighted Hardy spaces

Asymptotics of Laplace transform in weighted entire function spaces

Results applied to zero depth in Denjoy-Carleman classes and Levinson-Sjoberg majorant

Abstract

We study the asymptotics near the origin of the Fourier transform in weighted Hardy spaces of analytic functions in the upper half-plane, and of the Laplace transform in weighted spaces of entire functions of zero exponential type. These results are applied to two closely related problems posed by E. Dyn'kin: we find the asymptotics of the depth of zero in non-quasianalytic Denjoy-Carleman classes, and of the exact Levinson-Sjoberg majorant.