Asymptotic Fourier and Laplace transforms for vector-valued hyperfunctions

TL;DR

This paper develops asymptotic Fourier and Laplace transforms for vector-valued hyperfunctions, extending existing models and broadening their applicability in complex locally convex spaces.

Contribution

It introduces simplified notions of these transforms for vector-valued hyperfunctions, improving upon previous models by Komatsu, Bäumer, Lumer, Neubrander, and Langenbruch.

Findings

Extended the class of hyperfunctions with well-defined asymptotic transforms

Provided a unified framework for vector-valued hyperfunctions in locally convex spaces

Enhanced the theoretical foundation for analyzing hyperfunctions with Fourier and Laplace transforms

Abstract

We study Fourier and Laplace transforms for Fourier hyperfunctions with values in a complex locally convex Hausdorff space. Since any hyperfunction with values in a wide class of locally convex Hausdorff spaces can be extended to a Fourier hyperfunction, this gives simple notions of asymptotic Fourier and Laplace transforms for vector-valued hyperfunctions, which improves the existing models of Komatsu, B\"aumer, Lumer and Neubrander and Langenbruch.