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

TL;DR

This paper investigates spaces of analytic functions with rapid decay and their duals, hyperfunctions of fast growth, providing criteria for their non-triviality based on growth conditions, and explores boundary values of holomorphic functions in ultradistribution spaces.

Contribution

It develops an analytic representation theory for dual spaces of hyperfunctions of fast growth and characterizes their non-triviality through growth conditions of weight functions.

Findings

Non-triviality of Gelfand-Shilov spaces characterized by growth conditions.

Established criteria linking weight function growth to space non-triviality.

Analyzed boundary values of holomorphic functions in ultradistribution spaces.

Abstract

We study function spaces consisting of analytic functions with fast decay on horizontal strips of the complex plane with respect to a given weight function. Their duals, so called spaces of (ultra)hyperfunctions of fast growth, generalize the spaces of Fourier hyperfunctions and Fourier ultrahyperfunctions. An analytic representation theory for their duals is developed and applied to characterize the non-triviality of these function spaces in terms of the growth order of the weight function. In particular, we show that the Gelfand-Shilov spaces of Beurling type $\mathcal{S}^{(p!)}_{(M_p)}$ and Roumieu type $\mathcal{S}^{\{p!\}}_{\{M_p\}}$ are non-trivial if and only if $$ \sup_{p \geq 2}\frac{(\log p)^p}{h^pM_p} < \infty, $$ for all $h > 0$ and some $h > 0$, respectively. We also study boundary values of holomorphic functions in spaces of ultradistributions of exponential type, which may be of quasianalytic type.