The nuclearity of Gelfand-Shilov spaces and kernel theorems

TL;DR

This paper investigates the nuclearity of Gelfand-Shilov spaces using weight systems, providing new characterizations and kernel theorems that unify various classes of these function spaces.

Contribution

It introduces new criteria for nuclearity in Gelfand-Shilov spaces and establishes generalized kernel theorems applicable to multiple types of these spaces.

Findings

Characterizations of nuclearity for Gelfand-Shilov spaces

Unified treatment of different Gelfand-Shilov and Beurling-Björck spaces

New kernel theorems for these function spaces

Abstract

We study the nuclearity of the Gelfand-Shilov spaces $\mathcal{S}^{(\mathfrak{M})}_{(\mathscr{W})}$ and $\mathcal{S}^{\{\mathfrak{M}\}}_{\{\mathscr{W}\}}$, defined via a weight (multi-)sequence system $\mathfrak{M}$ and a weight function system $\mathscr{W}$. We obtain characterizations of nuclearity for these function spaces that are counterparts of those for K\"{o}the sequence spaces. As an application, we prove new kernel theorems. Our general framework allows for a unified treatment of the Gelfand-Shilov spaces $\mathcal{S}^{(M)}_{(A)}$ and $\mathcal{S}^{\{M\}}_{\{A\}}$ (defined via weight sequences $M$ and $A$) and the Beurling-Bj\"orck spaces $\mathcal{S}^{(\omega)}_{(\eta)}$ and $\mathcal{S}^{\{\omega\}}_{\{\eta\}}$ (defined via weight functions $\omega$ and $\eta$). Our results cover anisotropic cases as well.