Decomposition theorems and kernel theorems for a class of functional spaces

TL;DR

This paper establishes new decomposition and kernel theorems for generalized functions in Gelfand-Shilov spaces, with implications for nonlocal quantum field theory, by analyzing spaces of entire functions with specific growth and decay properties.

Contribution

It introduces a new class of spaces $S^eta(U)$ related to Gelfand-Shilov spaces, proving their completeness, nuclearity, and a decomposition theorem for continuous functionals.

Findings

Spaces $S^eta(U)$ are complete and nuclear.

Every continuous functional has a unique minimal carrier cone.

Kernel theorems relate multilinear forms to generalized functions.

Abstract

We prove new theorems about properties of generalized functions defined on Gelfand-Shilov spaces $S^\beta$ with $0\le\beta<1$. For each open cone $U\subset\mathbb R^d$ we define a space $S^\beta(U)$ which is related to $S^\beta(\mathbb R^d)$ and consists of entire analytic functions rapidly decreasing inside U and having order of growth $\le 1/(1-\beta)$ outside the cone. Such sheaves of spaces arise naturally in nonlocal quantum field theory, and this motivates our investigation. We prove that the spaces $S^\beta(U)$ are complete and nuclear and establish a decomposition theorem which implies that every continuous functional defined on $S^\beta(\mathbb R^d)$ has a unique minimal closed carrier cone in $\mathbb R^d$. We also prove kernel theorems for spaces over open and closed cones and elucidate the relation between the carrier cones of multilinear forms and those of the generalized functions determined by these forms.