Decomposition theorems and kernel theorems for a class of functional spaces

Understand

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 $\mathbb R^d$.