Gelfand-Shilov spaces, Structural and Kernel theorems

Abstract

It was shown recently that the space isomorphic with an Gelfand Shilov space is well adapted for the use in quantum field theory with a fundamental length. It is our believe that all Gelfand Shilov spaces, especially those with quasianalytic test function spaces, are good domains for the quantum field theory. The theory requires technical results from the theory of generalized functions and not merely differential calculus and well defined Fourier transform, but also the kernel theorem and the structural theorem. In the paper we give the structural (regularity) theorem and kernel theorem for Gelfand-Shilov spaces, of Roumieu and Beurling type.