Hermite Expansions of Elements of Generalized Gelfand-Shilov space

We characterize the elements of generalized Gelfand Shilov spaces in terms of the coefficients of their Fourier-Hermite expansion. The technique we use can be applied both in quasianalytic and nonquasianalytic case. The characterizations imply the kernel theorems for the dual spaces. The cases when the test space is quasianalytic are important in quantum field theory with a fundamental length, see for example papers of E.Bruning and S.Nagamachi,where it was conjectured that the properties of the space of Fourier hyper functions, which is isomorphic with S^1_1 are well adapted for the use in the theory.