  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.
