Intersection of the Jaffe spaces and the Ruelle theorem

Traverse of Ssup(a) Gelfand-Shilov spaces containing finite functions is calculated and its linearly topological properties are studied. Such traverse in quantum field theory serves as universal space of test functions for strictly localized Jaffe fields. By means of this calculation it is proved that if two generalized functions coincide in an open cone GITA and carriers of their Fourier transformations are μ>O apart, then each of these functions decreases inside GITA by exponential way, the index of the exponent being proportional to μ. Thus a new, simpler and more general proof of the Ruelle theorem on spacely-similar asymptotics of vacuum averages applicable not only to local but also to non-local fields is constructed