Equivalence of the Gelfand-Shilov spaces

We prove that there is a one to one correspondence between the Gelfand-Shilov space W-M(Omega) of type W and the space S-MP(Np) of generalized type S. As an application pie prove the equality W-M boolean AND W-Omega = W-M(Omega), which is a generalization of the equality S, boolean AND S-s = S-r(s) found by I. M. Gelfand and G. E. Shilov (''Generalized Functions, II, III,'' Academic Press, New York/London, 1967). (C) 1996 Academic Press, Inc.