Bochner-Schwartz theorems for ultradistributions

We prove the Bochner-Schwartz theorem for the ultradistributions in the qua si-analytic case. In other words, every positive definite ultradistribution of class {M-p} is the Fourier transform of a positive {M}-tempered measure mu, that is, integral exp[ -M(epsilon\x\)] d mu < infinity for every epsil on > 0, where M(t) is the associated function of M-p. To prove this, we sho w that every positive element u in F-{Mp}(t) is a positive {M}-tempered mea sure, and that every positive definite ultradistribution of Roumieu type is nothing but a positive definite element in (F-Mp(Mp))' and hence is the Fo urier transform of a positive {M}-tempered measure. Our result includes the cases for Roumieu type and Beurling type and also both for all the non-qua si-analytic cases and most of the quasi-analytic cases. (C) 1998 Academic P ress.