TRANSLATION INVARIANT POSITIVE-DEFINITE HERMITIAN BILINEAR ULTRADISTRIBUTIONS

Every translation invariant positive definite Hermitian bilinear funct ional on the Gei'fand-Shilov space S-Mp(Mp) (R-n x R-n) of general typ e S is of the form B(phi, psi) = integral <(phi)over cap>(x)<(<(psi)ov er cap>)over bar>(x)d mu(x), phi, psi is an element of S-Mp(Mp) (R-n), where mu is a positive {M}-tempered measure, i.e., for every epsilon > 0 integral exp [-M(epsilon\x\)] d mu(x) < infinity. To prove this we prove Schwartz kernel theorem for {M}-tempered ultradistributions and need Bochner-Schwartz theorem for {M}-tempered ultradistributions. Ou r result includes most of the quasianalytic cases. Also, we obtain par allel results for the case of Beurling type (M-p).