Translation Invariant Positive Definite Hermitian Bilinear Ultradistributions

Every translation invariant positive definite Hermitian bilinear functional on the Gel'fand-Shilov space sMpMp(n×nK) of general type S is of the form B(,) = (x)(x)d(x), , sMpMp (n), where is a positive {M}-tempered measure, i.e., for every > 0 exp[-M(|x|)] d(x) < . To prove this we prove Schwartz kernel theorem for {M}-tempered ultradistributions and need Bochner-Schwartz theorem for {M}-tempered ultradistributions. Our result includes most of the quasianalytic cases. Also, we obtain parallel results for the case of Beurling type (Mp.