In this paper we study the Fourier-Laplace transform of tempered ultrahyperfunctions introduced by Sebastião e Silva and Hasumi. We establish a generalization of Paley-Wiener-Schwartz theorem for this setting. This theorem is interesting in connection with the microlocal analysis. For this reason, the paper also contains a description of the singularity structure of tempered ultrahyperfunctions in terms of the concept of analytic wave front set.

In the present paper, we intent to enlarge the axiomatic framework of non-commutative quantum field theories (QFT). We consider QFT on non-commutative spacetimes in terms of the tempered ultrahyperfunctions of Sebastião e Silva corresponding to a convex cone, within the framework formulated by Wightman. Tempered ultrahyperfunctions are representable by means of holomorphic functions. As is well known there are certain advantages to be gained from the representation of distributions in terms of holomorphic functions. In particular, for non-commutative theories the Wightman functions involving the ⋆-product, W ⋆ m , have the same form as the standard form Wm. We conjecture that the functions W ⋆ m satisfy a set of properties which actually will characterize a non-commutative QFT in terms of tempered ultrahyperfunctions. In order to support this conjecture, we prove for this setting the validity of some important theorems, of which the CPT theorem and the theorem on the Spin-Statistics connection are the best known. We assume the validity of these theorems for non-commutative QFT in the case of spatial non-commutativity only.Dedicated to Prof. Olivier Piguet on the occasion of his 65th birthday. Date: August 30, 2018. 1991 Mathematics Subject Classification. 46F15, 46F20, 81T05.Key words and phrases. Non-commutative theory, axiomatic field theory, tempered ultrahyperfunctions. J.A. Lourenço is supported by the Brazilian agency CNPq.

In the present paper, we intend to enlarge the axiomatic framework of non-commutative quantum field theories (QFT). We consider QFT on non-commutative spacetimes in terms of the tempered ultrahyperfunctions of Sebastião e Silva corresponding to a convex cone, within the framework formulated by Wightman. Tempered ultrahyperfunctions are representable by means of holomorphic functions. As is well known there are certain advantages to be gained from the representation of distributions in terms of holomorphic functions. In particular, for non-commutative theories the Wightman functions involving the ⋆-product, , have the same form as the standard form . We conjecture that the functions satisfy a set of properties which actually will characterize a non-commutative QFT in terms of tempered ultrahyperfunctions. In order to support this conjecture, we prove for this setting the validity of some important theorems, of which the CPT theorem and the theorem on the spin-statistics connection are the best known. We assume the validity of these theorems for non-commutative QFT in the case of spatial non-commutativity only.