A new type of Fourier hyperfunctions is introduced. The axiomatic quantum field theory in terms of Fourier hyperfunctions is shown to include Wightman's formulation of tempered fields and its generalizations. The complete equivalence is established between the axioms for Wightman Fourier hyperfunctions and those for Green's functions by eliminating from the latter the linear growth condition of Osterwalder and Schrader.
The theory of Fourier hyperfunctions was founded by M. Sato and developed by Kawai [1 ]. We have shown in two previous papers, quoted as NM I and II [2, 3], that Fourier hyperfunctions can be new ingredients for the construction of the axiomatic quantum field theory. The most important result in NM I is that in spite of the fact that the best function space of Fourier hyperfunctions contains no Co function the support concept of the hyperfunction enables us to formulate the locality axiom and the spectral condition for the quantum field theory in terms of Fourier hyperfunctions. In order to establish the axioms for Euclidean Green's functions which are free from the linear growth condition of Osterwalder and Schrader [4] and completely equivalent to the modified Wightman axioms, we have proposed in NM II an extension of Kawai's class of Fourier hyperfunctions and named the extended class the Fourier hyperfunctions of the second type in distinction from its original subclass, the Fourier hyperfunctions of the first type. Actually in NM II we have proved the complete equivalence of the relativistic and Euclidean field theories in a rather asymmetric case when the Wightman functions are assumed to be Fourier hyperfunctions of the second type for time-variables, while they are of the first type for space-variables.
The purpose of this letter is to give a short but systematic presentation of the second type Fourier hyperfunctions, in particular the concept of their supports, and to describe briefly the proof of a theorem on the equivalence for the quantum field theories in terms of pure second type Fourier hyperfunctions. A fuller account will be given in our forthcoming paper [5]. In what follows we say simply Fourier hyperfunctions by which we always mean Fourier hyperfunctions of the second type.