In this paper it is
shown that elements of a space of analytic functions defined in the tube
domain Tc=Rn+ iC, where C is an open convex cone of a certain type,
obtain distributional boundary values in the weak topology of the distribution
spaces (𝒮x)′,α = (α1,⋯,αn),αj≧ 1,j = 1,⋯,n; and representation results of
the analytic functions in terms of the boundary values are given. Converse
results are obtained in which an analytic function in the defined space is
constructed from a given distribution in (𝒮x)′, and some applications of the
distributional boundary value theorems are obtained. The main results are
proved with the aid of several new lemmas concerning the C∞ function spaces
of type 𝒮 and their dual spaces. The results obtained here are motivated
by known results used in the construction of local fields in quantum field
theory.
