Vol. 56, No. 2, 1975

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Distributional boundary values in the dual spaces of spaces of type 𝒮

Richard D. Carmichael and E. O. Milton

Vol. 56 (1975), No. 2, 385–422

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)= (α1,n)j 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.

Mathematical Subject Classification 2000
Primary: 46F10
Secondary: 32A10, 43A85
Received: 26 March 1974
Published: 1 February 1975
Richard D. Carmichael
E. O. Milton