1. Introduction. The present paper is a continuation of the work of Part I (see [5 ]) in which we showed how to define the Fourier transform of an arbitrary distribution of L. Schwartz. We were led, in a natural manner, to the consideration of linear functions on spaces of analytic functions. In ?2, below, we shall show how our previous methods apply also when dealing with indefinitely differentiable maps on Euclidean space of compact carrier with values in an arbitrary topological vector space V. The Fourier transform of such a map is an map of exponential of the Euclidean space into V, and any such entire map of exponential type is the Fourier transform of an entire map of compact carrier of the Euclidean space into V. Once this generalized Paley-Wiener theorem is established, all our previous method on the Fourier transform of distributions can be extended to this generalized situation. In this connection we should mention that there are two methods of defining vector-valued distributions; the general kernel theorem of L. Schwartz states that these two methods yield the same result. We present in ?2 a simple proof of this theorem. We consider also in ?2 the concepts of an analytic function on a topological vector space, and, more generally, of an analytic map of one topological vector space into another. Our notion of analytic map is a generalization of analyticity for Banach spaces (see [6, Chapter 41). We expect to use analytic functions on the space D (see [5; 7 ]) to extend the theory of distributions of L. Schwartz so as to apply to certain kinds of nonlinear differential equations. In certain cases we can prove that our analytic functions are limits of polynomials (i.e. belong to the ring generated by constants and linear functions). This situation is markedly different from that for infinite dimensional Banach spaces. In ?3 below, we consider some applications of the Fourier transform of distributions to the study of functions analytic in the interior of a strip in the complex plane, and having a very general type of boundary value on the sides of the strip. We show that the functions considered are uniquely determined by their boundary values. We also establish a generalization of Cauchy's formula, by means of which we establish a generalization of the following theorem of Phragmen and Lindelof: Let F(z) be analytic in the
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Open Access
Jan 1, 1972
C Ward Henson + 1
