1958/01/01 by Leon Ehrenpreis · 2 citations
Mathematics · Economics, Econometrics and Finance · #Mathematical and Theoretical Analysis #Stochastic processes and financial applications
paper · pdf · doi:10.1090/s0002-9947-1958-0105616-8
1. Introduction. The present paper is a continuation of the work of Part I (see 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 F. The Fourier transform of such a map is an "entire map of exponential type" 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 F. 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.