A Paley–Wiener theorem for generalized entire functions on infinite-dimensional spaces

We study entire functions on infinite-dimensional spaces. The basis is the study of spaces of Gateaux holomorphic functions that are bounded on certain subsets (bounded entire functions). The main goal is to characterize the Fourier image of the corresponding spaces of generalized entire functions (ultra-distributions) by an infinite-dimensional Paley–Wiener theorem. We introduce entire functions of exponential type and prove a generalization of the classical Paley–Wiener theorem. The crucial point of our theory is the dimension-invariant estimate given by Lemma 4.12.