On a space of smooth functions on a convex unbounded set in ? ~n admitting holomorphic extension in ? ~n

摘要

For some given logarithmically convex sequence M of positive numbers we construct a subspace of the space of rapidly decreasing infinitely differentiable functions on an unbounded closed convex set in ? ~n. Due to the conditions on M each function of this space admits a holomorphic extension in ? ~n. In the current article, the space of holomorphic extensions is considered and Paley-Wiener type theorems are established. To prove these theorems, some auxiliary results on extensions of holomorphic functions satisfying some weighted L ~2-bounds in a domain of holomorphy in ? ~n are obtained with the aid of L. H?rmander's method of L ~2-bounds for the ?? operator. Also, some new facts on the Fourier-Laplace transform of tempered distributions complementing some well-known results of V. S. Vladimirov are employed.