Paley-wiener type theorems for the temperature transform*

Abstract

For the temperature transform T defined by and its extension to generalized functions with compact support we prove the Paley-Wiener type theorems for infinitely differentiable functions, Denjoy-Carleman classes, distributions, ultradistributions, and hyperfunctions with compact support, respectively in a unified way. A special interest lies in the quasianalytic classes although the theorems do not exclude the non-quasianalytic classes. Also, considering the limit case of the above result as we can obtain the usual Paley-Wiener-Schwartz theorem for distributions, ultradistributions and hyperfunctions.