An integral transformation and its applications to harmonic analysis on the space of solutions of heat equation

We introduce an integral transformation T defined by -W f(y)dy, in order to do harmonic analysis on the space of C°° solutions of heat equation. First, the Paley-Wiener type theorem for the transformation T will be given for the C°° functions and distributions with compact support. Secondly, as an application of the transformation the solutions of heat equation given on the torus J" will be characterized. Finally, we represent solutions of heat equation as an infinite series of Hermite temperatures, which are to be defined as the images of Hermite polynomials under the transformation T. §