Holomorphic sobolev spaces, hermite and special hermite semigroups and a Paley-Wiener theorem for the windowed fourier transform

The images of Hermite and Laguerre-Sobolev spaces under the Hermite and special Hermite semigroups (respectively) are characterized. These are used to characterize the image of Schwartz class of rapidly decreasing functions f on R<SUP>n</SUP> and C<SUP>n</SUP> under these semigroups. The image of the space of tempered distributions is also considered and a Paley-Wiener theorem for the windowed (short-time) Fourier transform is proved