Holomorphic Sobolev spaces, Hermite ans special Hermite semigroups and a Paley-Wiener theorem for the windowed Fourier transform

TL;DR

This paper characterizes the images of Hermite and Laguerre Sobolev spaces under specific semigroups, uses these to describe Schwartz functions, and proves a Paley-Wiener theorem for the windowed Fourier transform.

Contribution

It introduces new characterizations of Sobolev spaces under Hermite and Laguerre semigroups and establishes a Paley-Wiener theorem for the windowed Fourier transform.

Findings

Characterization of Sobolev spaces under Hermite and Laguerre semigroups

Description of the Schwartz class via these characterizations

A new Paley-Wiener theorem for the windowed Fourier transform

Abstract

The images of Hermite and Laguerre Sobolev spaces under the Hermite and special Hermite semigroups (respectively) are characterised. These are used to characterise the Schwartz class of rapidly decreasing functions. The image of the space of all tempered distributions is also considered and a Paley-Wiener theorem for the windowed Fourier transform is proved.