Time frequency localization in the Fourier Symmetric Sobolev space

TL;DR

This paper investigates the spectral properties of concentration operators in the Fourier symmetric Sobolev space, revealing unexpected eigenvalue decay behavior through the use of the Bargmann transform and reproducing kernel analysis.

Contribution

It introduces a novel analysis of eigenvalue decay in the Fourier symmetric Sobolev space, contrasting classical Paley-Wiener space results, via the Bargmann transform and kernel identification.

Findings

Eigenvalues decay region is of same order as the near-one eigenvalue region.

Bargmann transform is a unitary operator from the Sobolev space to a weighted Fock space.

Contrasts with classical Paley-Wiener space eigenvalue decay behavior.

Abstract

We study concentration operators acting on the Fourier symmetric Sobolev space $H$ consisting of functions $f$ such that $\int_{\mathbb{R}} |f(x)|^2(1+x^2) dx + \int_{\mathbb{R}} |\hat{f}(\xi)|^2(1+\xi^2) d\xi < \infty $. We find that the Bargmann transform is a unitary operator from $H$ to a weighted Fock space. After identifying the reproducing kernel of $H$, we discover an unexpected phenomenon about the decay of the eigenvalues of a two-sided concentration operator, namely that the plunge region is of the same order of magnitude as the region where the eigenvalues are close to 1, contrasting the classical case of Paley--Wiener spaces.