Exponential lower bound for the eigenvalues of the time-frequency localization operator before the plunge region

TL;DR

This paper establishes an exponential lower bound on the eigenvalues of the time-frequency localization operator near the plunge region, improving previous bounds and utilizing properties of the Bargmann transform.

Contribution

It provides a sharper exponential lower bound for eigenvalues before the plunge region, extending prior results to smaller epsilon values.

Findings

Eigenvalues satisfy λ_n(c) > 1 - δ^c for n = [(1-ε)c]

Improved bounds for eigenvalues with ε < 0.42

Utilizes properties of the Bargmann transform

Abstract

We prove that the eigenvalues $\lambda_n(c)$ of the time-frequency localization operator satisfy $\lambda_n(c) > 1 - \delta^c$ for $n = [(1-\varepsilon)c]$, where $\delta = \delta(\varepsilon) < 1$ and $\varepsilon > 0$ is arbitrary, improving on the result of Bonami, Jaming and Karoui, who proved it for $\varepsilon \ge 0.42$. The proof is based on the properties of the Bargmann transform.