The Fourier and Hilbert transforms under the Bargmann transform

TL;DR

This paper investigates how the Bargmann transform maps classical integral operators like the fractional Fourier and Hilbert transforms from real space to the Fock space, revealing their transformed behaviors.

Contribution

It provides a detailed analysis of the action of the Bargmann transform on key integral operators, offering new insights into their structure in the Fock space.

Findings

Characterization of the transformed fractional Fourier transform

Analysis of the fractional Hilbert transform under the Bargmann transform

Insights into the wavelet transform's behavior in Fock space

Abstract

There is a canonical unitary transformation from $L^2(\R)$ onto the Fock space $F^2$, called the Bargmann transform. We study the action of the Bargmann transform on several classical integral operators on $L^2(\R)$, including the fractional Fourier transform, the fractional Hilbert transform, and the wavelet transform.