Towards a dictionary for the Bargmann transform

TL;DR

This paper develops a framework to translate key operators and results from the real line $L^2( )$ space to the Fock space via the Bargmann transform, facilitating analysis in quantum and signal processing contexts.

Contribution

It introduces methods to transfer important operators and principles from $L^2( )$ to the Fock space, expanding analytical tools in this setting.

Findings

Established translation of Fourier and Hilbert transforms to Fock space

Extended Gabor frames and pseudo-differential operators to Fock space

Reformulated the uncertainty principle within the Fock space context

Abstract

There is a canonical unitary transformation from $L^2(\R)$ onto the Fock space $F^2$, called the Bargmann transform. The purpose of this article is to translate some important results and operators from the context of $L^2(\R)$ to that of $F^2$. Examples include the Fourier transform, the Hilbert transform, Gabor frames, pseudo-differential operators, and the uncertainty principle.