Towards a dictionary for the Bargmann transform

Abstract

There is a canonical unitary transformation from $L^2(\mathbb{R})$ 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 $L^2(\mathbb{R})$ to that of $F^2$. Examples include the Fourier transform, the Hilbert transform, Gabor frames, pseudo-differential operators, and the uncertainty principle.