The Bargmann Transform and Windowed Fourier Localization

We consider the relationship between Gabor-Daubechies windowed Fourier localization operators \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ L^{w}_{\varphi } $$ \end{document} and Berezin-Toeplitz operators Tφ, using the Bargmann isometry β. For “window” w a finite linear combination of Hermite functions, and a very general class of functions φ, we prove an equivalence of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ \beta L^{w}_{\varphi } \beta ^{{ - 1}} = C^{*} M_{\varphi } C = T_{{(1 + D)\varphi }} $$ \end{document} by obtaining the exact formulas for the operator C and the linear differential operator D.