The Bargmann isometry and Gabor-Daubechies wavelet localization operators

Abstract

I consider the relationship between Berezin-Toeplitz operators Tφ and the Gabor-Daubechies wavelet localization operators L ^w_φ . For Gaussian“window” wo, it is easy to check that $\beta L_{\varphi }^{{{{w}_{0}}}}{{\beta }^{{ - 1}}} = {{T}_{\varphi }}$, with β the Bargmann isometry. For more general w, and some interesting classes of “symbols” φ, I discuss some new equivalences of the form $\beta L_\varphi ^w{\beta ^{ - 1}} = {T_{\left( {I + D} \right)\varphi ,}}$ where D = D(w) is a constant-coefficient linear differential operator with constant term 0. One consequence is that there is a φ of modulus one with L ^w1_φ =0 a for w₁ normalized Hermite function with polynomial part of degree one. A detailed discussion of the Bargmann isometry is provided. Proof sketches of new results are provided with details to appear elsewhere.