# Research Track Record Overview

Currently I’m still working beyond the interplay between localization operators on the phase space and the structure of the Berezin symbol calculus in the context of polyanalytic Fock spaces.

Although localization operators has a long history of physics, they only became popular after the framework of Berezin (Berezin, 1971).  From the mathematical point of view Localization operators were used at a first time by I. Daubechies (1988) in order to localize a signal simultaneously on time and on frequency.  In the case that the symbol (window) is a Gaussian, the localization operators adopted by I. Daubechies are anti-Wick operators derived from Berezin quantization rules .

On the books:

MR1639461 (2000c:47098) Wong, M. W. Weyl transforms. Universitext. Springer-Verlag, New York, 1998. viii+158 pp. ISBN: 0-387-98414-3 (Reviewer: Khélifa Trimèche), 47G30 (42B10 43A32 44A15 81S30)

PDF Clipboard Series Book
MR1918652 (2003i:42003) Wong, M. W. Wavelet transforms and localization operators. Operator Theory: Advances and Applications, 136. Birkhäuser Verlag, Basel, 2002. viii+156 pp. ISBN: 3-7643-6789-X (Reviewer: Steen Pedersen), 42-02 (42C40 43A80 43A85 47B38 47G10)

a complete framework about Gabor-Daubechies localization operators was established using the representation theory backdrop, namely the (Weyl-)Heisenberg group.

The main interest around this subject that seems to be useful applications in time-frequency analysis, namely on the study of modulation spaces (e.g. the Feichtinger algebra $S_0$) using essentially methods from analytic function spaces (e.g. $BMO^p$ and $Q_p$ (?)) is the Coburn conjecture announced in:

Coburn, L. A. The Bargmann isometry and Gabor-Daubechies wavelet localization operators. Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000), 169–17

Roughly speaking, Coburn’s conjecture states that any localization operator $L_f^{(w)}$ with symbol $f$ and window $w$ coincides with a Toeplitz operator $T_{D f}$ with polynomial differential symbol $D f$.

This conjecture were already proved in the papers

Lo, Min-Lin(1-CASSB)
The Bargmann transform and windowed Fourier localization.
Integral Equations Operator Theory 57 (2007), no. 3, 397–412.

Engliš, Miroslav(CZ-SIL-IM)
Toeplitz operators and localization operators.
Trans. Amer. Math. Soc. 361 (2009), no. 2, 1039–1052.

in the context of  Segal-Bargmann-Fock spaces for nicer class of windows under the constraint that the symbol $f$ belongs to $L^\infty(\mathbb{C})$. Recently, Joshua Isralowitz and Bo Li use the results of the papers of Lo (2007) and Englis (2009) to study regularity conditions on the symbol under certain (BMO) regularity constraints as well as the boundedness and compactness of Toeplitz operators on the Segal-Bargmann-Fock spaces. This results appeared recently in Transactions of Mathematics.

In the sequel we are still developed on an extension of this result to polyanalytic Fock spaces in the context of Gabor analysis as a continuation of the framework started in the papers below. Forthcoming results should be announced soon…

Abreu, Luís Daniel On the structure of Gabor and super Gabor spaces. Monatsh. Math. 161 (2010), no. 3, 237–253, 46Exx (33C45 42C15 81S30 94A12)

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Abreu, Luís Daniel Sampling and interpolation in Bargmann-Fock spaces of polyanalytic functions. Appl. Comput. Harmon. Anal. 29 (2010), no. 3, 287–302,42C15 (94A20)

1. A kind of $\psi DO$ are still in consideration for the construction of $D$. Namely, if $W_\zeta F(z,\overline{z})=e^{-\frac{\pi}{2}|\zeta|^2}e^{\pi \overline{\zeta}z}F(z-\zeta,\overline{z}-\overline{\zeta})$ is the Weyl representation on $\mathbb{C}$ and $\widehat{F}(\zeta,\overline{\zeta})$ the corresponding symplectic Fourier transform,
$D:=\widehat{W}_D =\int_{\mathbb{C}}\widehat{D}(\zeta,\overline{\zeta})W_\zeta \frac{d \overline{\zeta} \wedge d\zeta}{2i}$
is the so-called Weyl operator and corresponds to a quantization of $D(z,\overline{z})$ on $\mathbb{C}$.
2. Gelfand-Shilov spaces $\mathcal{S}^\alpha_\alpha(\mathbb{C})$ seem to be a wide class of symbols for the localization operators $L_\sigma^{\Psi,\Theta}$. It’s worthwile to see that for $\alpha=1/2$, this space include the Gaussian, special Hermite and and special Laguerre functions on the variables $z$ and $\overline{z}$ as basic elements.