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:
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 ) using essentially methods from analytic function spaces (e.g. and (?)) 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 with symbol and window coincides with a Toeplitz operator with polynomial differential symbol .
This conjecture were already proved in the papers
in the context of Segal-Bargmann-Fock spaces for nicer class of windows under the constraint that the symbol belongs to . 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…