Time-Frequency Localization Operators and a Berezin Transform

TL;DR

This paper investigates the properties of time-frequency localization operators and introduces a new Berezin transform to analyze their range and dependence on window functions, bridging time-frequency analysis and operator theory.

Contribution

It develops a new Berezin transform for operators on L^2(R^d) and characterizes the range of localization operators based on window function properties.

Findings

Range is dense in Schatten p-classes when windows' STFT has no zeros

New methods are developed for non-holomorphic settings

Results connect time-frequency analysis with operator theory

Abstract

Time-frequency localization operators are a quantization procedure that maps symbols on $R^{2d}$ to operators and depends on two window functions. We study the range of this quantization and its dependence on the window functions. If the short-time Fourier transform of the windows does not have any zero, then the range is dense in the Schatten $p$-classes. The main tool is new version of the Berezin transform associated to operators on $L^2(R^d)$. Although some results are analogous to results about Toeplitz operators on spaces of holomorphic functions, the absence of a complex structure requires the development of new methods that are based on time-frequency analysis.