Five ways to recover the symbol of a non-binary localization operator

TL;DR

This paper introduces five constructive methods for recovering non-binary symbols of time-frequency localization operators, expanding previous binary-focused techniques with new estimators and spectral approaches, supported by numerical implementation.

Contribution

It presents three novel methods for non-binary symbol recovery and adapts existing techniques, providing a comprehensive framework for symbol reconstruction in localization operators.

Findings

All five methods are numerically implemented and evaluated.

The new estimators effectively recover non-binary symbols.

Spectral methods offer targeted symbol recovery capabilities.

Abstract

Five constructive methods for recovering the symbol of a time-frequency localization operator with non-binary symbol are presented, two based on earlier work and three novel methods. For the two derivative methods which have previously been applied to binary symbols, we propose a changed symbol estimator and provide additional estimates that show how we can recover non-binary symbols. The three novel methods each have their own advantages and are all applicable to non-binary symbols. Two of them rely on prescribing the input of the localization operator and examining the output, allowing for targeting of the part of the symbol one wishes to recover while the last one relies on spectral information about the operator. All five methods are also implemented numerically and evaluated with the code available.