An Inverse Problem for Localization Operators

TL;DR

This paper proves a converse to Daubechies' theorem, showing that Hermite eigenfunctions imply circular localization domains for time-frequency operators, and extends the inverse problem analysis to wavelet localization.

Contribution

It establishes that Hermite eigenfunctions correspond to disc-shaped localization domains and extends inverse problem analysis to wavelet localization operators.

Findings

Hermite functions imply disc-shaped localization domains.

Extended inverse problem analysis to wavelet localization.

Provided conditions under which eigenfunctions determine localization domains.

Abstract

A classical result of time-frequency analysis, obtained by I. Daubechies in 1988, states that the eigenfunctions of a time-frequency localization operator with circular localization domain and Gaussian analysis window are the Hermite functions. In this contribution, a converse of Daubechies' theorem is proved. More precisely, it is shown that, for simply connected localization domains, if one of the eigenfunctions of a time-frequency localization operator with Gaussian window is a Hermite function, then its localization domain is a disc. The general problem of obtaining, from some knowledge of its eigenfunctions, information about the symbol of a time-frequency localization operator, is denoted as the inverse problem, and the problem studied by Daubechies as the direct problem of time-frequency analysis. Here, we also solve the corresponding problem for wavelet localization, providing the inverse problem analogue of the direct problem studied by Daubechies and Paul.