**ABSTRACT:**

The modern theory of time-frequency localization operators is strongly influenced by a 1988 result of Ingrid Daubechies, who proved that if the localization domain of a time-frequency localization operator with gaussian window is a disc, then its eigenfunctions are the Hermite functions.

We will prove the converse of Daubechies theorem. More precisely, we will show that, 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 finding information about the localization domain (or, more generaly, the localization symbol) of a time-frequency localization operator, given information about its eigenfunctions will be named as "The Inverse Problem", and the problem studied by Daubechies and later extended in several directions as "The Direct Problem".

We will also solve the corresponding problem for wavelet localization, providing the "Inverse problem analogue" of the direct problem studied by Daubechies and Paul.

The results to be presented were obtained in collaboration with Monika DÃ¶rfler.