Recent results on the norm of localization operators

Time-frequency localization operators are operators introduced by Ingrid Daubechies in 1988, with the aim of optimally localizing a signal simultaneously in the time and frequency domains. These operators are defined by two functions, one called "window function" and one "weight function." In short, the former allows one to adjust the temporal resolution while the latter acts on the time-frequency distribution of the signal being analyzed. Subsequent to their introduction, these operators have been the subject of intense study regarding their properties such as boundedness, compactness, membership in Schatten classes and asymptotic estimates on eigenvalues. In this thesis, after introducing the localization operators and presenting their properties, an open problem concerning the norm of these operators in the particular case where the window is Gaussian is studied. Specifically, the problem considered is as follows: how large can the norm of a localization operator be knowing that the window satisfies two integrability constraints? Which weight functions reach the maximum? Both questions are answered, and in particular, for optimal weight functions it is observed that there are various regimes depending on the relationship between the integrability constraints.