This is my master's thesis.
We prove a sharp quantitative version of the Faber-Krahn inequality for the short-time Fourier transform in dimension 2, with its extension to the general d-dimensional case and the wavelet transform analogue, the Faber-Krahn inequality for wavelet transforms. In doing so, we give a brief historical overview and recall the main concepts from time-frequency and time-scale analysis, and Faber-Krahn concentration estimates.
My master's thesis was a joint project with Paolo Tilli, André Guerra and João P.G. Ramos, resulting in a research paper. It was supervised by Alessio Figalli and João P.G. Ramos at ETH Zurich.