Semester paper on a recent result by F. Nicola and P. Tilli.
In this semester paper we go through the work of F. Nicola and P. Tilli in [The Faber-Krahn inequality for the Short-time Fourier transform], providing detailed explanations of both the necessary preliminaries and the results shown in the paper. We begin with an introduction to time-frequency analysis from the point of view of signal analysis and explain the intuition behind the Short-time Fourier transform, as well as its properties and similarities with the classic Fourier transform.
We continue with the Bargmann-Fock space and the Bargmann transform, one of the key ingredients in the proof of the main result of the paper. After this, we finally deal with the main result and explain the ingenious and beautiful proof that shows how incredibly intertwined the geometry and the complex analysis are in the problem.