Time-Frequency Analysis and Metaplectic Operators

We perform a time-frequency analysis using metaplectic operators as main protagonists of this field. By means of metaplectic operators we are able to define the most famous time-frequency representations, such as the short-time Fourier transform (STFT), the \(\tau \) -Wigner distributions, the ambiguity function, and many others. We call them the metaplectic Wigner distributions. In particular, the ones enjoying the so-called shift-invertibility property characterize modulation and Wiener amalgam spaces, replacing the STFT in their norms. Finally, we introduce new frames, called metaplectic Gabor frames, as natural generalizations of Gabor frames in the framework of metaplectic Wigner distributions. These metaplectic distributions (and related frames) provide meaningful definitions of local frequencies: they can be used to effectively measure the local frequency content of signals. Further applications are the study of the phase-space concentration of Schrödinger equations with bounded perturbations. This approach could pave the way to new quantization procedures.