2024/01/08 by Cordero, Elena, Giacchi, Gianluca, Rodino, Luigi · 3 citations
#42A38 #46F10 #46F12 #81S30 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2401.03882
To overcome the impossibility of representing the energy of a signal simultaneously in time and frequency, many time-frequency representations have been introduced in the literature. Some of these are recalled in the Introduction. In this work we propose a unified approach of the previous theory by means of metaplectic Wigner distributions WA, with A symplectic matrix in Sp(2d,ℝ), which were introduced by Cordero, Rodino (2022) and then widely studied in subsequent papers. Namely, the short-time Fourier transform and the most popular members of the Cohen's class can be represented via metaplectic Wigner distributions. In particular, we introduce A-metaplectic spectrograms which contain the classical ones and their variations arising from the τ-Wigner distributions of Boggiatto, De Donno, and Oliaro (2010). We provide a complete characterization of those A-Wigner distributions which give rise to generalized spectrograms. This characterization is related to the block decomposition of the symplectic matrix A. Moreover, a characterization of the Lp-boundedness of both A-Wigner distributions and related metaplectic pseudodifferential operators is provided.