2022/04/29 by Elena Cordero, Cordero, Elena, Luigi Rodino +1 · 2 citations
Computer Science · Mathematics · #35J10 #42B35 #FOS: Mathematics #Functional Analysis (math.FA) #Image and Signal Denoising Methods #Mathematical Analysis and Transform Methods
paper · pdf · doi:10.48550/arxiv.2204.14124
openalex publication_date 2022/04/29 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
In the last twenty years modulation spaces, introduced by H. G. Feichtinger in 1983, have been successfully addressed to the study of signal analysis, PDE's, pseudodifferential operators, quantum mechanics, by hundreds of contributions. In 2011 M. de Gosson showed that the time-frequency representation Short-time Fourier Transform (STFT), which is the tool to define modulation spaces, can be replaced by the Wigner distribution. This idea was further generalized to τ-Wigner representations in [9]. In this paper time-frequency representations are viewed as images of symplectic matrices via metaplectic operators. This new perspective highlights that the protagonists of time-frequency analysis are metaplectic operators and symplectic matrices A ∈ Sp(2d,ℝ). We find conditions on A for which the related symplectic time-frequency representation WA can replace the STFT and give equivalent norms for weighted modulation spaces. In particular, we study the case of covariant matrices A, i.e., their corresponding WA are members of the Cohen class. Finally, we show that symplectic time-frequency representations WA can be efficiently employed in the study of Schrödinger equations. This new approach may have further applications in quantum mechanics and PDE's.