Symplectic Analysis of Time-Frequency Spaces

Abstract

We present a different symplectic point of view in the definition of weighted modulation spaces $M_m^{p,q}({\mathbb{R}}^d)$ and weighted Wiener amalgam spaces $W(\mathcal{F}{L}_{m_1}^p,L_{m_2}^q)({\mathbb{R}}^d)$. All of the classical time-frequency representations, such as the short-time Fourier transform (STFT), the $\tau$-Wigner distributions and the ambiguity function, can be written as metaplectic Wigner distributions $\mu (\mathcal{A})(f\otimes \bar{g})$, where $\mu (\mathcal{A})$ is the metaplectic operator and $\mathcal{A}$ is the associated symplectic matrix. Namely, time-frequency representations can be represented as images of metaplectic operators, which become the real protagonists of time-frequency analysis. In [E. Cordero and L. Rodino (2022) "Characterization of Modulation Spaces by symplectic representations and applications to Schrödinger equations", arXiv:2204.14124], the authors suggest that any metaplectic Wigner distribution that satisfies the so-called "shift-invertibility" condition can replace the STFT in the definition of modulation spaces. In this work, we prove that shift-invertibility alone is not sufficient, but it has to be complemented by an upper-triangularity condition for this characterization to hold, whereas a lower-triangularity property comes in to play for Wiener amalgam spaces. The shift-invertibility property is necessary: Ryhaczek and and conjugate Ryhaczek distributions are not shift-invertible and they fail the characterization of the above spaces. We also exhibit examples of shift-invertible distributions without upper-tryangularity condition which do not define modulation spaces. Finally, we provide new families of time-frequency representations that characterize modulation spaces, with the purpose of replacing the time-frequency shifts with other atoms that allow to decompose signals differently, with possible new outcomes in applications.

Cited by in corpus (5)

• Metaplectic Gabor frames of Wigner-Decomposable Distributions
• Metaplectic Gabor Frames and Symplectic Analysis of Time-Frequency Spaces
• On the determination of Lagrange Multipliers for a weighted LASSO problem using geometric and convex analysis techniques
• Hardy's Uncertainty principle for Schrödinger equations with quadratic Hamiltonians
• Boundedness of metaplectic operators within $L^p$ spaces, applications to pseudodifferential calculus, and time-frequency representations