A characterization of modulation spaces by symplectic rotations

TL;DR

This paper introduces a novel way to characterize modulation spaces using symplectic rotations, replacing traditional time-frequency measures with integrals over symplectic groups, akin to polar coordinates in phase space.

Contribution

It provides a new characterization of modulation spaces via symplectic rotations and metaplectic operators, expanding the geometric understanding of time-frequency analysis.

Findings

Characterization of modulation spaces using symplectic rotations.

Gaussian functions remain invariant under symplectic rotations.

Extension of the framework to the torus group $ ext{T}^n$.

Abstract

This note contains a new characterization of modulation spaces $M^p(\mathbb{R}^n)$, $1\leq p\leq \infty$, by symplectic rotations. Precisely, instead to measure the time-frequency content of a function by using translations and modulations of a fixed window as building blocks, we use translations and metaplectic operators corresponding to symplectic rotations. Technically, this amounts to replace, in the computation of the $M^p(\mathbb{R}^n)$-norm, the integral in the time-frequency plane with an integral on $\mathbb{R}^n\times U(2n,\mathbb{R})$ with respect to a suitable measure, $U(2n,\mathbb{R})$ being the group of symplectic rotations. More conceptually, we are considering a sort of polar coordinates in the time-frequency plane. In this new framework, the Gaussian invariance under symplectic rotations yields to choose Gaussians as suitable window functions. We also provide a similar characterization with the group $U(2n,\mathbb{R})$ being reduced to the $n$-dimensional torus $\mathbb{T}^n$.