A characterization of modulation spaces by symplectic rotations

Abstract

This note contains a new characterization of modulation spaces $M^p({\mathbb{R}}^n)$, $1\le p\le \infty$, 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 ${\mathbb{T}}^n$.

18 pages, 1 figure