A characterization of modulation spaces by symplectic rotations

This note contains a new characterization of modulation spaces Mmp(Rn), 1≤p≤∞, 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 Mmp(Rn)-norm, the integral in the time-frequency plane with an integral on Rn×U(2n,R) with respect to a suitable measure, U(2n,R) being the group of symplectic rotations. More conceptually, we are considering a sort of polar coordinates in the time-frequency plane. To have invariance under symplectic rotations we choose a Gaussian as suitable window function. We also provide a similar (and easier) characterization with the group U(2n,R) being reduced to the n-dimensional torus Tn.