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.

A characterization of modulation spaces by symplectic rotations

Cordero E.;
2020-01-01

Abstract

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.
2020
278
11
1
19
https://arxiv.org/abs/1707.06862
Metaplectic operators; Modulation spaces; Short-time Fourier transform; Symplectic group
Cordero E.; de Gosson M.; Nicola F.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/1782031
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