A characterization of modulation spaces by symplectic rotations

2017/07/21 by Cordero, Elena, De Gosson, Maurice, Nicola, Fabio
#FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1707.06862

Abstract

This note contains a new characterization of modulation spaces M^p(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 M^p(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. 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,ℝ) being reduced to the n-dimensional torus \mathbbTⁿ.

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