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 $\mathbb{T}^n$.

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通过辛旋转表征调制空间

本笔记包含一个新的调制空间表征 $M^p(\mathbb{R}^n)$, $1\leq p\leq \infty$，通过辛旋转。准确地说，不是通过使用固定窗口的平移和调制作为构建块来测量函数的时频内容，我们使用与辛旋转相对应的平移和 metaplectic 算子。从技术上讲，这相当于在 $M^p(\mathbb{R}^n)$-norm 的计算中，用 $\mathbb{R}^n\ 上的积分替换时频平面中的积分乘以 U(2n,\mathbb{R})$ 相对于合适的度量，$U(2n,\mathbb{R})$ 是辛旋转群。从概念上讲，我们正在考虑时频平面中的一种极坐标。在这个新框架中，辛旋转下的高斯不变性产生选择高斯作为合适的窗函数。我们还提供了类似的表征，将群 $U(2n,\mathbb{R})$ 简化为 $n$ 维环面 $\mathbb{T}^n$。