调制和多解析Fock空间的Donoho-Logan大筛原理
Donoho-Logan Large Sieve Principles for Modulation and Polyanalytic Fock Spaces
摘要：
我们获得的估计的$L^{p}$ -范数的短时傅里叶变换（STFT）的调制空间中的功能，提供信息的浓度在一个给定的子集上的$\mathbb{R}^{2}$，导致确定性的保证完美的重建使用凸优化方法。更准确地说，我们将获得Donoho-Logan型的大筛不等式，但不是使用傅里叶变换将信号定位在时间-频率平面的区域$T\times W$中以交错时间和频率，而是将信号的表示定位在具有任意几何形状的集合$Δ$中的短时傅里叶变换方面。在技术层面上，由于没有适当的模拟Beurling的极值函数的STFT设置，我们引入了一种新的方法，它依赖于一个参数的组合类似舒尔的测试与扩展的赛普的本地再生公式一般厄米窗口。当窗口是Hermite函数时，我们得到了多解析Fock空间的局部再生公式，从而得到了显式的大筛常数估计，作为副产品，得到了$f\in L^{2}(\mathbb{R})$在任意圆盘上的STFT值的重建公式。关于最优性的讨论如下，沿着Donoho-Stark关于不确定性原理和信号恢复的论文。我们还考虑了离散Gabor系统，向量值STFT变换的情况下，并改写的多解析Bargmann-Fock变换的结果。
英文摘要：
We obtain estimates for the $L^{p}$-norm of the short-time Fourier transform (STFT) for functions in modulation spaces, providing information about the concentration on a given subset of $\mathbb{R}^{2}$, leading to deterministic guarantees for perfect reconstruction using convex optimization methods. More precisely, we will obtain large sieve inequalities of the Donoho-Logan type, but instead of localizing the signals in regions $T\times W$ of the time-frequency plane using the Fourier transform to intertwine time and frequency, we will localize the representation of the signals in terms of the short-time Fourier transform in sets $Δ$ with arbitrary geometry. At the technical level, since there is no proper analogue of Beurling's extremal function in the STFT setting, we introduce a new method, which rests on a combination of an argument similar to Schur's test with an extension of Seip's local reproducing formula to general Hermite windows. When the windows are Hermite functions, we obtain local reproducing formulas for polyanalytic Fock spaces which lead to explicit large sieve constant estimates and, as a byproduct, to a reconstruction formula for $f\in L^{2}(\mathbb{R})$ from its STFT values on arbitrary discs. A discussion on optimality follows, along the lines of Donoho-Stark paper on uncertainty principles and signal recovery. We also consider the case of discrete Gabor systems, vector-valued STFT transforms and rephrase the results in terms of the polyanalytic Bargmann-Fock transforms.
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