论文标题
时频和小波本地化操作员的规范
The norm of time-frequency and wavelet localization operators
论文作者
论文摘要
时频定位操作员（带有高斯窗口）$ l_f：l^2（\ Mathbb {r}^d）\ to l^2（\ Mathbb {r}^d）$，其中$ f $是$ \ m mathbb {r}^{r}^{2d} $中的$ f $ in Signal in I. daub in. daub in. daub in.daub in Inib in in in in in in in I.相空间的视角。对于信号恢复，量子物理学和不确定性原理的研究，这些操作员的规范（和奇异值）的尖锐上限被认为是一个具有挑战性的问题。在本注中，我们为操作员规范提供最佳的上限l^\ infty（\ mathbb {r}^{2d}）$，$ 1 \ leq p <\ infty $。事实证明，出现了两个制度，具体取决于数量$ \ | f \ | _ {l^p}/\ | f \ | _ {l^\ infty} $比某个临界值小还是大。在第一个制度中，极端权重$ f $在估计中发生平等是某些高斯人，而在第二个制度中，它们被证明是被截断的高斯人，以$ p = 1 $的球的特征功能的倍数脱落。通过截短的高斯人的这种相过渡似乎是时频浓度问题的一种新现象。对于小波本地化操作员的类似问题（库奇小波扮演上述高斯窗口的作用）也提供了完整的解决方案。
Time-frequency localization operators (with Gaussian window) $L_F:L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)$, where $F$ is a weight in $\mathbb{R}^{2d}$, were introduced in signal processing by I. Daubechies in 1988, inaugurating a new, geometric, phase-space perspective. Sharp upper bounds for the norm (and the singular values) of such operators turn out to be a challenging issue with deep applications in signal recovery, quantum physics and the study of uncertainty principles. In this note we provide optimal upper bounds for the operator norm $\|L_F\|_{L^2\to L^2}$, assuming $F\in L^p(\mathbb{R}^{2d})$, $1<p<\infty$ or $F\in L^p(\mathbb{R}^{2d})\cap L^\infty(\mathbb{R}^{2d})$, $1\leq p<\infty$. It turns out that two regimes arise, depending on whether the quantity $\|F\|_{L^p}/\|F\|_{L^\infty}$ is less or greater than a certain critical value. In the first regime the extremal weights $F$, for which equality occurs in the estimates, are certain Gaussians, whereas in the second regime they are proved to be truncated Gaussians, degenerating in a multiple of a characteristic function of a ball for $p=1$. This phase transition through truncated Gaussians appears to be a new phenomenon in time-frequency concentration problems. For the analogous problem for wavelet localization operators -- where the Cauchy wavelet plays the role of the above Gaussian window -- a complete solution is also provided.
