论文标题
注意在复合磁盘上近似高斯的拉普拉斯变换
Note on approximating the Laplace transform of a Gaussian on a complex disk
论文作者
论文摘要
在此简短的说明中，我们研究了$ [ - a，a] $支持的分布如何近似高斯分布。也许自然的猜想是，对于大$ a $，几乎最佳的选择是通过将高斯截断为$ [ - a，a] $给出的。实际上，这种近似值就特征函数之间的$ l_ \ infty $ distance而言，达到了$ e^{ - θ（a^2）} $的最佳速率。但是，如果我们考虑LapLace之间的$ L_ \ infty $ distance在复杂磁盘上变换，则最佳速率为$ e^{ - θ（a^2 \ log a）} $，而截断仍然只能达到$ e e^{ - θ（a^2）} $。高斯 - 温矿正流可以达到最佳速率。作为推论，我们还构建了$θ（a^2）$组件的``super-flat''高斯混合物，并在$ [ - a，a] $中使用均值，并且其密度具有所有衍生物，所有衍生词都由$ e^{ - ω（a^2 \ log（a）} $（$ o（a^2 \ log（a））} $（1）$ o（1）$ - o（1）$ - 社区$ - -
In this short note we study how well a Gaussian distribution can be approximated by distributions supported on $[-a,a]$. Perhaps, the natural conjecture is that for large $a$ the almost optimal choice is given by truncating the Gaussian to $[-a,a]$. Indeed, such approximation achieves the optimal rate of $e^{-Θ(a^2)}$ in terms of the $L_\infty$-distance between characteristic functions. However, if we consider the $L_\infty$-distance between Laplace transforms on a complex disk, the optimal rate is $e^{-Θ(a^2 \log a)}$, while truncation still only attains $e^{-Θ(a^2)}$. The optimal rate can be attained by the Gauss-Hermite quadrature. As corollary, we also construct a ``super-flat'' Gaussian mixture of $Θ(a^2)$ components with means in $[-a,a]$ and whose density has all derivatives bounded by $e^{-Ω(a^2 \log(a))}$ in the $O(1)$-neighborhood of the origin.
