We study the Wasserstein metric \(W_p\), a notion of distance between two probability distributions, from the perspective of Fourier Analysis and discuss applications. In particular, we bound the Earth Mover Distance \(W_1\) between the distribution of quadratic residues in a finite field \({\mathbb {F}}_p\) and uniform distribution by \(\lesssim p^{-1/2}\) (the Polya–Vinogradov inequality implies \(\lesssim p^{-1/2} \log {p}\)). We also show that for continuous \(f:{\mathbb {T}} \rightarrow {\mathbb {R}}_{}\) with mean value 0

Moreover, we show that for a Laplacian eigenfunction \(-\Delta _g \phi _{\lambda } = \lambda \phi _{\lambda }\) on a compact Riemannian manifold \(W_p\left( \max \left\{ \phi _{\lambda }, 0\right\} dx, \max \left\{ -\phi _{\lambda }, 0\right\} dx\right) \lesssim _p \sqrt{\log {\lambda }/\lambda } \Vert \phi _{\lambda }\Vert _{L^1}^{1/p}\), which is at most a factor \(\sqrt{\log {\lambda }}\) away from sharp. Several other problems are discussed.

我们从傅立叶分析的角度研究Wasserstein度量\（W_p \），这是两个概率分布之间的距离的概念，并讨论了其应用。尤其是，我们将有限域\（{{mathbb {F}} _ p \）中二次残差的分布与均匀分布之间的对地移动距离\（W_1 \）和\（\ lesssim p ^ {-1 / 2} \）（Polya–Vinogradov不等式暗示\（\ lesssim p ^ {-1/2} \ log {p} \））。我们还表明，对于平均值为0的连续\（f：{\ mathbb {T}} \ rightarrow {\ mathbb {R}} _ {} \）

此外，我们表明，拉普拉斯本征函数\（ - \三角洲_g \披_ {\拉姆达} = \拉姆达\披_ {\拉姆达} \）在一个紧凑的黎曼流形\（W_p \左（\最大\左\ {\ phi _ {\ lambda}，0 \ right \} dx，\ max \ left \ {-\ phi _ {\ lambda}，0 \ right \} dx \ right）\ lesssim _p \ sqrt {\ log {\ lambda} / \ lambda} \ Vert \ phi _ {\ lambda} \ Vert _ {L ^ 1} ^ {1 / p} \），这最多是一个因素\（\ sqrt {\ log {\ lambda}} \）远离尖锐。讨论了其他几个问题。