Title: What is the Wigner function closest to a given square integrable function? Show Chinese title

Title: 给定一个平方可积函数，与之最接近的Wigner函数是什么？

Abstract: We consider an arbitrary square integrable function $F$ on the phase space and look for the Wigner function closest to it with respect to the $L^2$ norm. It is well known that the minimizing solution is the Wigner function of any eigenvector associated with the largest eigenvalue of the Hilbert-Schmidt operator with Weyl symbol $F$. We solve the particular case of radial functions on the two-dimensional phase space exactly. For more general cases, one has to solve an infinite dimensional eigenvalue problem. To avoid this difficulty, we consider a finite dimensional approximation and estimate the errors for the eigenvalues and eigenvectors. As an application, we address the so-called Wigner approximation suggested by some of us for the propagation of a pulse in a general dispersive medium. We prove that this approximation never leads to a {\it bona fide} Wigner function. This is our prime motivation for our optimization problem. As a by-product of our results we are able to estimate the eigenvalues and Schatten norms of certain Schatten-class operators. The techniques presented here may be potentially interesting for estimating eigenvalues of localization operators in time-frequency analysis and quantum mechanics. Show Chinese abstract

Abstract: 我们考虑相空间上一个任意的平方可积函数$F$，并寻找与之最接近的 Wigner 函数，接近程度由$L^2$范数衡量。 众所周知，最小化解是与Hilbert-Schmidt算子的最大特征值相对应的任何特征向量的Wigner函数，该算子的Weyl符号为$F$。 我们精确地解决了二维相空间上的径向函数的特殊情况。 对于更一般的情况，必须解决无限维的特征值问题。 为了避免这个困难，我们考虑有限维近似并估计特征值和特征向量的误差。 作为应用，我们讨论了一些人提出的用于一般色散介质中脉冲传播的所谓 Wigner 近似方法。 我们证明这种近似永远不会导致一个{\it 真正的}的 Wigner 函数。 这是我们优化问题的主要动机。 作为我们的结果的一个副产品，我们能够估计某些Schatten类算子的特征值和Schatten范数。 这里介绍的技术可能对时频分析和量子力学中定位算子的特征值估计具有潜在的兴趣。