In this paper, we describe the Cauchy data that gives rise to slowly oscillating solutions to the Levin equation. We present a general result on the existence of a unique minimizer of $\|Bx\|$ subject to the constraint $Ax=y$, where $A,B$ are linear, but not necessarily bounded operators on a complex Hilbert space. This result is used to obtain the solution to the Levin equation, both in the univariate and multivariate case, which minimizes the mean-square of the derivative over the domain. The Cauchy data that generates this solution is then obtained, and this can be used to supplement the Levin equation in the computation of highly oscillatory integrals in the presence of stationary points.

中文翻译：

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Levin 方法的柯西数据

在本文中，我们描述了导致莱文方程缓慢振荡解的柯西数据。我们提出了关于 $\|Bx\|$ 的唯一最小化器的存在性的一般结果，该最小化器受到 $Ax=y$ 约束，其中 $A,B$ 是线性的，但不一定是复杂希尔伯特空间上的有界运算符。该结果用于在单变量和多变量情况下获得 Levin 方程的解，从而最小化域上导数的均方。然后获得生成该解的柯西数据，这可以用来补充莱文方程在存在驻点的情况下计算高振荡积分。