This paper deals with the equation $-\Delta u+\mu u=f$ on high-dimensional spaces $\mathbb{R}^m$ where $\mu$ is a positive constant. If the right-hand side $f$ is a rapidly converging series of separable functions, the solution $u$ can be represented in the same way. These constructions are based on approximations of the function $1/r$ by sums of exponential functions. The aim of this paper is to prove results of similar kind for more general right-hand sides $f(x)=F(Tx)$ that are composed of a separable function on a space of a dimension $n$ greater than $m$ and a linear mapping given by a matrix $T$ of full rank. These results are based on the observation that in the high-dimensional case, for $\omega$ in most of the $\mathbb{R}^n$, the euclidian norm of the vector $T^t\omega$ in the lower dimensional space $\mathbb{R}^m$ behaves like the euclidian norm of $\omega$.

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关于类拉普拉斯方程的解展开为可分离高维函数的迹

本文处理高维空间 $\mathbb{R}^m$ 上的方程 $-\Delta u+\mu u=f$ ，其中 $\mu$ 是一个正常数。如果右侧 $f$ 是一系列快速收敛的可分离函数，则解 $u$ 可以用相同的方式表示。这些构造基于函数$1/r$ 通过指数函数之和的近似值。本文的目的是证明更一般的右侧 $f(x)=F(Tx)$ 的类似结果，这些右侧由维度 $n$ 大于 $m 的空间上的可分离函数组成$ 和由满秩矩阵 $T$ 给出的线性映射。这些结果是基于观察到的，在高维情况下，对于大部分 $\mathbb{R}^n$ 中的 $\omega$，