On the expansion of solutions of Laplace-like equations into traces of separable higher dimensional functions

TL;DR

This paper extends the representation of solutions to Laplace-like equations in high-dimensional spaces for more general right-hand sides, using exponential sum approximations and properties of linear mappings.

Contribution

It generalizes solution representations to cases where the right-hand side is composed of a separable function after a linear transformation, based on high-dimensional norm behavior.

Findings

Solutions can be represented as sums of exponential functions for generalized right-hand sides.

High-dimensional norm behavior under linear transformations enables these representations.

Results apply to a broader class of functions than previously known.

Abstract

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 $\omega$.