In $d$ dimensions, approximating an arbitrary function oscillating with frequency $\lesssim k$ requires $\sim k^d$ degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber $k$) suffers from the pollution effect if, as $k\to \infty$, the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold. While the $h$-version of the finite element method (FEM) (where accuracy is increased by decreasing the meshwidth $h$ and keeping the polynomial degree $p$ fixed) suffers from the pollution effect, the celebrated papers [Melenk, Sauter 2010], [Melenk, Sauter 2011], [Esterhazy, Melenk 2012], and [Melenk, Parsania, Sauter 2013] showed that the $hp$-FEM (where accuracy is increased by decreasing the meshwidth $h$ and increasing the polynomial degree $p$) applied to a variety of constant-coefficient Helmholtz problems does not suffer from the pollution effect. The heart of the proofs of these results is a PDE result splitting the solution of the Helmholtz equation into "high" and "low" frequency components. In this expository paper we prove this splitting for the constant-coefficient Helmholtz equation in full space (i.e., in $\mathbb{R}^d$) using only integration by parts and elementary properties of the Fourier transform; this is in contrast to the proof for this set-up in [Melenk, Sauter 2010] which uses somewhat-involved bounds on Bessel and Hankel functions. The proof in this paper is motivated by the recent proof in [Lafontaine, Spence, Wunsch 2022] of this splitting for the variable-coefficient Helmholtz equation in full space; indeed, the proof in [Lafontaine, Spence, Wunsch 2022] uses more-sophisticated tools that reduce to the elementary ones above for constant coefficients.

翻译：在 $d$ 维空间中，逼近一个频率 $\lesssim k$ 的任意振荡函数需要 $\sim k^d$ 个自由度。求解亥姆霍兹方程（波数为 $k$）的数值方法若在 $k\to\infty$ 时，维持精度所需的总自由度数量增长快于这一自然阈值，则存在污染效应。尽管 $h$ 版本的有限元方法（通过减小网格宽度 $h$ 并保持多项式阶数 $p$ 固定来提高精度）存在污染效应，但著名论文 [Melenk, Sauter 2010]、[Melenk, Sauter 2011]、[Esterhazy, Melenk 2012] 和 [Melenk, Parsania, Sauter 2013] 表明，将 $hp$-FEM（通过减小网格宽度 $h$ 并增加多项式阶数 $p$ 来提高精度）应用于各种常系数亥姆霍兹问题时，不会出现污染效应。这些结果证明的核心是一个偏微分方程定理，该定理将亥姆霍兹方程的解分解为“高”频和“低”频分量。在本说明性论文中，我们仅使用分部积分和傅里叶变换的基本性质，证明全空间（即 $\mathbb{R}^d$）中常系数亥姆霍兹方程的该分解；这与 [Melenk, Sauter 2010] 中对该设置的证明形成对比，后者使用了较为复杂的贝塞尔函数和汉克尔函数有界性估计。本文的证明受 [Lafontaine, Spence, Wunsch 2022] 中关于全空间变系数亥姆霍兹方程该分解的最新证明的启发；实际上，[Lafontaine, Spence, Wunsch 2022] 的证明使用了更复杂的工具，这些工具在常系数情形下简化为上述基本方法。