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In $d$ dimensions, approximating an arbitrary function oscillating withfrequency $\lesssim k$ requires $\sim k^d$ degrees of freedom. A numericalmethod for solving the Helmholtz equation (with wavenumber $k$) suffers fromthe pollution effect if, as $k\to \infty$, the total number of degrees offreedom needed to maintain accuracy grows faster than this natural threshold.  While the $h$-version of the finite element method (FEM) (where accuracy isincreased by decreasing the meshwidth $h$ and keeping the polynomial degree $p$fixed) suffers from the pollution effect, the celebrated papers [Melenk, Sauter2010], [Melenk, Sauter 2011], [Esterhazy, Melenk 2012], and [Melenk, Parsania,Sauter 2013] showed that the $hp$-FEM (where accuracy is increased bydecreasing the meshwidth $h$ and increasing the polynomial degree $p$) appliedto a variety of constant-coefficient Helmholtz problems does not suffer fromthe pollution effect.  The heart of the proofs of these results is a PDE result splitting thesolution of the Helmholtz equation into "high" and "low" frequency components.In this expository paper we prove this splitting for the constant-coefficientHelmholtz equation in full space (i.e., in $\mathbb{R}^d$) using onlyintegration by parts and elementary properties of the Fourier transform; thisis in contrast to the proof for this set-up in [Melenk, Sauter 2010] which usessomewhat-involved bounds on Bessel and Hankel functions. The proof in thispaper is motivated by the recent proof in [Lafontaine, Spence, Wunsch 2022] ofthis splitting for the variable-coefficient Helmholtz equation in full space;indeed, the proof in [Lafontaine, Spence, Wunsch 2022] uses more-sophisticatedtools that reduce to the elementary ones above for constant coefficients.

A simple proof that the $hp$-FEM does not suffer from the pollution effect for the constant-coefficient full-space Helmholtz equation