Fourier modes on the periodic three-torus and the Galerkin projection P_N

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let ${\mathbb{T}}_{2\pi }^3$, $\mu$, ${\chi }_k$ and the spaces $H^s({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^m)$ be as in The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s ($m\ge 1$).

(a) Mode sets. For $N\ge 0$ put ${\mathrm{\Lambda }}_N:=\{k\in {\mathbb{Z}}^3:\mathrm{\mid }k\mathrm{\mid }\le N\}$ with $\mathrm{\mid }k\mathrm{\mid }$ the Euclidean norm, and $X_N:=\{{\sum }_{k\in {\mathrm{\Lambda }}_N}{a}_k{\chi }_k:a_k\in {\mathbb{C}}^m,a_{-k}=\overline{a_k}\}\subseteq C^{\mathrm{\infty }}({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^m),$ the Galerkin space of modes up to $N$; it is a finite-dimensional real vector space of dimension $d_N:=m\mathrm{\mid }{\mathrm{\Lambda }}_N\mathrm{\mid }$, and $X_N\subseteq X_{N^{\mathrm{\prime }}}$ for $N\le N^{\mathrm{\prime }}$. (The reality condition $a_{-k}=\overline{a_k}$ is exactly the condition that the trigonometric polynomial be real-valued.)

(b) Projection. For $u\in L^2({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^m)$ define $P_{N}u:={\sum }_{k\in {\mathrm{\Lambda }}_N}\widehat{u}(k){\chi }_k,(P_{N}u)(x)={\sum }_{k\in {\mathrm{\Lambda }}_N}\widehat{u}(k){\chi }_k(x),$ with $\widehat{u}(k)$ the Fourier coefficients of clause (c) of The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s; the second expression is a finite sum, so $P_{N}u\in X_N$; and $P_N:X_N\to X_N$ is the identity. $P_N$ is called the Galerkin (Fourier) projection.

The definition fixes the projection only. Its self-adjointness, derivative-commutation and operator-norm properties are proved in The Galerkin projection is self-adjoint, commutes with derivatives, is a contraction in every H^r, converges to the identity and has the sharp Fourier-tail bound.