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

(Algebraic properties (a).) Linearity is immediate since coefficients are linear in $u$. For the pairing, [L1] and the finiteness of ${\mathrm{\Lambda }}_N$ give $⟨P_{N}u,v{⟩}_2={\sum }_{k\in {\mathrm{\Lambda }}_N}\widehat{u}(k)\cdot \overline{\widehat{v}(k)}$ and $⟨u,P_{N}v{⟩}_2={\sum }_{l\in {\mathrm{\Lambda }}_N}\widehat{u}(l)\cdot \overline{\widehat{v}(l)}$ after reindexing $l\mapsto -l$ and using [L3]; the two sums are equal. Since the coefficients in ${\mathrm{\Lambda }}_N$ are unchanged, $P_N^2=P_N$; the kernel and range descriptions are read off from the coefficient isometry.

(Derivative commutation (b).) For $\mathrm{\mid }\alpha \mathrm{\mid }\le s$ and $u\in H^s$, both $D^{\alpha }u\in H^{s-\mathrm{\mid }\alpha \mathrm{\mid }}\subseteq L^2$ and $P_{N}u\in X_N$ are defined; by [L2] the Fourier coefficient of $P_N{D}^{\alpha }u$ at $k$ is $(ik{)}^{\alpha }\widehat{u}(k)$ for $k\in {\mathrm{\Lambda }}_N$ and $0$ otherwise, and that of $D^{\alpha }{P}_{N}u$ is $(ik{)}^{\alpha }\widehat{u}(k)$ for $k\in {\mathrm{\Lambda }}_N$ and $0$ otherwise, since $P_{N}u$ is a finite trigonometric polynomial whose derivatives have the stated coefficients. Equality of coefficients gives (b).

(Contraction (c) and convergence (d).) For $u\in H^r$, [L2] and the definition of $P_N$ give $\mathrm{\parallel }{P}_{N}u{\mathrm{\parallel }}_{H^r}^2={\sum }_{k\in {\mathrm{\Lambda }}_N}{q}_k^r\mathrm{\mid }\widehat{u}(k){\mathrm{\mid }}^2\le {\sum }_k{q}_k^r\mathrm{\mid }\widehat{u}(k){\mathrm{\mid }}^2=\mathrm{\parallel }u{\mathrm{\parallel }}_{H^r}^2$, which is (c). Similarly $\mathrm{\parallel }u-P_{N}u{\mathrm{\parallel }}_{H^r}^2={\sum }_{k\mathrm{\notin }{\mathrm{\Lambda }}_N}{q}_k^r\mathrm{\mid }\widehat{u}(k){\mathrm{\mid }}^2$ is the tail of the convergent series ${\sum }_k{q}_k^r\mathrm{\mid }\widehat{u}(k){\mathrm{\mid }}^2$, hence tends to $0$ as $N\to \mathrm{\infty }$ with ${\mathrm{\Lambda }}_N$ increasing to ${\mathbb{Z}}^3$; this is (d).

(Tail bound (e).) For $\mathrm{\mid }k\mathrm{\mid }>N$ and $\mathrm{\mid }\alpha \mathrm{\mid }\le s-1$, $\mathrm{\mid }{k}^{\alpha }{\mathrm{\mid }}^2\le (1+\mathrm{\mid }k{\mathrm{\mid }}^2{)}^{s-1}$, so $q_k^{s-1}\le (s+1{)}^3(1+\mathrm{\mid }k{\mathrm{\mid }}^2{)}^{s-1}\le (s+1{)}^{3}2(1+N^2{)}^{-1}(1+\mathrm{\mid }k{\mathrm{\mid }}^2{)}^s$ because $(1+\mathrm{\mid }k{\mathrm{\mid }}^2)\ge 1+N^2$; hence $\mathrm{\parallel }u-P_{N}u{\mathrm{\parallel }}_{H^{s-1}}^2\le 2(s+1{)}^3(1+N^2{)}^{-1}{\sum }_k(1+\mathrm{\mid }k{\mathrm{\mid }}^2{)}^s\mathrm{\mid }\widehat{u}(k){\mathrm{\mid }}^2\le C_s^2(1+N^2{)}^{-1}\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s}^2$ by the norm equivalence [L2]. Taking square roots gives (e). The decay rate is sharp up to constants: for integers $n\ge 2$ take $N=n-1$ and the real polynomial $u(x)=\cos (n{x}^1)$, whose modes $\pm (n,0,0)$ lie outside ${\mathrm{\Lambda }}_N$. By [L2], $\mathrm{\parallel }u-P_{N}u{\mathrm{\parallel }}_{H^{s-1}}^2\mathrm{/}\mathrm{\parallel }u{\mathrm{\parallel }}_{H^s}^2=({\sum }_{a=0}^{s-1}{n}^{2a})\mathrm{/}({\sum }_{a=0}^s{n}^{2a})\ge 1\mathrm{/}((s+1)n^2)$. Hence no bound uniform in $u$ can replace $(1+N^2{)}^{-1\mathrm{/}2}$ by a sequence $o(N^{-1})$. This is the rate sharpness in the title; no optimal constant is asserted.