Periodic uniform PIC grid and inner product

Fix a spatial dimension $d\in \{1,2,3\}$, side length $L>0$, and odd integer $N\ge 3$. Put $h=L\mathrm{/}N$ and ${\mathbb{T}}_L^d=(\mathbb{R}\mathrm{/}L\mathbb{Z}{)}^d$. The periodic grid has index set $J_N=(\mathbb{Z}\mathrm{/}N\mathbb{Z}{)}^d$ and nodes $x_j=hj$ modulo $L$. A grid scalar is a real array $u=(u_j{)}_{j\in J_N}$; a grid vector has $d$ real components at each node. Define $⟨u,v{⟩}_h=h^d{\sum }_{j\in J_N}{u}_j{v}_j,\mathrm{\parallel }u{\mathrm{\parallel }}_h=⟨u,u{⟩}_h^{1\mathrm{/}2},\bar{u}=L^{-d}{h}^d{\sum }_{j\in J_N}{u}_j\mathrm{.}$ The same pairing is componentwise for grid vectors. For the coordinate vector $e_a$, define the centered field difference and positive periodic Poisson stencil by $(D_a^{0}u{)}_j=\frac{u_{j+e_a}-u_{j-e_a}}{2h},(A_{h}u{)}_j={\sum }_{a=1}^d\frac{2u_j-u_{j+e_a}-u_{j-e_a}}{h^2}\mathrm{.}$ All indices wrap modulo $N$. The discrete Fourier index set is ${\mathcal{K}}_N=\{-\frac{N-1}{2},\dots ,\frac{N-1}{2}{\}}^d$; the character at $k\in {\mathcal{K}}_N$ is $j\mapsto e^{2\pi ik\cdot j\mathrm{/}N}$. This odd-grid convention has no Nyquist mode.