The Laplacian of the Trigonometric System, and the Fourier Coefficients of a Laplacian on the Torus

We work in the setting of The Flat Torus: Standing Notation, used here with a natural number $n$ satisfying $1\le n$; the initial segments $[n]$, Euclidean space $\mathbb{R}^{n}$ with its norm $\lVert\,\cdot\,\rVert$, the integer lattice $\mathbb{Z}^{n}$, the cell $Q$, the class $\mathcal{L}^{2}(\mathbb{T}^{n})$ and the space $L^{2}(\mathbb{T}^{n})$ with the class map $[\,\cdot\,]$, the periodic classes $C_{\mathrm{per}}$, $C^{1}_{\mathrm{per}}$ and $C^{2}_{\mathrm{per}}$, the restriction $u|_{Q}$, and the partial derivatives $\partial_{i}$, the iterated partial derivatives and the classes $C^{k}$ on a Euclidean open set are the ones fixed there; $\mathbb{R}^{n}$ is open in itself by claim 1 of Euclidean Space Is Open in Itself, so those calculus notions apply to maps on all of $\mathbb{R}^{n}$. Let $\pi$ be the real number of The Number Pi (i) (the wrapping map of The Flat Torus: Standing Notation (ii) is not used here), let $2=1+1$ and $4=2+2$, and for a real number $t$ write $t^{2}=tt$. Let $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}}$ be the inner product of $L^{2}(\mathbb{T}^{n})$. Let $e_{k}$ for $k\in\mathbb{Z}^{n}$ be the trigonometric system on $\mathbb{T}^{n}$ and $E_{k}=[\,e_{k}|_{Q}\,]$ its classes, as introduced in The Trigonometric System on the Torus is Orthonormal (i). Let $\Delta$ denote the Laplacian of a map of class $C^{2}$ on $\mathbb{R}^{n}$. Then the following hold.

1. (The trigonometric system is twice continuously differentiable)(i) Let $k\in\mathbb{Z}^{n}$ and $i\in[n]$. Then $e_{k}$ is of class $C^{2}$ on $\mathbb{R}^{n}$ and belongs to $C^{2}_{\mathrm{per}}$, and

$$\partial_{i}\partial_{i}e_{k}(x)=-4\pi^{2}k_{i}^{2}\,e_{k}(x)\qquad\text{for every }x\in\mathbb{R}^{n}.$$

2. (Eigenfunctions of the Laplacian)(ii) Let $k\in\mathbb{Z}^{n}$. Then

$$\Delta e_{k}(x)=-4\pi^{2}\lVert k\rVert^{2}\,e_{k}(x)\qquad\text{for every }x\in\mathbb{R}^{n}.$$

3. (Fourier coefficients of a Laplacian)(iii) Let $u\in C^{2}_{\mathrm{per}}$. Then $u$ and $\Delta u$ belong to $C_{\mathrm{per}}$, their restrictions $u|_{Q}$ and $(\Delta u)|_{Q}$ belong to $\mathcal{L}^{2}(\mathbb{T}^{n})$, and for every $k\in\mathbb{Z}^{n}$

$$\bigl\langle\,[\,(\Delta u)|_{Q}\,],\,E_{k}\,\bigr\rangle_{L^{2}} =-4\pi^{2}\lVert k\rVert^{2}\,\bigl\langle\,[\,u|_{Q}\,],\,E_{k}\,\bigr\rangle_{L^{2}} .$$