The Trigonometric System on the Torus is Orthonormal

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 $\mathbb{R}^{n}$, the integer lattice $\mathbb{Z}^{n}$, the half-open unit cell $Q$, the class $C_{\mathrm{per}}$, the space $\mathcal{L}^{2}(\mathbb{T}^{n})$, the real Hilbert space $L^{2}(\mathbb{T}^{n})$ with the class $[v]$ of $v\in\mathcal{L}^{2}(\mathbb{T}^{n})$, and the restriction $v|_{Q}$ are the ones fixed there; $\langle\,\cdot\,,\cdot\,\rangle_{L^{2}(\mathbb{T}^{n})}$ denotes the inner product of that Hilbert space, as adopted in The Flat Torus: Standing Notation (iv). Let $\mathbb{Z}$ be the set of integers, let $\pi$ be the real number of The Number Pi (i), let $2=1+1$, let $\phi_{m}$ for $m\in\mathbb{Z}$ be the one-dimensional trigonometric maps and $e_{k}$ for $k\in\mathbb{Z}^{n}$ the trigonometric system on $\mathbb{T}^{n}$, and let $\sqrt{2}$ denote, as there, the unique nonnegative real number whose square is $2$. Let $J$ and $(J,\mathcal{B}_{J},\lambda_{J})$ be as in The Integral over the Unit Cell of a Product of One-Variable Functions, and for a map $w:\mathbb{R}\to\mathbb{R}$ write $\int_{J}w\,d\lambda_{J}$ for the integral of $w|_{J}$, as in Cell Integrals of the Trigonometric Monomials. For maps $v,w$ on a common domain, $vw$ denotes the pointwise product, and $|t|$ denotes the absolute value of a real number $t$. Differentiability at a point, with a given derivative, of a map $\mathbb{R}\to\mathbb{R}$ is that notion on the interval $\mathbb{R}$, every point of which is an interior point of it by Basic Facts about Intervals of the Real Line (i). Then the following hold.

1. (Regularity, and the classes of the system)(i) Let $k\in\mathbb{Z}^{n}$. Then $e_{k}\in C_{\mathrm{per}}$; consequently $e_{k}|_{Q}$ is measurable with respect to $\mathcal{B}_{Q}$ and belongs to $\mathcal{L}^{2}(\mathbb{T}^{n})$, by Continuous Periodic Functions are Power-Integrable and Dense on the Torus (i). We write

$$E_{k}=\bigl[\,e_{k}|_{Q}\,\bigr]\in L^{2}(\mathbb{T}^{n}).$$

2. (Orthonormality in one variable)(ii) Let $a,b\in\mathbb{Z}$. Then $\phi_{a}\phi_{b}$ is continuous on $\mathbb{R}$ and its restriction to $J$ is $\lambda_{J}$-integrable, and $\int_{J}\phi_{a}\phi_{b}\,d\lambda_{J}$ equals $1$ if $a=b$ and equals $0$ if $a\ne b$.

3. (Orthonormality on the torus)(iii) Let $k,m\in\mathbb{Z}^{n}$. Then $\langle E_{k},E_{m}\rangle_{L^{2}(\mathbb{T}^{n})}$ equals $1$ if $k=m$ and equals $0$ if $k\ne m$.

4. (Derivatives and bounds in one variable)(iv) Let $m\in\mathbb{Z}$. Then $\phi_{m}$ is differentiable at every $t\in\mathbb{R}$ with

$$\phi_{m}'(t)=-2\pi m\,\phi_{-m}(t),$$

and $|\phi_{m}(t)|\le\sqrt{2}$ for every $t\in\mathbb{R}$.