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 §lebesgue. Let $\mathbb{Z}$ be the set of integers, let $\pi$ be the real number of The Number Pi §pi, 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 and Their Interior Points §whole-line. Then the following hold.

1. (Regularity, and the classes of the system)¶ 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 §member. We write

2. (Orthonormality in one variable)¶ 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)¶ 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)¶ Let $m\in\mathbb{Z}$. Then $\phi_{m}$ is differentiable at every $t\in\mathbb{R}$ with

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