The Trigonometric System 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 $\mathbb{R}^{n}$ and the integer lattice $\mathbb{Z}^{n}$ are the ones fixed there. Let $\mathbb{Z}$ be the set of integers and, for $m\in\mathbb{Z}$, let $C_{m},S_{m}:\mathbb{R}\to\mathbb{R}$ be the trigonometric monomials $C_{m}(t)=\cos(2\pi mt)$ and $S_{m}(t)=\sin(2\pi mt)$ introduced there. Let $2=1+1$ and let $\sqrt{2}$ denote the unique nonnegative real number whose square is $2$, which exists by Existence and Uniqueness of the Nonnegative Square Root of a Nonnegative Real Number, applied to the real number $2$, which is nonnegative by claim 8 of Elementary Order Arithmetic in an Ordered Field. Finite products of real numbers are the finite products of that definition, formed in the field of real numbers.

1. (The one-dimensional system)(i) For $m\in\mathbb{Z}$ the map $\phi_{m}:\mathbb{R}\to\mathbb{R}$ is defined by

$$\phi_{0}(t)=1,\qquad \phi_{m}(t)=\sqrt{2}\,C_{m}(t)\quad\text{if }0<m,\qquad \phi_{m}(t)=\sqrt{2}\,S_{-m}(t)\quad\text{if }m<0,$$

for $t\in\mathbb{R}$. This is well defined: the three cases are mutually exclusive and cover every $m\in\mathbb{Z}$ by claim 1 of Arithmetic, Order, Discreteness and Intervals of the Integers, and $-m\in\mathbb{Z}$ by claim 2 of that lemma.

2. (The trigonometric system on the torus)(ii) For $k\in\mathbb{Z}^{n}$ the map $e_{k}:\mathbb{R}^{n}\to\mathbb{R}$ is defined by

$$e_{k}(x)=\prod_{i=1}^{n}\phi_{k_{i}}(x_{i})\qquad(x\in\mathbb{R}^{n}),$$

which is meaningful because $k_{i}\in\mathbb{Z}$ for every $i\in[n]$ by Lattice-Periodic Functions and the Periodic Function Classes (i). The family of all these maps, indexed by $k\in\mathbb{Z}^{n}$, is called the trigonometric system on $\mathbb{T}^{n}$.