---
tag: T4JEMK1549
kind: definition
name: Fourier series on the torus
msc: ["42B05", "42A16"]
defines: AAECHE30BD
uses: [089KWVYD69, X7GTF3D98F, 0NM626FH6Q, G12KXDHPPS]
sources:
  - url: https://en.wikipedia.org/wiki/Fourier_series
---

## Definition

Let $n \ge 1$ and let $\mathbb{T}^n = \R^n / \Z^n$ be the $n$-dimensional torus.
Identify functions on $\mathbb{T}^n$ with functions $f \colon \R^n \to \C$ that are [[periodic|089KWVYD69]] with period $1$ in each variable.
Let $f$ be such a function that is Lebesgue [[integrable|X7GTF3D98F]] on the cube $[0, 1]^n$.
For $k \in \Z^n$, the $k$-th *Fourier coefficient* of $f$ is
$$\hat f(k) = \int_{[0,1]^n} f(x)\, e^{-2\pi i k \cdot x}\, dx,$$
where $k \cdot x = k_1 x_1 + \dots + k_n x_n$.
The *Fourier series* of $f$ is the formal [[series|0NM626FH6Q]]
$$\sum_{k \in \Z^n} \hat f(k)\, e^{2\pi i k \cdot x}.$$

## Remarks

Example: let $n = 2$ and $f(x) = e^{2\pi i (x_1 + 2 x_2)}$.
Then $f$ is periodic in both variables, and $\hat f(k) = \int_0^1 e^{2\pi i (1 - k_1) x_1}\, dx_1 \int_0^1 e^{2\pi i (2 - k_2) x_2}\, dx_2$, which is $1$ for $k = (1, 2)$ and $0$ otherwise.
More generally, if $f(x) = g(x_1) h(x_2)$, then $\hat f(k) = \hat g(k_1) \hat h(k_2)$.

Non-example: $f(x) = x_1$ on $\R^2$ is not periodic in $x_1$, so it is not a function on $\mathbb{T}^2$ and the definition does not apply.

For $n = 1$ this is the classical [[Fourier series|G12KXDHPPS]], written for period $1$ instead of $2\pi$.
The series is formal: for $n \ge 2$ its partial sums can be taken over cubes $\max_j \abs{k_j} \le N$, over balls $\abs{k} \le N$ or over other shapes, and convergence results depend on that choice.
For $f \in L^2(\mathbb{T}^n)$ the series converges to $f$ in $L^2$ for each of these choices, and $\sum_k \abs{\hat f(k)}^2 = \int_{[0,1]^n} \abs{f}^2$.