Fourier series on the torus

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, 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 $\sum_k \abs{\hat f(k)}^2 = \int_{[0,1]^n} \abs{f}^2$.