Fourier series on the torus

Summary

A Fourier series on the torus $\mathbb{T}^n = \R^n / \Z^n$ expands a function that is periodic in each of $n$ variables into waves. There is one wave $e^{2\pi i k \cdot x}$ for each integer vector $k \in \Z^n$. For $n = 1$ it is the classical Fourier series of a periodic function. The coefficients form a function on $\Z^n$. The passage from $f$ to its coefficients is a case of the Fourier transform on locally compact abelian groups, for the compact group $\mathbb{T}^n$. Convergence in several variables depends on how the partial sums are formed, for example over cubes or over balls in $\Z^n$. See the definition.