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  "body": "\n## Summary\n\nA 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.\nThere is one wave $e^{2\\pi i k \\cdot x}$ for each integer vector $k \\in \\Z^n$.\nFor $n = 1$ it is the classical [[Fourier series|G12KXDHPPS]] of a periodic function.\nThe coefficients form a function on $\\Z^n$.\nThe passage from $f$ to its coefficients is a case of the [[Fourier transform on locally compact abelian groups|VHN5VW5Q92]], for the compact group $\\mathbb{T}^n$.\nConvergence in several variables depends on how the partial sums are formed, for example over cubes or over balls in $\\Z^n$.\nSee the [[definition|T4JEMK1549]].\n\n## Explanations\n\n### L1 human\n\nThink of a function on a square whose values on opposite edges match, like a wallpaper pattern that tiles the plane.\nSuch a pattern can be written as a sum of plane waves, each oscillating a whole number of times across the square in each direction.\nThe wave with frequency vector $k = (k_1, k_2)$ is $e^{2\\pi i (k_1 x_1 + k_2 x_2)}$.\nFor instance, $\\cos(2\\pi x_1) \\cos(2\\pi x_2)$ is the sum of the four waves with $k = (\\pm 1, \\pm 1)$, each with coefficient $\\frac14$.\n",
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