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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\mathbb{T}^n = \\R^n / \\Z^n$ be the $n$-dimensional torus.\nIdentify functions on $\\mathbb{T}^n$ with functions $f \\colon \\R^n \\to \\C$ that are [[periodic|089KWVYD69]] with period $1$ in each variable.\nLet $f$ be such a function that is Lebesgue [[integrable|X7GTF3D98F]] on the cube $[0, 1]^n$.\nFor $k \\in \\Z^n$, the $k$-th *Fourier coefficient* of $f$ is\n$$\\hat f(k) = \\int_{[0,1]^n} f(x)\\, e^{-2\\pi i k \\cdot x}\\, dx,$$\nwhere $k \\cdot x = k_1 x_1 + \\dots + k_n x_n$.\nThe *Fourier series* of $f$ is the formal [[series|0NM626FH6Q]]\n$$\\sum_{k \\in \\Z^n} \\hat f(k)\\, e^{2\\pi i k \\cdot x}.$$\n\n## Remarks\n\nExample: let $n = 2$ and $f(x) = e^{2\\pi i (x_1 + 2 x_2)}$.\nThen $f$ 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.\nMore generally, if $f(x) = g(x_1) h(x_2)$, then $\\hat f(k) = \\hat g(k_1) \\hat h(k_2)$.\n\nNon-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.\n\nFor $n = 1$ this is the classical [[Fourier series|G12KXDHPPS]], written for period $1$ instead of $2\\pi$.\nThe 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.\nFor $f \\in L^2(\\mathbb{T}^n)$ the series converges to $f$ in $L^2$ for each of these choices, and $\\sum_k \\abs{\\hat f(k)}^2 = \\int_{[0,1]^n} \\abs{f}^2$.\n",
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