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  "body": "\n## Statement\n\nLet $\\mathbb{T} = \\R / 2\\pi\\Z$ and let $f \\in L^2(\\mathbb{T})$, the [[Lebesgue space|SASP14MA1Z]] of square integrable functions with the norm $\\norm{g}_2 = \\left( \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} \\abs{g(x)}^2 \\, dx \\right)^{1/2}$.\nLet $\\hat f(n)$ be the [[Fourier coefficients|5NN2YZZR3V]] of $f$ and let $S_N f(x) = \\sum_{n=-N}^{N} \\hat f(n) e^{inx}$ be the partial sums of its [[Fourier series|G12KXDHPPS]].\nThen\n$$\\lim_{N \\to \\infty} \\norm{f - S_N f}_2 = 0 \\qquad \\text{and} \\qquad \\frac{1}{2\\pi} \\int_{-\\pi}^{\\pi} \\abs{f(x)}^2 \\, dx = \\sum_{n=-\\infty}^{\\infty} \\abs{\\hat f(n)}^2.$$\n\n## Remarks\n\nBoth claims follow from the [[Fourier basis theorem|NXQX1KFMDB]], which says that the monomials $e_n$ form an orthonormal basis of $L^2(\\mathbb{T})$.\n$S_N f$ is the orthogonal projection of $f$ onto the span of $e_{-N}, \\dots, e_N$, so $\\norm{f - S_N f}_2^2 = \\norm{f}_2^2 - \\sum_{\\abs{n} \\le N} \\abs{\\hat f(n)}^2$.\nCompleteness of the basis makes the right side tend to $0$; the second formula is Parseval's identity for this basis.\nExample: for $f(x) = x$ on $(-\\pi, \\pi)$, $\\hat f(0) = 0$ and $\\abs{\\hat f(n)} = 1/\\abs{n}$ for $n \\ne 0$, and Parseval gives $\\frac{\\pi^2}{3} = 2 \\sum_{n \\ge 1} \\frac{1}{n^2}$, which is the value $\\frac{\\pi^2}{6}$ of the [[Basel problem|0DMY75D6FV]].\nConvergence in $L^2$ does not give convergence at each point; pointwise convergence almost everywhere for $L^2$ functions is the much deeper [[Carleson's theorem|FBH6KG484Z]].\nThe converse direction is the Riesz–Fischer theorem: every sequence $(c_n) \\in \\ell^2(\\Z)$ is the sequence of Fourier coefficients of some $f \\in L^2(\\mathbb{T})$.\n\n## Explanations\n\n### L1 human\n\nThe finite sums of the Fourier series of a square integrable function get closer and closer to the function, when distance is measured by the average of the squared error.\nThe energy of the function, the average of $\\abs{f}^2$, equals the sum of the energies of its frequency components.\n\n### L2 human\n\nThis is the simplest and most robust convergence theorem for Fourier series.\nIt needs no smoothness, only square integrability, and it holds for every function in $L^2(\\mathbb{T})$.\nThe analogous statement in $L^p(\\mathbb{T})$ for $1 \u003c p \u003c \\infty$ is true but harder; it rests on the boundedness of the conjugate function, a theorem of M. Riesz.\nFor $p = 1$ and in $C(\\mathbb{T})$ with the supremum norm the analogous statement is false, because the Lebesgue constants of the Dirichlet kernel are unbounded.\n",
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