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  "body": "\n## Definition\n\nLet $f \\colon \\R \\to \\C$ be $2\\pi$-periodic and integrable on $[-\\pi, \\pi]$, and let $S_N f(x) = \\sum_{n=-N}^{N} \\hat f(n) e^{inx}$ be the symmetric [[partial sums|09RJ859N7A]] of its [[Fourier series|G12KXDHPPS]].\n\n1. The Fourier series of $f$ *converges at* $x \\in \\R$ if the limit $\\lim_{N \\to \\infty} S_N f(x)$ exists in $\\C$.\nIt *converges to $f$ at $x$* if this limit equals $f(x)$.\n2. It *converges pointwise* on a set $E \\subseteq \\R$ if it converges at each $x \\in E$ (see [[pointwise convergence|0GFGK40582]]).\nIt converges *almost everywhere* if it converges to $f$ at each $x$ outside a set of measure zero.\n3. It *converges uniformly* to $f$ on $E$ if $\\sup_{x \\in E} \\abs{S_N f(x) - f(x)} \\to 0$ as $N \\to \\infty$ (see [[uniform convergence|08VZ57PT26]]).\n4. For $1 \\le p \u003c \\infty$ and $f \\in L^p(\\mathbb{T})$, it *converges in $L^p$* if $\\int_{-\\pi}^{\\pi} \\abs{S_N f - f}^p \\, dx \\to 0$ as $N \\to \\infty$, in the [[Lebesgue space|SASP14MA1Z]] $L^p(\\mathbb{T})$.\n\n## Remarks\n\nThe partial sums are taken symmetric in $n$; with this convention the real form $\\frac{a_0}{2} + \\sum_{n=1}^{N} (a_n \\cos nx + b_n \\sin nx)$ has the same partial sums.\nExample: for $f(x) = \\abs{x}$ on $[-\\pi, \\pi]$, extended periodically, $\\abs{\\hat f(n)} \\le \\frac{2}{\\pi n^2}$, so the series converges uniformly to $f$ on $\\R$.\nNon-example: for the square wave equal to $\\operatorname{sgn} x$ on $(-\\pi, \\pi)$, the series converges at every point but not uniformly on $\\R$.\nIndeed a uniform limit of continuous functions would be continuous.\nConvergence of $S_N f(x)$ is a different question from summability, where averages such as the Cesàro means of $S_N f(x)$ are used.\n",
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