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  "body": "\n## Statement\n\nLet $f, g \\colon \\R \\to \\C$ be [[periodic functions|0ATF6Q0633]] of period $2\\pi$ that are integrable on $[-\\pi, \\pi]$.\nLet $S_N f$ and $S_N g$ be the partial sums of their [[Fourier series|G12KXDHPPS]].\nLet $x_0 \\in \\R$ and $\\delta \u003e 0$, and suppose that $f(x) = g(x)$ for almost every $x$ in $(x_0 - \\delta, x_0 + \\delta)$.\nThen\n$$\\lim_{N \\to \\infty} \\big(S_N f(x_0) - S_N g(x_0)\\big) = 0.$$\nIn particular the Fourier series of $f$ converges at $x_0$ if and only if that of $g$ does, and then to the same limit.\n\n## Remarks\n\nPut $h = f - g$; then $h$ vanishes near $x_0$, and $S_N h(x_0) = \\frac{1}{2\\pi} \\int_{\\delta \\le \\abs{t} \\le \\pi} h(x_0 - t) D_N(t) \\, dt$ with the [[Dirichlet kernel|3JY4BVE4DW]] $D_N$.\nOn $\\delta \\le \\abs{t} \\le \\pi$ the factor $\\frac{1}{\\sin(t/2)}$ is bounded, so $t \\mapsto \\frac{h(x_0 - t)}{\\sin(t/2)}$ is integrable there.\nThe [[Riemann–Lebesgue lemma|WVKDC011KP]] applied to $\\sin((N + \\frac{1}{2}) t)$ gives $S_N h(x_0) \\to 0$.\nThe same argument shows that $S_N h \\to 0$ uniformly on every closed interval inside the interval where $h$ vanishes.\nExample: a function equal to $x^2$ near $x_0 = 1$ has a Fourier series converging to $1$ at $x_0$, whatever its values far from $x_0$.\nBy contrast, each Fourier coefficient depends on the values of $f$ on the whole period.\n\n## Explanations\n\n### L1 human\n\nWhether a Fourier series converges at a point depends only on the function near that point.\nChanging the function far away changes every coefficient, but these changes cancel in the limit at the point.\n\n### L2 human\n\nLocalization reduces convergence questions to local behaviour, which is why the Dini and Dirichlet–Jordan tests involve only values of $f$ near $x$.\nIt contrasts with the global nature of the coefficients.\n",
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  "msc": [
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  "name": "Riemann localization principle",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/HC3/HC3T4EY6WZ.md",
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