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  "body": "\n## Definition\n\nLet $f \\colon \\R \\to \\R$ be a $2\\pi$-periodic function that is piecewise continuously differentiable.\nLet $x_0$ be a point of [[jump discontinuity|004FH91BN5]] of $f$, with jump $J = f(x_0^+) - f(x_0^-) \\ne 0$.\nLet $S_N f$ be the partial sums of the [[Fourier series|G12KXDHPPS]] of $f$.\nThe *Gibbs phenomenon* is the following behaviour of $S_N f$ near $x_0$:\n$$\\lim_{N \\to \\infty} S_N f\\left(x_0 + \\frac{\\pi}{N}\\right) = f(x_0^+) + J \\left( \\frac{1}{\\pi} \\int_0^{\\pi} \\frac{\\sin t}{t} \\, dt - \\frac{1}{2} \\right).$$\nThe symmetric formula holds at $x_0 - \\frac{\\pi}{N}$, where $f(x_0^-)$ and $-J$ replace $f(x_0^+)$ and $J$.\n\n## Remarks\n\nThe constant $\\frac{1}{\\pi} \\int_0^\\pi \\frac{\\sin t}{t} \\, dt - \\frac{1}{2}$ is about $0.0895$, so the partial sums overshoot the jump by about $9\\%$ of $\\abs{J}$ on each side.\nSo $\\sup_{x_0 \u003c x \u003c x_0 + \\delta} S_N f(x)$ stays away from $f(x_0^+)$ for every $\\delta \u003e 0$, and the [[convergence|V6Q24CHJG8]] is not uniform on any interval with $x_0$ as an endpoint, although $S_N f(x) \\to f(x)$ at each point of continuity.\nExample: for the square wave equal to $\\operatorname{sgn} x$ on $(-\\pi, \\pi)$, with $J = 2$ at $x_0 = 0$, one gets $S_N f(\\pi/N) \\to \\frac{2}{\\pi} \\int_0^\\pi \\frac{\\sin t}{t} \\, dt \\approx 1.179$.\nNon-example: for a continuous piecewise smooth function such as $\\abs{x}$ on $[-\\pi, \\pi]$, there is no jump, the Fourier series converges uniformly, and there is no overshoot of this kind.\nThe Fejér means of the Fourier series show no Gibbs phenomenon.\nThey are averages against a nonnegative kernel of mass $1$, so they stay between the infimum and supremum of $f$.\n",
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