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  "body": "\n## Summary\n\nThe Lanczos sigma factors are the weights $\\sigma_k = \\frac{\\sin(\\pi k/m)}{\\pi k/m}$ by which one multiplies the terms of a truncated [[Fourier series|G12KXDHPPS]] of order $m$.\nThe weighted sum, called the sigma approximation, equals the average of the partial sum over a window of length $2\\pi/m$ around each point.\nThis local averaging greatly reduces the overshoot near jumps known as the [[Gibbs phenomenon|VMF169AT67]].\nThe factors are named after Cornelius Lanczos.\n\n## Explanations\n\n### L1 human\n\nA truncated Fourier series of a function with a jump overshoots near the jump and ripples around it.\nAveraging the truncated series over a short window, exactly one period of the first frequency that was cut off, removes most of the ripple.\nIn terms of the coefficients, this averaging multiplies the term of frequency $k$ by $\\frac{\\sin(\\pi k/m)}{\\pi k/m}$.\nFor $m = 2$ the term of frequency $1$ is multiplied by $2/\\pi \\approx 0.64$.\n",
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