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  "body": "\n## Definition\n\nLet $m \\ge 1$ be an integer.\nLet $k$ be an integer with $\\abs{k} \u003c m$.\nThe *Lanczos sigma factor* of $k$ is $\\sigma_{m,0} = 1$ for $k = 0$, and for $k \\ne 0$ it is\n$$\\sigma_{m,k} = \\frac{\\sin(\\pi k / m)}{\\pi k / m}.$$\nFor a $2\\pi$-periodic integrable function $f$ with [[Fourier coefficients|5NN2YZZR3V]] $\\hat f(k)$, the *sigma approximation* of order $m$ of $f$ is\n$$\\Sigma_m f(x) = \\sum_{\\abs{k} \u003c m} \\sigma_{m,k} \\, \\hat f(k) e^{ikx}.$$\n\n## Remarks\n\nLet $S_{m-1} f(x) = \\sum_{\\abs{k} \\le m-1} \\hat f(k) e^{ikx}$ be the partial sum of the [[Fourier series|G12KXDHPPS]].\nSince $\\frac{m}{2\\pi} \\int_{x - \\pi/m}^{x + \\pi/m} e^{ikt} \\, dt = \\frac{\\sin(\\pi k/m)}{\\pi k/m} e^{ikx}$ for $k \\ne 0$,\n$$\\Sigma_m f(x) = \\frac{m}{2\\pi} \\int_{x - \\pi/m}^{x + \\pi/m} S_{m-1} f(t) \\, dt.$$\nSo the sigma approximation is the moving average of the partial sum over a window of length $2\\pi/m$.\nIn the notation $\\operatorname{sinc} u = \\frac{\\sin \\pi u}{\\pi u}$, the factors are $\\sigma_{m,k} = \\operatorname{sinc}(k/m)$.\n\nExample: for $m = 2$ and $f(x) = \\cos x$, the factor is $\\sigma_{2, \\pm 1} = \\frac{\\sin(\\pi/2)}{\\pi/2} = \\frac{2}{\\pi}$, so $\\Sigma_2 f(x) = \\frac{2}{\\pi} \\cos x$.\nThis agrees with the window average $\\frac{1}{\\pi} \\int_{x - \\pi/2}^{x + \\pi/2} \\cos t \\, dt = \\frac{2}{\\pi} \\cos x$.\n\nNon-example: the [[Fejér weights|MA8X1G11DH]] $1 - \\abs{k}/m$ also fade out high frequencies, but they are not the sigma factors; for $m = 2$ and $k = 1$ they give $1/2$, not $2/\\pi$.\n",
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