{
  "id": "enwiki-001320720670",
  "slug": "sigma-approximation",
  "title": "Sigma approximation",
  "qid": "",
  "lead": "In mathematics, σ-approximation adjusts a Fourier summation to greatly reduce the Gibbs phenomenon, which would otherwise occur at discontinuities. An m-1-term, σ-approximated summation for a series of period T can be written as follows: s(\\theta) = \\frac{1}{2} a_0 + \\sum_{k=1}^{m-1} \\left(\\operatorname{sinc} \\frac{k}{m}\\right)^{p} \\cdot \\left[a_{k} \\cos \\left( \\frac{2 \\pi k}{T} \\theta \\right) + b_k \\sin \\left( \\frac{2 \\pi k}{T} \\theta \\right) \\right], in terms of the normalized sinc function: \\operatorname{sinc} x = \\frac{\\sin \\pi x}{\\pi x}. a_{k} and b_{k} are the typical Fourier Series coefficients, and p, a non negative parameter, determines the amount of smoothening applied, where higher values of p further reduce the Gibbs phenomenon but can overly smoothen the representation of the function. The term \\left(\\operatorname{sinc} \\frac{k}{m}\\right)^{p} is the Lanczos σ factor, which is responsible for eliminating most of the Gibbs phenomenon. This is sampling the right side of the main lobe of the \\operatorname{sinc} function to rolloff the higher frequency Fourier Series coefficients. As is known by the uncertainty principle, having a sharp cutoff in the frequency domain (cutting off the Fourier series abruptly without adjusting coefficients) causes a wide spread of information in the time domain (equivalent to large amounts of ringing).…",
  "body": "In mathematics, σ-approximation adjusts a Fourier summation to greatly reduce the Gibbs phenomenon, which would otherwise occur at discontinuities. An m-1-term, σ-approximated summation for a series of period T can be written as follows: s(\\theta) = \\frac{1}{2} a_0 + \\sum_{k=1}^{m-1} \\left(\\operatorname{sinc} \\frac{k}{m}\\right)^{p} \\cdot \\left[a_{k} \\cos \\left( \\frac{2 \\pi k}{T} \\theta \\right) + b_k \\sin \\left( \\frac{2 \\pi k}{T} \\theta \\right) \\right], in terms of the normalized sinc function: \\operatorname{sinc} x = \\frac{\\sin \\pi x}{\\pi x}. a_{k} and b_{k} are the typical Fourier Series coefficients, and p, a non negative parameter, determines the amount of smoothening applied, where higher values of p further reduce the Gibbs phenomenon but can overly smoothen the representation of the function. The term \\left(\\operatorname{sinc} \\frac{k}{m}\\right)^{p} is the Lanczos σ factor, which is responsible for eliminating most of the Gibbs phenomenon. This is sampling the right side of the main lobe of the \\operatorname{sinc} function to rolloff the higher frequency Fourier Series coefficients. As is known by the uncertainty principle, having a sharp cutoff in the frequency domain (cutting off the Fourier series abruptly without adjusting coefficients) causes a wide spread of information in the time domain (equivalent to large amounts of ringing). This can also be understood as applying a window function to the Fourier series coefficients to balance maintaining a fast rise time (analogous to a narrow transition band) and small amounts of ringing (analogous to stopband attenuation). ==References==",
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    "title": "Sigma approximation",
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        "family": "Anticloud FZ LLE",
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    "publisher": "Anticloud FZ LLE",
    "publisher-place": "Dubai, United Arab Emirates",
    "URL": "https://encyclopegia.org/a/sigma-approximation",
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      "date-parts": [
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          2026,
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          15
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