thumb|Animation of the additive synthesis of a square wave with an increasing number of harmonics by way of the σ-approximation with p=1 In mathematics, '''σ-approximation''' adjusts a Fourier summation to greatly reduce the Gibbs phenomenon, which would otherwise occur at discontinuities.<ref>{{Cite journal |last=Chhoa |first=Jannatul Ferdous |date=2020-08-01 |title=An Adaptive Approach to Gibbs' Phenomenon |url=https://aquila.usm.edu/masters_theses/762 |journal=Master's Theses}}</ref><ref>{{Cite journal |last1=Recktenwald |first1=Steffen M. |last2=Wagner |first2=Christian |last3=John |first3=Thomas |date=2021-06-29 |title=Optimizing pressure-driven pulsatile flows in microfluidic devices |journal=Lab on a Chip |language=en |volume=21 |issue=13 |pages=2605–2613 |doi=10.1039/D0LC01297A |pmid=34008605 |issn=1473-0189|doi-access=free }}</ref>

An ''m-1''-term, σ-approximated summation for a series of period ''T'' can be written as follows: <math display="block">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],</math> in terms of the normalized sinc function: <math display="block"> \operatorname{sinc} x = \frac{\sin \pi x}{\pi x}.</math> <math> a_{k} </math> and <math> b_{k} </math> 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 <math display="block">\left(\operatorname{sinc} \frac{k}{m}\right)^{p}</math> 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 <math>\operatorname{sinc}</math> 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).

== See also == * Lanczos resampling

==References== {{Reflist}} Category:Fourier series Category:Numerical analysis

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