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 ( θ ) = 1 2 a 0 + ∑ k = 1 m − 1 ( sinc k m ) p ⋅ [ a k cos ( 2 π k T θ ) + b k sin ( 2 π k T θ ) ] , {\displaystyle 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:
sinc x = sin π x π x . {\displaystyle \operatorname {sinc} x={\frac {\sin \pi x}{\pi x}}.}
a k {\displaystyle a_{k}} and b k {\displaystyle 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
( sinc k m ) p {\displaystyle \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 sinc {\displaystyle \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).
See also Lanczos resampling
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