In mathematics, σ-approximation adjusts a Fourier summation to eliminate the Gibbs phenomenon which would otherwise occur at discontinuities.

A σ-approximated summation for a series of period T can be written as follows:

$s(\backslash theta)\; =\; \backslash frac\{1\}\{2\}\; a\_0\; +\; \backslash sum\_\{k=1\}^\{m-1\}\; \backslash mathrm\{sinc\}\backslash left(\backslash frac\{k\}\{m\}\backslash right)\backslash cdot\; \backslash left\backslash cos\; \backslash left(\; \backslash frac\{2\; \backslash pi\; k\}\{T\}\; \backslash theta\; \backslash right)\; +b\_k\backslash sin\backslash left(\; \backslash frac\{2\; \backslash pi\; k\}\{T\}\; \backslash theta\; \backslash right)\; \backslash right$,   in terms of the normalized sinc function ($\backslash mathrm\{sinc\}\; x\; =\; \backslash frac\{\backslash sin\; \backslash pi\; x\}\{\backslash pi\; x\}$).

Here, the term

$\backslash mathrm\{sinc\}\backslash left(\backslash frac\{k\}\{m\}\backslash right)$

is the Lanczos σ factor, which is responsible for eliminating most of the Gibbs ringing phenomenon. It does not do so entirely, however, but one can square or even cube the expression to serially attenuate Gibbs Phenomenon in the most extreme cases.

Fourier analysis | Mathematical series | Numerical analysis