Sigma approximation

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

A σ-approximated summation can be written as follows,

${\displaystyle s(\theta )={\frac {1}{2}}a_{0}+\sum _{k=1}^{m-1}\mathrm {sinc} \left({\frac {k\pi }{m}}\right)\left[a_{k}\cos \left(k\theta \right)+b_{k}\sin \left(k\theta \right)\right].}$

Here, the term

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

is the Lanczos σ factor, which is responsible for eliminating the Gibbs ringing phenomenon.