Sigma approximation

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,

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

Here, the term ${\displaystyle \mathrm {sinc} ({\frac {k\pi }{m}})}$ is the Lanczos σ factor which is responsible for eliminating the Gibbs ringing phenomenon.