Sigma approximation
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 .