Sigma approximation
In mathematics, σ-approximation adjusts a Fourier summation to greatly reduce the Gibbs phenomenon, which would otherwise occur at discontinuities.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>
An m-1-term, σ-approximated summation for a series of period T can be written as follows: <math display="block">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],</math> in terms of the normalized sinc function: <math display="block"> \operatorname{sinc} x = \frac{\sin \pi x}{\pi x}.</math> <math> a_{k} </math> and <math> b_{k} </math> 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 <math display="block">\left(\operatorname{sinc} \frac{k}{m}\right)^{p}</math> 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 <math>\operatorname{sinc}</math> 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 (lots 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).