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 in terms of the normalized sinc function 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.

is the Lanczos σ factor 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 (equivalent to large amounts 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).

See also

• Lanczos resampling

References