Sigma approximation explained

In mathematics, σ-approximation adjusts a Fourier summation to greatly reduce the Gibbs phenomenon, which would otherwise occur at discontinuities.^{[1] [2]}

A σ-approximated summation for a series of period T can be written as follows:$$s(\backslash theta)\; =\; \backslash frac\; a\_0\; +\; \backslash sum\_^\; \backslash operatorname\; \backslash frac\; \backslash cdot\; \backslash left[a\_\{k\}\; \backslash cos\; \backslash left(\backslash frac\{2\; \backslash pi\; k\}\{T\}\; \backslash theta\; \backslash right)\; +\; b\_k\; \backslash sin\; \backslash left(\backslash frac\{2\; \backslash pi\; k\}\{T\}\; \backslash theta\; \backslash right)\; \backslash right],$$in terms of the normalized sinc function$$\backslash operatorname\; x\; =\; \backslash frac.$$

The term$$\backslash operatorname\; \backslash frac$$is the Lanczos σ factor, which is responsible for eliminating most of the Gibbs phenomenon. It does not do so entirely, however, but one can square or even cube the expression to serially attenuate Gibbs phenomenon in the most extreme cases.

See also

• Lanczos resampling

Notes and References

1. Chhoa . Jannatul Ferdous . 2020-08-01 . An Adaptive Approach to Gibbs' Phenomenon . Master's Theses.
2. Recktenwald . Steffen M. . Wagner . Christian . John . Thomas . 2021-06-29 . Optimizing pressure-driven pulsatile flows in microfluidic devices . Lab on a Chip . en . 21 . 13 . 2605–2613 . 10.1039/D0LC01297A . 34008605 . 1473-0189. free .