Sigma approximation

In mathematics, 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,

$s(\theta) = \frac{1}{2} a_0 + \sum_{k=1}^{m-1} \mathrm{sinc}\left(\frac{k\pi}{m}\right) \left[a_{k} \cos \left( k\theta \right) +b_k\sin\left(k \theta \right) \right].$

Here, the term

$\mathrm{sinc}\left(\frac{k\pi}{m}\right)$

is the Lanczos σ factor, which is responsible for eliminating the Gibbs ringing phenomenon.

Categories: Fourier analysis | Mathematical series | Numerical analysis

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07-14-2008 23:18:10
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