Sigma approximation: Difference between revisions

Revision as of 09:11, 1 February 2005

Revision as of 19:13, 18 March 2005

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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,

A σ-approximated summation can be written as follows,

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

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

Here, the term

Here, the term <math>\mathrm{sinc}(\frac{k\pi}{m})</math> is the Lanczos &sigma; factor which is responsible for eliminating the Gibbs ringing phenomenon.

:<math>\mathrm{sinc}(\frac{k\pi}{m})</math>

Here, the term <math>\mathrm{sinc}(\frac{k\pi}{m})</math> is the '''Lanczos &sigma; factor''', which is responsible for eliminating the Gibbs ringing phenomenon.

[[Category:Mathematical analysis]][[Category:Numerical analysis]]