Sigma approximation: Difference between revisions

Revision as of 21:49, 29 August 2006

Revision as of 11:56, 25 September 2006

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A σ-approximated summation for a series of period T can be written as follows''':'''

:<math>s(\theta) = \frac{1}{2} a_0 + \sum_{k=1}^{m-1} \mathrm{sinc}\left(\frac{k}{m}\right)\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>, &nbsp;

in terms of the normalized [[sinc function]] (

:<math> \mathrm{sinc} x = \frac{\sin \pi x}{\pi x}</math>).

Here, the term

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is the '''Lanczos &sigma; factor''', which is responsible for eliminating most of the Gibbs ringing 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.

[[Category:Fourier analysisseries]]

[[Category:Mathematical series]]

[[Category:Numerical analysis]]