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update sinc notation |
Fixed error in argument of sinc function. Explicitly showed applicability to series of arbitrary period. |
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A σ-approximated summation can be written as follows''':'''
:<math>s(\theta) = \frac{1}{2} a_0 + \sum_{k=1}^{m-1} \mathrm{sinc}\left(\frac{\pi 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>, in terms of the normalized [[sinc function]] (where T is the period of the Fourier Series).
Here, the term
:<math>\mathrm{sinc}\left(\frac{\pi k}{m}\right)</math>
is the '''Lanczos σ 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.
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