mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, σ-approximation adjusts a Fourier summation to greatly reduce the

Gibbs phenomenon In mathematics, the Gibbs phenomenon, discovered by Available on-line at:National Chiao Tung University: Open Course Ware: Hewitt & Hewitt, 1979. and rediscovered by , is the oscillatory behavior of the Fourier series of a piecewise continuousl ...

, which would otherwise occur at discontinuities. A σ-approximated summation for a series of period ''T'' can be written as follows:

s(\theta) = \frac a_0 + \sum_^ \operatorname \frac \cdot \left_ \cos \left( \frac \theta \right) + b_k \sin \left( \frac \theta \right) \right

in terms of the normalized

sinc function In mathematics, physics and engineering, the sinc function, denoted by , has two forms, normalized and unnormalized.. In mathematics, the historical unnormalized sinc function is defined for by \operatornamex = \frac. Alternatively, the u ...

\operatorname x = \frac.

The term

\operatorname \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.

References

Fourier series Numerical analysis {{mathanalysis-stub

See also

References