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Lanczos sigma Factor

A Lanczos sigma factor is a sinc function multiplier used to reduce the Gibbs phenomenon in a truncated Fourier series. For a positive integer m, the filtered approximation is

f_m(theta)=1/2a_0+sum_(n=1)^(m-1)sinc((npi)/m)[a_ncos(ntheta)+b_nsin(ntheta)]

(Hamming 1986, p. 535), where sinc(x)=sinx/x with sinc(0)=1. Here m-1 is the highest included harmonic. Including the term n=m gives the same result because its sinc function multiplier is 0.

See also

Fourier Series, Gibbs Phenomenon, Sinc Function

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References

Acton, F. S. Numerical Methods That Work, 2nd printing. Washington, DC: Math. Assoc. Amer., p. 228, 1990.Hamming, R. W. "Lanczos' sigma Factors" and "The sigma Factors in the General Case." §32.6 and 32.7 in Numerical Methods for Scientists and Engineers, 2nd ed. New York: Dover, pp. 534-536, 1986.Lanczos, C. Applied Analysis. Princeton, NJ: Van Nostrand, 1956.

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Lanczos sigma Factor

Cite this as:

Weisstein, Eric W. "Lanczos sigma Factor." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LanczosSigmaFactor.html

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