Lanczos sigma Factor
Writing a Fourier series as
![画像: f(theta)=1/2a_0+sum_(n=1)^(m-1)sinc((npi)/(2m))[a_ncos(ntheta)+b_nsin(ntheta)],](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/LanczosSigmaFactor/NumberedEquation1.svg){kind=link}
where m is the last term, reduces the Gibbs phenomenon. The sinc(x) terms are the known as the Lanczos sigma factors. Note that (Acton 1990, p. 228) incorrectly lists the upper index of the sum as m, while Hamming (1986, p. 535) gives the correct form reproduced above.
See also
Fourier Series, Gibbs Phenomenon, Sinc FunctionExplore with Wolfram|Alpha
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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.Referenced on Wolfram|Alpha
Lanczos sigma FactorCite this as:
Weisstein, Eric W. "Lanczos sigma Factor." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LanczosSigmaFactor.html