The Gibbs phenomenon is an overshoot (or "ringing") of Fourier series and other eigenfunction series occurring at simple discontinuities. It can be reduced with the Lanczos sigma factor. The phenomenon is illustrated above in the Fourier series of a square wave.
Gibbs Phenomenon
See also
Apodization, Fourier Series, Fourier Series--Square Wave, Lanczos sigma Factor, Wilbraham-Gibbs ConstantExplore with Wolfram|Alpha
References
Arfken, G. "Gibbs Phenomenon." §14.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 783-787, 1985.Foster, J. and Richards, F. B. "The Gibbs Phenomenon for Piecewise-Linear Approximation." Amer. Math. Monthly 98, 47-49, 1991.Gibbs, J. W. "Fourier Series." Nature 59, 200 and 606, 1899.Hewitt, E. and Hewitt, R. "The Gibbs-Wilbraham Phenomenon: An Episode in Fourier Analysis." Arch. Hist. Exact Sci. 21, 129-160, 1980.Jeffreys, H. and Jeffreys, B. S. "The Gibbs Phenomenon." §14.07 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, pp. 445-446, 1988.Jerri, A. J. The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations. Dordrecht, Netherlands: Kluwer, 1998.Sansone, G. "Gibbs' Phenomenon." §2.10 in Orthogonal Functions, rev. English ed. New York: Dover, pp. 141-148, 1991.Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, pp. 31-32, 2004. http://www.mathematicaguidebooks.org/.Referenced on Wolfram|Alpha
Gibbs PhenomenonCite this as:
Weisstein, Eric W. "Gibbs Phenomenon." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GibbsPhenomenon.html