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.
See also
Apodization,
Fourier Series,
Fourier Series--Square Wave,
Lanczos sigma Factor,
Wilbraham-Gibbs
Constant
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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 Phenomenon
Cite this as:
Weisstein, Eric W. "Gibbs Phenomenon."
From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GibbsPhenomenon.html
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