View source for Gibbs phenomenon

A characteristic of the behaviour of the partial sums (or their averages) of a [[Fourier series|Fourier series]].

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/g044410a.gif" />

Figure: g044410a

First noted by H. Wilbraham [[#References|[1]]] and rediscovered by J.W. Gibbs [[#References|[2]]] at a much later date. Let the partial sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g0444101.png" /> of the Fourier series of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g0444102.png" /> converge to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g0444103.png" /> in some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g0444104.png" /> of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g0444105.png" /> at which

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g0444106.png" /></td> </tr></table>

The Gibbs phenomenon takes place for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g0444107.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g0444108.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g0444109.png" /> where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441010.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441011.png" /></td> </tr></table>

The geometrical meaning of this is that the graphs (cf. Fig.) of the partial sums <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441012.png" /> do not approach the  "expected"  interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441013.png" /> on the vertical line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441014.png" />, but approach the strictly-larger interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441015.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441017.png" />. The Gibbs phenomenon is defined in an analogous manner for averages of the partial sums of a Fourier series when the latter is summed by some given method.

For instance, the following theorems are valid for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441018.png" />-periodic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441019.png" /> of bounded variation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441020.png" /> [[#References|[3]]].

1) At points of non-removable discontinuity, and only at such points, the Gibbs phenomenon occurs for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441021.png" />. In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441022.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441023.png" />, then for the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441024.png" /> the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441025.png" />, while the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441026.png" /> where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441027.png" /></td> </tr></table>

2) There exists an absolute constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441029.png" />, such that the Cesàro averages <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441030.png" /> do not have the Gibbs phenomenon if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441031.png" />, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441032.png" /> the phenomenon is observed at all points of non-removable discontinuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441033.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Wilbraham,   ''Cambridge and Dublin Math. J.'' , '''3'''  (1848)  pp. 198–201</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.W. Gibbs,   ''Nature'' , '''59'''  (1898)  pp. 200</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)</TD></TR></table>



====Comments====
In a more explicit form the definitions of the constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441035.png" /> above are:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441036.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441037.png" /></td> </tr></table>

At an isolated jump discontinuity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441038.png" />, the ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441039.png" /> equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441040.png" />. This means that the Fourier series approximation establishes an overshoot of about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441041.png" /> of the length of the jump at either end of the jump interval.

Actually, it was only in a second letter to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g044/g044410/g04441042.png" /> ([[#References|[a1]]]) that Gibbs stated the phenomenon correctly, though without any proof. For details see [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.W. Gibbs,   ''Nature'' , '''59'''  (1899)  pp. 606</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S. Carslaw,   "Introduction to the theory of Fourier's series and integrals" , Dover, reprint  (1930)</TD></TR></table>