Gibbs phenomenon

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

Figure: g044410a

First noted by H. Wilbraham [1] and rediscovered by J.W. Gibbs [2] at a much later date. Let the partial sums $ s _ {n} $ of the Fourier series of a function $ f $ converge to $ f $ in some neighbourhood $ \{ {x } : {0 < | x - x _ {0} | < h } \} $ of a point $ x _ {0} $ at which

$$ a \equiv f( x _ {0} - ) \leq f ( x _ {0} + ) \equiv b. $$

The Gibbs phenomenon takes place for $ s _ {n} $ at $ x _ {0} $ if $ A < a \leq b < B $ where

$$ A = \ {fnnme \underline{lim} } _ {\begin{array}{c} n \rightarrow \infty \\ x \uparrow x _ {0} \end{array} } \ s _ {n} ( x), $$

$$ B = \overline{\lim\limits}\; _ {\begin{array}{c} n \rightarrow \infty \\ x \downarrow x _ {0} \end{array} } s _ {n} ( x). $$

The geometrical meaning of this is that the graphs (cf. Fig.) of the partial sums $ s _ {n} $ do not approach the "expected" interval $ [ a, b ] $ on the vertical line $ x = x _ {0} $, but approach the strictly-larger interval $ [ A, B ] $ as $ x \rightarrow x _ {0} $ and $ n \rightarrow \infty $. 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 $ 2 \pi $- periodic functions $ f $ of bounded variation on $ [ - \pi , \pi ] $[3].

1) At points of non-removable discontinuity, and only at such points, the Gibbs phenomenon occurs for $ s _ {n} $. In particular, if $ f( x) = ( \pi - x)/2 $ for $ 0 < x < 2 \pi $, then for the point $ x = 0 $ the segment $ [ a, b] = [- \pi /2 , \pi /2] $, while the segment $ [ A, B] = [- l, l] $ where

$$ l = \int\limits _ { 0 } ^ \pi \frac{\sin t }{t } \ dt \approx 1.85 \dots > { \frac \pi {2} } . $$

2) There exists an absolute constant $ \alpha _ {0} $, $ 0 < \alpha _ {0} < 1 $, such that the Cesàro averages $ \sigma _ {n} ^ \alpha $ do not have the Gibbs phenomenon if $ \alpha \geq \alpha _ {0} $, while if $ \alpha < \alpha _ {0} $ the phenomenon is observed at all points of non-removable discontinuity of $ f $.

References

[1]	H. Wilbraham, Cambridge and Dublin Math. J. , 3 (1848) pp. 198–201
[2]	J.W. Gibbs, Nature , 59 (1898) pp. 200
[3]	A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)

Comments

In a more explicit form the definitions of the constant $ A $ and $ B $ above are:

$$ A = \lim\limits _ {x \uparrow x _ {0} } \ \lim\limits _ {n \rightarrow \infty } \ \inf _ {x < y < x _ {0} } \ s _ {n} ( y) , $$

$$ B = \lim\limits _ {x \downarrow x _ {0} } \lim\limits _ {n \rightarrow \infty } \sup _ {x _ {0} < y < x } s _ {n} ( y) . $$

At an isolated jump discontinuity of $ f $, the ratio $ ( B - A ) / ( b - a ) $ equals $ ( 2 / \pi ) l = 1.17898 \dots $. This means that the Fourier series approximation establishes an overshoot of about $ 8.95\pct $ of the length of the jump at either end of the jump interval.

Actually, it was only in a second letter to $ Nature $([a1]) that Gibbs stated the phenomenon correctly, though without any proof. For details see [a2].

References

[a1]	J.W. Gibbs, Nature , 59 (1899) pp. 606
[a2]	H.S. Carslaw, "Introduction to the theory of Fourier's series and integrals" , Dover, reprint (1930)