# Accuracy of Fourier series at discontinuities

What could I say when asked to "comment on the accuracy of Fourier series at discontinuities"? It is very vague, though I reckon it alludes to the W-G phenomenon. I have read the wiki page on Gibbs phenomenon, but I don't know what to say about the accuracy. Thanks in advance!

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Well, if it wiggles so much in the vicinity of the discontinuities of the function being approximated, then... –  Ｊ. Ｍ. Sep 23 '11 at 13:47
@J.M.: I know that the accuracy at the discontinuities would be quite bad, but is there a more quantitative comment? –  jose Sep 23 '11 at 13:50
Ah, then you'll like this... –  Ｊ. Ｍ. Sep 23 '11 at 13:56
–  Américo Tavares Sep 23 '11 at 14:06

HINT: Consider step function $f(x)$ which is $1$ for $x > 0$ and zero for $x<0$ and equal to some value $a$ at $x=0$, find its Fourier series, and compare its value to the value of the function at the discontinuity, does it depend on $a$ ?
Would it be $1\over 2$? So independent of the value of $a$... But how might I comment on the effect of the wiggling near the discontinuity? –  jose Sep 23 '11 at 13:56
Yes, it is $\frac{1}{2}$ and independent of $a$. The full series for interval $(-\pi, \pi)$ is $\frac{1}{2}+\sum _{n=0}^{\infty } \frac{2 \sin ((2 n+1) x)}{(2 n+1) \pi }$. The wiggling shows up only when you truncate the Fourier series. The value of the series at the discontinuity is $\frac{1}{2} \left( f(a^-) + f(a^+) \right)$, and thus may disagree with $f(a)$. –  Sasha Sep 23 '11 at 14:00