1
$\begingroup$

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!

  • 0
    Have you seen [Wikipedia section's Formal mathematical description of the phenomenon](http://en.wikipedia.org/wiki/Gibbs_phenomenon#Formal_mathematical_description_of_the_phenomenon)?2011-09-23

1 Answers 1

1

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 $a$ ?

  • 0
    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 $f(a)$.2011-09-23