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Can someone explain in complete detail with the appropriate convergence arguments of the Gibbs Phenomenon for Fourier Series? I know that the overshoot near a jump does not die out as the frequency increases, but approaches a finite limit. We derived the overshoot in an engineering class and there was a lot of hand-waving. I understand it but I would like to see a complete picture explained here for a general function $f(x)$ on a bounded domain.

Further, the Gibbs Phenomena doesn't occur for some other eigen function expansions like Chebyshev expansion etc. The explanation given in the class was again very crude. Again for these cases could someone explain in detail the connection to the Sturm-Liouville equation and which of these eigenfunction expansions lead to overshoot and which ones do not.

Thanks

Willie Wong
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    I hate to do this, but Wikipedia http://en.wikipedia.org/wiki/Gibbs_phenomenon has a pretty comprehensive mathematical explanation along with the full computation of the square wave example (what I would have posted anyway). The key bizarre thing is that the series converges pointwise everywhere (except the discontinuity?), yet the overshooting actually gets worse and worse. It is just that the overshooting moves closer and closer to the discontinuity and hence there is no contradiction. – Matt Feb 06 '11 at 00:11
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    Th Gibbs phenomenon occurs not only for Fourier series, but for all systens of orthogohal functions, including the classical Chebyshev, Legendre and Hermite polynomials. – Julián Aguirre Feb 06 '11 at 01:51
  • @Matt I am curious to know why you hate to refer to a Wikipedia page. – Did Feb 06 '11 at 10:01
  • @Didier Piau It wasn't so much the Wikipedia. It was the fact that I was referring someone to something rather than constructing my own answer. There was a good chance they already looked at that page and was looking for a new point of view which I could have tried to provide but was mostly feeling lazy. – Matt Feb 06 '11 at 18:49
  • @Matt Thanks for the answer, now I see your point. (How strongly your premises on math.SE are empirically supported is another matter.) – Did Feb 07 '11 at 07:24
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    @Matt, @Didier: I posted the question here after looking up wiki. I am expecting something more. For instance, why does he look at $\displaystyle \lim_{N \rightarrow \infty} f(x_0 + \frac{L}{2N})$ and not say $\displaystyle \lim_{N \rightarrow \infty} f(x_0 + \frac{L}{2N^2})$. Also do these things hold for any eigen function expansion in general?. Also I am not sure about the mode of convergence of these eigen functions. –  Feb 07 '11 at 15:45

3 Answers3

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Chapter 3 of the book The Gibbs phenomenon in Fourier analysis, splines, and wavelet approximations by Abdul J. Jerri seems to cover the case of a general orthogonal expansion along eigenfunctions of a Sturm-Liouville problem.

Did
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The periodic jump function defined by $j(t):=(\pi -t)/2$ on $]0,2\pi[$ has $\sum_{k=1}^\infty \sin(kt)/k$ as its Fourier series. We now compute the value of the partial sum $s_N$ at the point $t_N:=\pi/N$: $$s_N(t_N)=\sum_{k=1}^N{\sin(k\pi/N)\over k}= \sum_{k=1}^N{\pi\over N}{\sin(k\pi/N)\over k\pi/N} \doteq \int_0^\pi {\rm sinc}(t)dt\doteq 1.852$$ (the last sum can be interpreted as a Riemann sum for the sinc-integral). On the other hand $\lim_{t\to 0+} j(t)=\pi/2\doteq 1.571$ so that we have a guaranteed overshoot of about 17%.

The choice of the point $t_N$ is nearly "optimal", but I won't go into this here.

Christian Blatter
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Here is my attempted proof, sorry I don't know how to convert Word into Latex Here

Chuyin Wang
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