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I have approximation of some arbitrary function with Fourier series. On the image I have circled areas of discontinuity, affected by Gibbs phenomenon

How is it possible to calculate the "Radius of the yellow circle", inside of which Gibbs phenomenon is mainly taking place? I am interested in length of the interval on x axis, around point of discontinuity that behaves "badly"

Thanks

enter image description here

blindProgrammer
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1 Answers1

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This is not a complete answer but is too long for a comment and hopefully provides some insight.

There are no discontinuities in the Fourier series

$$\tilde{g}(x)=\underset{\epsilon\to 0}{\text{lim}}\left(\frac{g(x-\epsilon)+g(x+\epsilon)}{2}\right)$$ $$=\frac{\text{a0}}{2}+\underset{f\to\infty}{\text{lim}}\left(\sum\limits_{n=1}^f \left(a(n) \cos\left(\frac{2 \pi n x}{L}\right)+b(n) \sin\left(\frac{2 \pi n x}{L}\right)\right)\right)\tag{1}$$

for finite values of the evaluation frequency $f$, the only discontinuities are in $g(x)$ itself. The seeming discontinuities in $\tilde{g}(x)$ in the figure in your question are probably related to the plot resolution, and if you focus the plot on the range of $x$ values within the yellow circles perhaps it'd be more apparent that $\tilde{g}(x)$ is a smooth function with no discontinuities.

You need to define what you mean by behaves "badly". The oscillation you see is still present outside of your yellow circles, but it's just less noticeable as you move away from the discontinuity in $g(x)$ since the amplitude decreases as you move away from the discontinuity in $g(x)$. The amplitude of the oscillation is a function of the evaluation frequency $f$ as well as a function of $x$.

In conclusion, you need to be more precise in your question.

One example way to do this would be to ask:

What is the evaluation frequency $f$ required to achieve $\left|\tilde{g}(x)-g(x)\right|<c$ on the interval $a<x<b$?

where you specify the numeric values of $a$, $b$, and $c$.

Another example way to do this would be to ask:

For the evaluation frequency $f$, what is the interval $a<x<b$ where $\left|\tilde{g}(x)-g(x)\right|<c$?

where you specify the numeric values of $f$ and $c$.

Steven Clark
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  • Wow Thanks for answer. We know that Fourier series does not have points of discontinuation. Under "behaves badly" I mean large deviation from values of original function at points of discontinuity in Komparation of other continuous parts of g(x). how this differences behave as we make number of n terms in Fourier series large? Also I would like to choose acceptable deviation value and based on that, i hoped to determine the interval around point of discontinuity. Would this be possible? – blindProgrammer Apr 01 '22 at 19:52
  • @blindProgrammer As more terms are added, the error of the partial Fourier series for $\tilde{g}(x)$ converges to a fixed height so you can't ever eliminate the error. But the error in $\tilde{g}(x)$ moves closer to the discontinuity in $g(x)$ as the evaluation limit of the Fourier series for $\tilde{g}(x)$ increases. – Steven Clark Apr 02 '22 at 15:44
  • @blindProgrammer The Wikipedia article at https://en.wikipedia.org/wiki/Gibbs_phenomenon#The_square_wave_example focuses on the square wave example where the figure represents the period $T=1$ but the analysis is actually for the case $T=2 \pi$. The MathWorld article https://mathworld.wolfram.com/Wilbraham-GibbsConstant.html specifically addresses the Fourier series for the function $y=x$ in a paragraph towards the end of the article. – Steven Clark Apr 02 '22 at 15:46
  • @blindProgrammer The Wikipedia and MathWorld articles seem to primarily address the first local minimum or maximum of $\tilde{g}(x)$ on either side of the discontinuity in $g(x)$. I haven't seen analysis of the subsequent adjacent local minimums and maximums associated with the ringing, but perhaps the analysis in the Wikipedia and MathWorld articles can be extended to address these. – Steven Clark Apr 02 '22 at 15:46
  • @blindProgrammer The MathWorld and Wikipedia articles at https://mathworld.wolfram.com/LanczosSigmaFactor.html and https://en.wikipedia.org/wiki/Sigma_approximation describe a method to reduce the Gibbs phenomenom. – Steven Clark Apr 02 '22 at 15:47