How is it possible to calculate the length of the interval on x axis, around point of discontinuity that is affected by Gibbs phenomenon?

Question

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

Answer 1

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

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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.

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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 $x$.

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

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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 $c$.

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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 $c$.