4. In this problem, you will devise a computer experiment to investigate Gibb's phenomenon, which is the presence of spurious oscillations in the graph of a truncated Fourier series near the places where the full Fourier series is discontinous.
Choose any function you like that demonstrates Gibb's phenomenon. Your goal is to answer these two questions:
(a) You should find that the amount of overshoot only depends on the height of the discontinuity of your function. Expressed as a ratio to the height of the discontinuity, what is the approximate amount of overshoot/undershoot?
(b) What happens to the amount of overshoot/undershoot as you increadse the number of terms in your truncated Fourier series?
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Here is my solution.
I used Maple instead of Mathematica, so the code is in an additional Word file.
Make sure to check out:
The Gibbs' phenomenon states that as the overshoot in the truncated Fourier series representation of a function in the discontinuity point converges to 0.09 of "bump" of the function at the discontinuity point.
For example look at the function:
The function looks like:
This function is even; therefore we have to worry only about the coefficients of ...
The Gibbs phenomenon occurs at points where the derivative of the function is discontinuous.
This assignment demonstrates the Gibbs phenomenon with graphs and numerical analysis.