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In this page, we will study what happens to the convergence of the Fourier series for a piecewise smooth function. A piecewise smooth function is a function defined over an interval which is continuous along with its first derivative except for a finite number of first-order discontinuity points (jumps); the left and right limits for the function and its first derivative at the discontinuity points must exist.
An example of this is the sawtooth function , , extended periodically on the real line; this function is discontinuous at for all interger values of k.
Let us define as the sum of the fist n Fourier series terms for the function :
The interactive plot above displays the function (green line) and its Fourier series approximation (red line). You can explore what happens for different values of n by moving the slider called "partials". Use the mouse's scroll wheel to zoom and drag the plot to look at the details around a discontinuity point.
After playing a while, you should have noted the following facts:
This behavior is called Gibbs Phenomenon. It shows that the Fourier series does not converge uniformly to a discontinuos function in an arbitrary small interval of the discontinuity point. Formally, the following theorem can be proven [1, pag. 63,]:
Interestingly, the the overshoot factor depends only on the type of discontinuity and not on the values of the function.
The Gibbs phenomenon has interesting physical consequences. Consider an electric circuit in which by means of a switch, an arbitrarily fast voltage transition is created. The response of the linear components of the circuit will exhibit an overshoot, which could be potentially disruptive.