We begin this discussion by taking a signal with a finite number of discontinuities (like a square pulse) and finding its Fourier Series representation. We then attempt to reconstruct it from these Fourier coefficients. What we find is that the more coefficients we use, the more the signal begins to resemble the original. However, around the discontinuities, we observe rippling that does not seem to subside. As we consider even more coefficients, we notice that the ripples narrow, but do not shorten. As we approach an infinite number of coefficients, this rippling still does not go away. This is when we apply the idea of almost everywhere. While these ripples remain (never dropping below 9% of the pulse height), the area inside them tends to zero, meaning that the energy of this ripple goes to zero. This means that their width is approaching zero and we can assert that the reconstruction is exactly the original except at the points of discontinuity. Since the Dirichlet conditions assert that there may only be a finite number of discontinuities, we can conclude that the principle of almost everywhere is met. This phenomenon is a specific case of nonuniform convergence.
Below we will use the square wave, along with its Fourier Series representation, and show several figures that reveal this phenomenon more mathematically.
The Fourier series representation of a square signal below says that the left and right sides are "equal." In order to understand Gibbs Phenomenon we will need to redefine the way we look at equality.
When comparing the square wave to its Fourier series representation in Figure 1, it is not clear that the two are equal. The fact that the square wave's Fourier series requires more terms for a given representation accuracy is not important. However, close inspection of Figure 1 does reveal a potential issue: Does the Fourier series really equal the square wave at all values of ? In particular, at each step-change in the square wave, the Fourier series exhibits a peak followed by rapid oscillations. As more terms are added to the series, the oscillations seem to become more rapid and smaller, but the peaks are not decreasing. Consider this mathematical question intuitively: Can a discontinuous function, like the square wave, be expressed as a sum, even an infinite one, of continuous ones? One should at least be suspicious, and in fact, it can't be thus expressed. This issue brought Fourier much criticism from the French Academy of Science (Laplace, Legendre, and Lagrange comprised the review committee) for several years after its presentation on 1807. It was not resolved for also a century, and its resolution is interesting and important to understand from a practical viewpoint.
The extraneous peaks in the square wave's Fourier series never disappear; they are termed Gibb's phenomenon after the American physicist Josiah Willard Gibbs. They occur whenever the signal is discontinuous, and will always be present whenever the signal has jumps.
Let's return to the question of equality; how can the equal sign in the definition of the Fourier series be justified? The partial answer is that pointwise--each and every value of --equality is not guaranteed. What mathematicians later in the nineteenth century showed was that the rms error of the Fourier series was always zero.