Gibbs Phenomenon

http://www.sosmath.com/fourier/fourier3/gibbs.html

One shortcoming of Fourier series today known as the Gibbs phenomenon was first observed by H. Wilbraham in 1848 and then analyzed in detail by Josiah W. Gibbs (1839-1903). We will start with an example.

Example. Consider the function

Since this function is odd, we have a_n = 0, for . A direct calculation gives

for . The Fourier partial sums of f(x) are

The main Theorem implies that this sequence converges to f(x) except at the point x₀ = 0, which a point of discontinuity of f(x). Gibbs got interested to the behavior of the sequence of Fourier partial sums around this point.

Looking at the graphs of the partial sums, we see that a strange phenomenon is happening. Indeed, when x is close to the point 0, the graphs present a bump. Let us do some calculations to justify this phenomenon.
Consider the second derivative of f_2n-1, which will help us find the maximum points.

Using trigonometric identities, we get

So the critical points of f_2n-1 are

Since the functions are odd, we will only focus on the behavior to the right of 0. The closest critical point to the right of 0 is . Hence

In order to find the asymptotic behavior of the this sequence, when n is large, we will use the Riemann sums. Indeed, consider the function on the interval , and the partition of . So the Riemann sums

converges to . Esay calculations show that these sums are equal to

Hence

Using Taylor polynomials of at 0, we get

i.e. up to two decimals, we have

These bumps seen around 0 are behaving like a wave with a height equal to 0,18. This is not the case only for this function. Indeed, Gibbs showed that if f(x) is piecewise smooth on , and x₀ is a point of discontinuity, then the Fourier partial sums will exhibit the same behavior, with the bump's height almost equal to

To smooth this phenomenon, we introduce a new concept called the -approximation. Indeed, let f(x) be a function piecewise smooth on and f_N(x) its Fourier partial sums. Set

where

which are called the -factors. To see that the sequence of sums better approximate the function f(x) than the Fourier partial sums , we use the following result:

Theorem. We have

Proof. We have

Similarly, we have

Hence

which yields the conclusion above.

Using the above conclusion, we can easily see that indeed the sums approximate the function f(x) in a very smooth way. On the graphs, we can see that the Gibbs phenomenon has faded away.

Example. The picture

shows how the -approximation helps fade away the Gibbs phenomenon for the function

Note that in this case, we have

and