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

*x*

_{0}= 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

*x*

_{0}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

Note that in this case, we have

and