## This should have kept you up all night!

If you read the previous post on the Fourier Series, then you might have noticed that* this animation was kind of lying to you.*

It surely does seem to resemble a square wave but notice that the peaks in red : They are overshooting and undershooting the maximum and minimum amplitudes.

What on earth is happening here? This goes by the name ‘Gibbs Phenomenon’.

## We do not have enough terms

Remember that in Fourier Series you are trying to construct a square wave *(which has sharp edges)* with smooth and continuous sine and cosine waves.

**Fourier series promises us to reconstruct the waveform perfectly ONLY if we provide it with the entire spectrum of frequencies. **

But practically we can only work in a finite range of frequencies and when working in a finite domain this overshoot is unavoidable and does not die out.

And if you are an engineer working with a system whose maximum output must not exceed the limit, this can be quite frustrating.

## Is there a way out of this ?

In order to get much smoother Fourier series, methods such as Fejér summation or Riesz summation, or sigma-approximation are employed.

Here’s the Fejér summation in action:

* Without Fejér summation *

Have a good one!

** Read more about the consequences of Gibbs phenomenon here