# Lecture 4: The Gibbs Phenomenon

According to our theory, with a complete set of basis functions we can represent
any function exactly. We furthermore know how to obtain the best approximation
if we use only a finite set of functions. Interestingly, even the best approximation
can still have some substantial errors. Consider the error in the square-wave
series. Observe that there is a jump just before the point of discontinuity.
As it turns out,
**
no matter how large
**
is, this error remains, and it has an amplitude of about 9% of the discontinuity.
As
gets larger and larger, this wiggle becomes narrower and moves closer to the
point of the discontinuity, but it
**
never goes away.
**
This overshoot phenomenon
is known as the
**
Gibbs phenomenon
**
.

One of the important ramifications of this is in how we define functions to be equal. It is true that