##

The Gibbs Phenomenon

Near a point, where *f* has a jump discontinuity, the partial sums *S*_{n} of a Fourier series
exhibit a substantial overshoot near these endpoints, and an increase in *n* will not diminish the amplitude of the overshoot,
although with increasing *n* the the overshoot occurs over smaller and smaller intervals. This phenomenon is called Gibbs phenomenon.
In this section we examine some detail in the behaviour of the partial sums *S*_{n} of
.

The next step is to replace the partial sums *S*_{n} with integrals

For
we have a typically "overshoot". This will be the next step to show.
Let
.

**Theorem 7** (The Gibbs phenomenon)

Let

and

.

and

Since
for *x* near 0, we see that an "overshoot" by approximately
is maintained as
(but over smaller and smaller intervals centred at *x*=0).

*Developed by Marian Prutscher*: Email