## The Gibbs Phenomenon

Near a point, where f has a jump discontinuity, the partial sums Sn 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 Sn of .

Theorem 6

The next step is to replace the partial sums Sn 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).

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