Convergence of Fourier Series discontinuities

Basic Concepts and Principles

Fourier series convergence to the midjump-value in jump-discontinuities points.
We will see at this section what is the behavior of the Fourier series for those functions with jump-discontinuities.
Exactly, the Dirichlet theorem assures us that the jump discontinuities, the Fourier series converges to the middle value of jump.
We also study the so-called Gibbs Phenomenon, which says that we are near a jump discontinuity, the Fourier series firing a 9% jump in the value of the jump.

Extended Theory

Theorem (Dirichlet)
Let f be a continuous function except at a point

Suppose there exits

Then the Fourier series converges to the midpoint of the jump, ie

We will now see the so-called Gibbs Phenomenon
Theorem (Gibbs Phenomenon)
Let f be as in the previous theorem, then the Fourier series of f near of

It shoots with a value of approximately 9% of the value of the jump of f, ie

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