1. If  and both left and right derivatives of  exist at , the Fourier series of  on [-L, L] converges at  to

2. If both  and  exist, the Fourier series converges at both L and –L to

The figure below is an illustration of the theorem. Let us talk a little more in depth about it.

At each point between –L and L where the right and left derivatives exist, the Fourier series converges to the average of the left and right limits of . This fact is illustrated above. In particular if  is continuous at  for the whole domain of x, the left and right limits both equal  and the series converges at  to

 or

While you may be quick to think that since all functions are usually continuous in the domain without any jump discontinuity, pay close attention to the domain for which the theorem applies namely , a strict inequality. Convergence for the endpoints is tested using the second part of the theorem.

At both –L and ­L, the Fourier series converges to the average of the right limit at –L and the left limit at L, assuming that the respective derivatives exist at both points. It should not be surprising that the Fourier series converges to the same value at L and at –L. Upon letting x = L in the Fourier series of , we get

because . If we let x = -L in the Fourier series, we get exactly the same series because  and .

In terms of the graph, the Fourier series converges at –L and at L to the point midway between the ends of the graph at those points.

In light of the above theorem, we can often tell what the Fourier series of  converges to just by looking at the graph of . Really, it is a matter of taking the averages of the value of the function at the jump discontinuous and the average of the function at the end points.

Using our previous graph of an example, I think this is how the Fourier series will converge.

Last point to note: while it may be obvious to some, notice that the information of the convergence of the Fourier series comes from the function  and NOT the Fourier series itself. Yes, there may be a few convergence test we can use on the Fourier series itself, but the above theorem is applied to the function. In other words, the function  tells us whether its Fourier series converges to itself.