The Fourier Series is the representation of continuous-time, periodic signals in terms of complex exponentials. The Dirichlet conditions suggest that discontinuous signals may have a Fourier Series representation so long as there are a finite number of discontinuities. This seems counter-intuitive, however, as complex exponentials are continuous functions. It does not seem possible to exactly reconstruct a discontinuous function from a set of continuous ones. In fact, it is not. However, it can be if we relax the condition of 'exactly' and replace it with the idea of 'almost everywhere'. This is to say that the reconstruction is exactly the same as the original signal except at a finite number of points. These points, not necessarily surprisingly, occur at the points of discontinuities.

**History**

In the late 1800s, many machines were built to calculate Fourier coefficients and re-synthesize:

Comments:"My introduction to signal processing course at Rice University."