What kind of periodic signals cannot be represented with the Fourier Series? Oppenheim et al. state in Signals and Systems that there exist periodic signals which cannot be represented with Fourier series. What signals are these?


  Although Euler and Lagrange would have been happy with the results of Examples 3.3 and 3.4, they would have objected to Example 3.5, since x(t) is discontinuous while each of its harmonic components is continuous. Fourier, on the other hand, considered the same example and maintained that the Fourier series representation of the square wave is valid. In fact, Fourier maintained that any periodic signal could be represented by a Fourier series.


Oppenheim et al, Signals and Systems, Prentice Hall, Second Edition, Page 195.
Oppenheim et al. give an answer on page 196. Paraphrased:

Define the approximation error as:

$l_N(t) = x(t) - \sum_{k=-N}^{k=+N} a_ke^{jk\omega_0t}$

There exist a class of signals for which:

$lim_{N\rightarrow \infty} l_N(t) \neq 0$

For these signals, the Fourier Series is, at best, an approximation.

Oppenheim et al. reference R. V. Churchill Fourier Series and Boundary Value Problems.