# 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.

• Not sure if I should remove this question, since I found the answer on the next page. Please refer to the answer given below. Commented Feb 26, 2020 at 3:30

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.

• Further, refer to Dirichlet conditions, which are conditions for Fourier Series being a suitable representation of a signal. Commented Feb 26, 2020 at 3:43
• where there is a jump discontinuity, the Fourier series will converge to the midpoint of the jump discontinuity. they're just saying that, for some pathological periodic functions, there may not be a Fourier series that converges to it. Commented Feb 26, 2020 at 5:33