Dirichlet conditions for Fourier series

The Dirichlet conditions are a set of sufficient conditions on a periodic signal $x(t)$ (period $T$) which guarantee that its Fourier series converges to $x(t)$ at every point of continuity.

1. Absolute integrability over one period: the signal must be absolutely integrable over a period,

$$\int_{T} |x(t)|\, dt < \infty$$

This makes every coefficient finite, since $|a_k| \le \frac{1}{T}\int_T |x(t)|\,dt$.

• Violating example: $x(t) = 1/t$ for $0 < t \le 1$, repeated with period 1.

2. Finite number of maxima and minima: in any one period, $x(t)$ has only a finite number of maxima and minima (bounded variation).
   • Violating example: $x(t) = \sin(2\pi/t)$ for $0 < t \le 1$.
3. Finite number of discontinuities: in any finite interval there are only a finite number of discontinuities, and each one is finite (a finite jump).
   • Violating example: a staircase signal whose steps get halved again and again within one period, giving infinitely many jumps.

Result: if these conditions hold, the Fourier series equals $x(t)$ wherever $x(t)$ is continuous. At a jump it converges to the average of the left and right limits, $\frac{1}{2}[x(t^-) + x(t^+)]$. Near a jump, the partial sums of the series still overshoot by about 9% of the jump (the Gibbs phenomenon).

The conditions are sufficient but not necessary. Signals that break them are mostly artificial, so almost every practical signal has a convergent Fourier series.