Dirichlet conditions
/ dee-ree-SHLAY /
It is one thing to write down a Fourier series for f; it is another to know the series actually adds back up to f. The Dirichlet conditions are a friendly, checkable set of hypotheses — easy to verify by eye — that guarantee a Fourier series converges and tells you exactly what value it converges to at every point. They are not the sharpest possible conditions, but they cover essentially every function an engineer or physicist meets: pulses, ramps, square waves, anything piecewise nice.
Stated for a periodic f: if over one period f is absolutely integrable, has only a finite number of maxima and minima, and only a finite number of jump discontinuities (each a finite jump), then the Fourier series of f converges everywhere. Where f is continuous, the series converges to f(x). Where f has a jump, the series converges to the midpoint of the jump, the average (f(x+) + f(x-))/2 of the left-hand and right-hand limits. That single, clean verdict at jumps is the practically important part: the series does not pick a side, it splits the difference.
These conditions are why Fourier methods are so dependable in applications. You rarely need the heavy machinery of L^2 theory to justify a computation; you just confirm your signal is piecewise smooth with finite jumps and you immediately know the series is legitimate and where it lands. The honest framing: Dirichlet conditions are sufficient, not necessary — there exist convergent expansions for functions that violate them and pathological continuous functions whose Fourier series diverges at a point — but for the bread-and-butter functions of mathematical physics they are exactly the working criterion you want.
The square wave is piecewise constant with one jump per half-period, so it meets the Dirichlet conditions. At x = 0, where it jumps from -1 to +1, the series converges to (-1 + 1)/2 = 0 — and indeed every sine term is zero there.
At a jump the series lands on the average of the two sides — never on either value itself.
Dirichlet conditions guarantee pointwise convergence, not uniform convergence: near a jump the partial sums always overshoot (Gibbs phenomenon), so 'the series converges everywhere' does not mean 'the partial sums look like f everywhere'.