pointwise convergence of Fourier series
/ Dirichlet: dee-ree-SHLAY /
You build the Fourier series of a function, but does the infinite sum actually add up to the function you started with? Pointwise convergence asks the most basic version of this question: pick one specific point x, add up the series there, and see whether the running total settles down to the function's value at that very point. The surprising answer is: usually yes, but with sharp exceptions exactly where the function misbehaves.
Dirichlet's theorem gives clean, checkable conditions. If f is periodic, and on each period it is piecewise smooth — bounded, with only finitely many jumps and corners, and with one-sided limits and one-sided slopes everywhere — then its Fourier series converges at every point. Where f is continuous, the series converges to f(x) exactly. At a jump, it converges to the AVERAGE of the left and right limits, the midpoint of the gap — the series splits the difference, no matter what value you assigned to f there. These are the Dirichlet conditions, and they cover essentially every function that arises in a normal PDE.
Two honest cautions. First, pointwise convergence is weaker than it sounds: the sum can converge at every point yet behave badly in between, and near a jump it persistently overshoots (the Gibbs phenomenon) even as it converges pointwise away from the jump. Second, the general theory is subtle — there exist continuous functions whose Fourier series diverges at some points, so mere continuity is not enough; you really want the piecewise-smoothness of the Dirichlet conditions. For honest comfort there is also the stronger mean-square convergence, which holds for any square-integrable function.
The square wave that is +1 on (0, pi) and -1 on (-pi, 0) has a jump at x = 0. Its Fourier series there sums to (+1 + (-1))/2 = 0 — the midpoint of the jump — even though the function itself is +1 just to the right and -1 just to the left. Away from the jumps the series converges to the actual values +1 or -1.
At a jump the series ignores whatever value you assigned and converges to the midpoint of the two one-sided limits.
Continuity alone does NOT guarantee pointwise convergence everywhere — there are continuous functions whose Fourier series diverges at isolated points. The reliable sufficient condition is the piecewise-smoothness of the Dirichlet conditions, not bare continuity.