Dirichlet conditions

Remarks

A function of bounded variation is a difference of two monotone functions, so it suffices to treat a monotone $f$. One writes $S_N f(x) - s$ as an integral of $f(x + t) + f(x - t) - 2s$ against the Dirichlet kernel over $[0, \pi]$, with $s = \frac{1}{2}(f(x^+) + f(x^-))$. The second mean value theorem for integrals, together with the uniform boundedness of $\int_0^a \frac{\sin((N + 1/2) t)}{t} \, dt$ in $a$ and $N$, controls the part near $t = 0$, and the Riemann–Lebesgue lemma handles the rest. Dirichlet's original conditions are that $f$ is bounded on a period, with finitely many maxima and minima and finitely many discontinuities there. Such an $f$ has bounded variation, so Jordan's form above contains them. Example: for the square wave, equal to $\operatorname{sgn} x$ on $(-\pi, \pi)$, the series converges to $0$ at the jumps, the average of $-1$ and $1$, and to $f(x)$ elsewhere. Bounded variation cannot be replaced by continuity: there are continuous functions whose Fourier series diverge at a point.

Going deeper

For a student

This is the convergence theorem used in most applications, since piecewise monotone and piecewise smooth functions have bounded variation. It is a pointwise test, like the Dini test; the two tests are independent, and neither implies the other. Part 2 explains why the convergence for the square wave is uniform away from the jumps but not near them, where the Gibbs phenomenon appears. Camille Jordan stated the test in this form, for functions of bounded variation.