pointwise convergence of Fourier series

The most natural hope is that, at each fixed point x, the partial sums S_N(x) settle down to the function's value f(x). This is pointwise convergence — convergence checked one point at a time. It is more delicate than it sounds, and the honest history of analysis is largely the story of how delicate.

Dirichlet's theorem (1829) gives a clean sufficient condition: if f is 2*pi-periodic, piecewise continuous, and piecewise monotone (or more generally of bounded variation) near x, then S_N(x) converges to the average (f(x+) + f(x-))/2 of the one-sided limits. At a point of continuity this average is just f(x); at a jump it is the midpoint. The Dini test gives another sufficient condition in terms of an integrability hypothesis on the difference quotient of f near x.

Now the cautions, which are easy to misremember. Continuity alone does NOT guarantee pointwise convergence: du Bois-Reymond built a continuous function whose Fourier series diverges at a point. Worse, Kolmogorov (1923) built an integrable function whose Fourier series diverges everywhere. The redeeming result is Carleson's theorem (1966): if f is in L2 (or merely L^p for p > 1), its Fourier series converges to f almost everywhere — but ‘almost everywhere’ is the best one can claim for general such f, not ‘everywhere’.