Facts

Pointwise Convergence#

Theorem (Carleson). Let \(f:\mathbb{R}\to\mathbb{C}\) be a \(1\)-periodic function and let \(S_n f\) be the \(n\)-th Fourier sum of \(f\) . Then
1. If \(f\) is continuous then \((S_n f)(x) \to f(x)\) as \(n\to\infty\) for at least one \(x\in[0,1)\).
2. Better still, if \(f\) is Riemann integrable then \((S_n f)(x)\to f(x)\) as \(n\to \infty\) for almost all \(x\in[0,1)\).
3. Better still, if \(|f|^2\) is Lebesgue integrable then \((S_n f) (x)\to f(x)\) as \(n\to \infty\) for almost all \(x\in[0,1)\).

The theorem was proved only in 1964.