Dini criterion

Not to be confused with Dini–Lipschitz criterion.

In mathematics, Dini's criterion is a condition for the pointwise convergence of Fourier series, introduced by Dini (1880).

Statement

Dini's criterion states that if a periodic function f has the property that (f(t) + f(–t))/t is locally integrable near 0, then the Fourier series of f converges to 0 at t = 0.

Dini's criterion is in some sense as strong as possible: if g(t) is a positive continuous function such that g(t)/t is not locally integrable near 0, there is a continuous function f with |f(t)| ≤ g(t) whose Fourier series does not converge at 0.