Dini criterion explained

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

Statement

Dini's criterion states that if a periodic function has the property that

(f(t) + f( - t))/t is locally integrable near, then the Fourier series of converges to 0 at t =0 .

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