Dini criterion

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 $f$ 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 $0$.