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

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

• Dini, Ulisse (1880), Serie di Fourier e altre rappresentazioni analitiche delle funzioni di una variabile reale, Pisa: Nistri, ISBN 978-1429704083
• Golubov, B. I. (2001) [1994], "Dini criterion", Encyclopedia of Mathematics, EMS Press