Dini criterion

Not to be confused with Dini–Lipschitz criterion or Dini's theorem.

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

References

• 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