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 ${\displaystyle f}$ 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 ${\displaystyle 0}$.

References

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