Dini criterion

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 [math]\displaystyle{ (f(t)+f(-t))/t }[/math] is locally integrable near 0, then the Fourier series of f converges to 0 at [math]\displaystyle{ t=0 }[/math].

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, https://archive.org/details/seriedifourierea00diniuoft 
• Hazewinkel, Michiel, ed. (2001), "Dini criterion", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Dini_criterion&oldid=28457