# Dini criterion

Not to be confused with Dini–Lipschitz criterion.

In mathematics, **Dini's criterion** is a condition for the pointwise convergence of Fourier series, introduced by 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), "Dini criterion", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4

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