# Dini–Lipschitz criterion

Not to be confused with Dini criterion.

In mathematics, the **Dini–Lipschitz criterion** is a sufficient condition for the Fourier series of a periodic function to converge uniformly at all real numbers. It was introduced by Dini (1872), as a strengthening of a weaker criterion introduced by Lipschitz (1864). The criterion states that the Fourier series of a periodic function *f* converges uniformly on the real line if

where ω is the modulus of continuity of *f* with respect to δ.

## References

- Dini, U. (1872),
*Sopra la serie di Fourier*, Pisa - Golubov, B.I. (2001), "Dini–Lipschitz_criterion", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4

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