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 Ulisse Dini (1872), as a strengthening of a weaker criterion introduced by Rudolf Lipschitz (1864). The criterion states that the Fourier series of a periodic function f converges uniformly on the real line if

$\lim _{\delta \rightarrow 0^{+}}\omega (\delta ,f)\log(\delta )=0$

where $\omega$ is the modulus of continuity of f with respect to $\delta$.

References

• Dini, Ulisse (1872), Sopra la serie di Fourier, Pisa
• Golubov, B. I. (2001) [1994], "Dini-Lipschitz criterion", Encyclopedia of Mathematics, EMS Press