Dini–Lipschitz 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

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

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

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