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 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

$\lim_{\delta\rightarrow0^+}\omega(\delta,f)\log(\delta)=0$

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