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

[math]\displaystyle{ \lim_{\delta\rightarrow0^+}\omega(\delta,f)\log(\delta)=0 }[/math]

where [math]\displaystyle{ \omega }[/math] is the modulus of continuity of f with respect to [math]\displaystyle{ \delta }[/math].

References

• Dini, Ulisse (1872), Sopra la serie di Fourier, Pisa, https://books.google.com/books?id=bCD\_SAAACAAJ 
• Hazewinkel, Michiel, ed. (2001), "Dini-Lipschitz criterion", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page