# Dini-Lipschitz criterion

(Redirected from Dini–Lipschitz criterion)

for the convergence of Fourier series

2010 Mathematics Subject Classification: Primary: 42A20 [MSN][ZBL]

A criterion proved independently by Dini and Lipschitz for the uniform convergence of Fourier series, see [Di] and [Li].

Consider a continuous function $f:{\mathbb R} \to {\mathbb R}$ which is $2\pi$-periodic and denote by $\omega (\delta, I)$ its modulus of continuity, namely $\omega (\delta, I) := \sup\; \{|f(x)-f(y)| : x,y\in I \;\mbox{and}\; |x-y|\leq \delta\}\, .$ The Dini-Lipschitz criterion is then the following theorem (cp. with Theorems 10.3 and 10.5 of [Zy]):

Theorem 1 If on some open interval $I$ we have $\lim_{\delta\to 0}\; \omega (\delta, I) |\log \delta| = 0\,$ then the Fourier series of $f$ converges uniformly to $f$ on any closed interval $J\subset I$.

Note that, as an obvious corollary, if the interval $I$ has length larger than $2\pi$, then the Fourier series converges uniformly to $f$ on the entire real axis.

The Dini-Lipschitz criterion is sharp in the following sense. If $f: {\mathbb R}^+\to {\mathbb R}^+$ is any function such that $\limsup_{\delta\to 0}\; f (\delta) |\log \delta| > 0\,$ then there is a continuous function $f$ such that $\omega (\delta, {\mathbb R}) \leq f(\delta)$ for any $\delta$ and the corresponding Fourier series diverges at some point.

How to Cite This Entry:
Dini–Lipschitz criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dini%E2%80%93Lipschitz_criterion&oldid=22358