Dini-Lipschitz criterion

for the convergence of Fourier series

2020 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.

References

[Ba]	N.K. Bary, "A treatise on trigonometric series" , Pergamon, 1964.
[Di]	U. Dini, "Sopra la serie di Fourier" , Pisa (1872).
[Le]	H. Lebesgue, "Sur la répresentation trigonométrique approchée des fonctions satisfiasants à une condition de Lipschitz" Bull. Soc. Math. France , 38 (1910) pp. 184-210
[Li]	R. Lipschitz, "De explicatione per series trigonometricas instituenda functionum unius variabilis arbitrariarum, etc." J. Reine Angew. Math. , 63 : 2 (1864) pp. 296-308
[Ni]	S.M. Nikol'skii, "On the Dini–Lipschitz condition for convergence of Fourier series" Doklady Akad. Nauk SSSR , 73 : 3 (1950) pp. 457–460
[Zy]	A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) MR0933759 Zbl 0628.42001