Dini criterion

for the convergence of Fourier series

2020 Mathematics Subject Classification: Primary: 42A20 [MSN][ZBL]

A criterion first proved by Dini for the convergence of Fourier series in [Di].

Theorem Consider a summable $2\pi$ periodic function $f$ and a point $x\in \mathbb R$. If there is a number $S$ and a $\delta>0$ such that \begin{equation}\label{e:Dini} \int_0^\delta |f(x+u) + f(x-u)-2S| \frac{du}{u} < \infty \end{equation} then the Fourier series of $f$ converges to $S$ at $x$.

Cp. with Section 38 of Chapter I in volume 1 of [Ba] and Section 6 of Chapter II in volume 1 of [Zy]. Observe that, if \eqref{e:Dini} holds, then the right and left limits $f (x^+)$ and $f(x^-)$ of $f$ at $x$ exists and $S= \frac{f(x^+)+f(x^-)}{2}$.

From Dini's statement it is possible to conclude several classical corollaries, for instance

• the convergence of the Fourier series of $f$ to $f(x)$ at every point where $f$ is differentiable
• the convergence of the Fourier series of $f$ to $f$ when $f$ is H\"older continuous.

It is also a (sharp) statement in the following sense. If $\omega: ]0, \infty[\to ]0, \infty[$ is a continuous function such that $\frac{\omega (t)}{t}$ is not integrable in a neighborhood of the origin, then there is a continuous $2\pi$-periodic function $f:\mathbb R \to \mathbb R$ such that $|f(t)-f(0)|\leq \omega (|t|)$ for every $t$ and the Fourier series of $f$ diverges at $0$.

The Dini criterion is weaker then the De la Vallee-Poussin criterion and not comparable to the Jordan criterion, cp. with Sections 2 and 3 of Chapter III in [Ba].

References

[Ba]	N.K. Bary, "A treatise on trigonometric series" , Pergamon, 1964.
[Di]	U. Dini, "Serie di Fourier e altre rappresentazioni analitiche delle funzioni di una variabile reale" , Pisa (1880).
[Ed]	R. E. Edwards, "Fourier series". Vol. 1. Holt, Rineheart and Winston, 1967.
[Zy]	A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) MR0933759 Zbl 0628.42001