Lebesgue criterion

for the convergence of Fourier series

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

A criterion first proved by Lebesgue for the convergence of Fourier series in [Le].

Theorem Consider a summable $2\pi$-periodic function $f: \mathbb R \to \mathbb R$, a point $x\in \mathbb R$ and the function \[ \varphi (u):= f(x+u)+f(x-u) - 2 f(x)\, . \] If there is $\delta>0$ such that \[ \lim_{h\downarrow 0} \int_h^\delta \left|\frac{\varphi (u+h)}{u+h} - \frac{\varphi (u)}{u}\right|\, du \;=\; 0\, , \] then the Fourier series of $f$ converges to $f(x)$ at $x$.

Cp. with Section 6 of Chapter III in volume 1 of [Ba] and Section 11 of Chapter II in volume 1 of [Zy]. The Lebesgue criterion is stronger then the Dirichlet criterion, the Jordan criterion, the Dini criterion, the de la Vallée-Poussin criterion, and the Young criterion. Cp. with Section 7 of Chapter III in volume 1 of [Ba].

References

[Ba]	N.K. Bary, "A treatise on trigonometric series" , Pergamon, 1964.
[Ed]	R. E. Edwards, "Fourier series". Vol. 1. Holt, Rineheart and Winston, 1967.
[Le]	H. Lebesgue, "Récherches sur le convergence des séries de Fourier" Math. Ann. , 61 (1905) pp. 251–280.
[Zy]	A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) MR0933759 Zbl 0628.42001