View source for Lebesgue criterion

''for the convergence of Fourier series''

{{MSC|42A20}}

{{TEX|done}}

A criterion first proved by Lebesgue for the convergence of Fourier series in {{Cite|Le}}. 

'''Theorem'''
Consider a summable $2\pi$ periodic function $f$, 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 {{Cite|Ba}} and Section 11 of Chapter II in volume 1 of {{Cite|Zy}}. The Lebesgue criterion is stronger then the [[Dirichlet theorem|Dirichlet criterion]], the [[Jordan criterion|Jordan criterion]], the [[Dini criterion|Dini criterion]], the [[De la Vallée-Poussin criterion|de la Vallée-Poussin criterion]], and the [[Young criterion|Young criterion]]. Cp. with Section 7 of Chapter III in volume 1 of {{Cite|Ba}}.

====References====
{|
|-
|valign="top"|{{Ref|Ba}}|| N.K. Bary,   "A treatise on trigonometric series" , Pergamon, 1964.
|-
|valign="top"|{{Ref|Ed}}|| R. E. Edwards, "Fourier series". Vol. 1. Holt, Rineheart and Winston, 1967.
|-
|valign="top"|{{Ref|Le}}|| H. Lebesgue,   "Récherches sur le convergence des séries de Fourier"  ''Math. Ann.'' , '''61'''  (1905)  pp. 251–280.
|-
|valign="top"|{{Ref|Zy}}||   A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ.   Press  (1988)  {{MR|0933759}}  {{ZBL|0628.42001}} 
|-
|}