View source for De la Vallée-Poussin criterion

''for the convergence of Fourier series''

{{MSC|42A20}}

{{TEX|done}}

A criterion first proved by De la Vallée-Poussin for the convergence of Fourier series in {{Cite|De}}. 

'''Theorem'''
Consider a summable $2\pi$ periodic function $f$, a point $x\in \mathbb R$ and the function
\[
F (t) := \frac{1}{t} \int_0^t \left(f(x+u)+f(x-u) - 2 f(x)\right)\, du \qquad \mbox{for } t>0
\]
and $F(0)=0$. If $f$ has bounded variation on some interval $[0, \delta]$ with $\delta>0$, then the Fourier series of $f$ converges to $f(x)$ at $x$.


The de la Vallée-Poussin criterion is stronger than the [[Dini criterion|Dini criterion]], the [[Dirichlet theorem|Dirichlet criterion]], and the [[Jordan criterion|Jordan criterion]], it is weaker than the [[Lebesgue criterion]] and not comparable to the [[Young criterion]]. Cp. with Section 3 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|De}}||   Ch.J. de la Vallée-Poussin,   "Un nouveau cas de convergence des séries  de Fourier"  ''Rend. Circ. Mat. Palermo'' , '''31'''  (1911)  pp.  296–299.
|-
|valign="top"|{{Ref|Ed}}|| R. E. Edwards, "Fourier series". Vol. 1. Holt, Rineheart and Winston, 1967.
|-
|valign="top"|{{Ref|Zy}}||  A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ.  Press  (1988)  {{MR|0933759}}  {{ZBL|0628.42001}} 
|-
|}