View source for Young criterion

''for the convergence of Fourier series''

{{MSC|42A20}}

{{TEX|done}}

A criterion first proved by W. H. Young for the convergence of Fourier series in {{Cite|Yo}}. 

'''Theorem'''
Consider a summable $2\pi$ periodic function $f$, a point $x\in \mathbb R$ and the function
\[
\varphi (u):= f(x+u)+f(x-u) - 2 f(x)
\]
Assume that:
* $\varphi (u)\to 0$ as $u\downarrow 0$;
* $\theta (t) = t\varphi (t)$ is a function of bounded variation in some interval $]0, \delta[$
* the total variation of $\theta$ on the interval $]0, h[$ is $O(h)$.
Then the Fourier series of $f$ converges to $f(x)$ at $x$.

Cp. with Section 4 of Chapter III in volume 1 of {{Cite|Ba}}.

The Young's criterion is stronger than the [[Dirichlet theorem|Dirichlet criterion]], the [[Dini criterion]] and the [[Jordan criterion]], it is not comparable to the [[De la Vallee-Poussin criterion]] and it is weaker than the [[Lebesgue criterion]]. Cp. with Sections 5 and 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|Yo}}||  W.H. Young,   "On the convergence of the derived series of Fourier  series"  ''Proc. London Math. Soc.'' , '''17'''  (1916)  pp. 195–236
|-
|valign="top"|{{Ref|Zy}}||    A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ.    Press  (1988)  {{MR|0933759}}  {{ZBL|0628.42001}} 
|-
|}