# Young criterion

for the convergence of Fourier series

2010 Mathematics Subject Classification: Primary: 42A20 [MSN][ZBL]

A criterion first proved by W. H. Young for the convergence of Fourier series in [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 [Ba].

The Young's criterion is stronger than the 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 [Ba].

How to Cite This Entry:
Young criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Young_criterion&oldid=28452
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article