Young criterion
for the convergence of Fourier series
2020 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].
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. |
[Yo] | W.H. Young, "On the convergence of the derived series of Fourier series" Proc. London Math. Soc. , 17 (1916) pp. 195–236 |
[Zy] | A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) MR0933759 Zbl 0628.42001 |
Young criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Young_criterion&oldid=28452