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