De la Vallée-Poussin criterion

for the convergence of Fourier series

2020 Mathematics Subject Classification: Primary: 42A20 [MSN][ZBL]

A criterion first proved by De la Vallée-Poussin for the convergence of Fourier series in [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, the Dirichlet criterion, and the Jordan criterion. Cp. with Section 3 of chapter III in Volume 1 of [Ba].

References

[Ba]	N.K. Bary, "A treatise on trigonometric series" , Pergamon, 1964.
[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.
[Ed]	R. E. Edwards, "Fourier series". Vol. 1. Holt, Rineheart and Winston, 1967.
[Zy]	A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) MR0933759 Zbl 0628.42001