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, it is weaker than the Lebesgue criterion and not comparable to the Young 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