De la Vallée-Poussin criterion

for pointwise convergence of a Fourier series

If a -periodic function which is integrable on the segment is such that the function defined by

, is of bounded variation on some segment , then the Fourier series of converges at to the number

The de la Vallée-Poussin criterion is stronger than the Dini criterion, the Dirichlet criterion (convergence of series), and the Jordan criterion. It was demonstrated by Ch.J. de la Vallée-Poussin [1].

References

[1]	Ch.J. de la Vallée-Poussin, "Un nouveau cas de convergence des séries de Fourier" Rend. Circ. Mat. Palermo , 31 (1911) pp. 296–299
[2]	N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian)