On the de la Vallé-Poussin theorem on the uniqueness of the trigonometric series representing a function

The de la Vallé-Poussin theorem states that if a trigonometric series converges to a finite integrable function $f$ everywhere outside a countable set $E$, then it is the Fourier series of $f$. In this paper the theorem is shown to hold also if the exceptional set $E$ is a union of finitely many $H$-sets.