In the article "Ensembles exceptionnels" by A. Beurling, the author cites the following theorem of Fejér: Suppose that a $2\pi$ periodic function $ f $, Lebesgue integrable in $(0,2\pi)$ satisfies the property that $$ \sum_{n=0}^\infty n |\hat{f}(n)|^2 < + \infty, $$

then there exists a sequence $r_N \in (0,1)$ such that $$\Big| \sum_{n=0}^N \hat{f}(n) e^{i\theta n} - \sum_{n=0}^\infty \hat{f}(n)r_N^ne^{i\theta n}\Big| \to 0, \,\, \text{as} \,\, N \to \infty,$$ uniformly (in $\theta$).

The cited article is " Über Gewisse Potenzreihen an der Konvergenzgrenze, L. Fejér". I could not spot the above theorem in the cited paper (mainly because I do not know any german).

I am looking for a reference in english of the above theorem or a proof of the claim.