Let $\{a_n\}$ be an even sequence of non-negative numbers satisfying in $a_n\to 0$ and $a_{n-1}+a_{n+1}-2a_n\geq0$ for every $n>0$. We define $f=\sum_{n=1}^\infty n(a_{n-1}+a_{n+1}-2a_n)F_{n-1}$ where $F_n$ is the Fejer's kernel. Theorem 4.1 in Katzenelson's book (page 23) says that $\hat{f}(j)=a_{|j|}$. I do not get last the line of the proof:
$$a_{|j|}=\sum_{n=|j|+1}^\infty n(a_{n-1}+a_{n+1}-2a_n)(1-\frac{|j|}{n})$$
It will be great if somebody clarifies this part.
Is this theorem coming from a paper?
It is not too hard to prove by induction on $k>1$ that $$\sum_{n=|j|+1}^{|j|+k}n(a_{n-1}+a_{n+1}-2a_n)(1-\frac{|j|}{n})=a_{|j|}+ka_{|j|+k+1}-(k+1)a_{|j|+k}$$ and since by the first part of the proof we have $$\lim_{n\to\infty}n(a_n-a_{n+1})=0$$ and since by hypothesis of the theorem $a_n\to 0$ as $n\to\infty$, you get the desired result by letting $k\to\infty$.