Q:

Any convex sequence induces a sequence $\{a_n\}$ to be the Fourier coefficients of a function $f\in L^1[0,2\pi]$

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? 

A:

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 $k\to\infty$.

