I am interested by a real sequence $\{a_n\}_{n\in\mathbb{Z}}$ as $\sum_{n\in\mathbb{N}}\left(\vert a_n\vert + \vert a_{-n}\vert\right)$ converges. I want to find the asymptotic behavior of this sequence :
$$u_n=\frac{1}{2n+1}\sum_{m\in\mathbb{Z}}\left\vert\sum_{k=-n}^n a_{m+k}\right\vert$$ $$\lim_{n\to +\infty} u_n=?$$
I can't find the trick in order to conclude because all the stuff I use for the moment isn't fine enough.
The best result I've managed to prove is that $u_n$ is bounded by $\sum_{n\in\mathbb{Z}}\vert a_n\vert$ but I cannot tell if it converges because I don't know the variations of $u_n$. Indeed we have thanks to the triangle inequality :
$$\frac{1}{2n+1}\sum_{m=-n}^n\left\vert\sum_{k=-n}^n a_{m+k}\right\vert\leq\frac{1}{2n+1}\sum_{m=-n}^n\sum_{k=-n}^n\vert a_{m+k}\vert\\\leq\frac{1}{2n+1}\left((2n+1)\sum_{k=-2n}^{2n}\vert a_k\vert\right)\leq\sum_{n\in\mathbb{Z}}\vert a_n\vert$$
Thanks for any help !