Assume that we have two sequences $(a_n)_{n \in \mathbb Z}, (b_n)_{n \in \mathbb Z}$ such that
- for each $l\in \mathbb N $ the sequence $\left(|n|^l a_n\right)_{n \in \mathbb Z}$ is bounded,
- there exists $s \in \mathbb N$ such that $\displaystyle\sum_{n \in \mathbb Z} \frac{|b_n|^2}{(1+n^2)^s}< \infty$.
Let $\displaystyle c_n=\sum_{k \in \mathbb Z} a_k b_{n-k}$ for $n \in \mathbb Z$. Is it then true that $\sum_{n \in \mathbb Z} |c_n|^2 <\infty ?$