There is a conclusion: For any $x\in \mathbb R^\mathbb N$, we denote by $A_x$ the set $$A_x= \{a\in \mathbb R^\mathbb N:\sum_n x(n)\alpha(n)~\text{converges}\},$$ then for $y,x_1,x_2,\dots,x_k \in \mathbb R^\mathbb N$ we have: $$ \bigcap_{n=1}^k A_{x_n} \subset A_y~ \Leftrightarrow ~ y(n)=\sum_{m=1}^k x_m(n)L_m(n) \quad(n\geq N_0) $$ where $L_m \in \mathbb R^\mathbb N $ is a sequence of bounded variation ($\sum_n |L_m(n+1)-L_m(n)| < \infty$) for any $m=1,2,\dots,k$ and $N_0$ is a large enough constant.
Actually, I think this conclusion has already been proved by someone; who has seen anything related to this conclusion?? Thanks.
