If I have a sequence $W_k$ that belongs to $L¹$, then that means the series $\sum_{}^{}W_k$ is absolutely convergent. Now I need to find such a sequence, in a way that it is orthogonal to the sequence defined by $\frac{1}{|k|}$.
So, by the definition of orthogonality, in need $$\sum_{k =-\infty }^{+\infty }\frac{1}{|k|}W_k = 0, k \neq 0$$ (This exercise does not specify the space, so I'm assuming it's infinite).
So I can write this as $$2\sum_{k = 1 }^{+\infty }\frac{1}{k}W_k = 0$$
Now, how do I find a non trivial solution to $W_k$?