Do these spaces have a name?

Question

Consider the Hilbert space $l^2(\mathbb{R})$ of square-summable sequences. It's easy to show that sequences that decay at a given rate e.g. $o(n^{-2})$ form a subspace of $l^2(\mathbb{R})$. Do these spaces have a name? Do you come across them somewhere?

Answer 1

This could be considered as a weighted Lebesgue space (which are well studied). The space of sequences $a_n$ such that $n^2a_n\in l^\infty$ is a subset of $l^2$.