Density of certain rational functions in the Hilbert space $L^2(-\infty,0)$

Question

It is easy to check that the functions $$f_{n,z}(x):=(z-x)^{-n},\quad n\geq 1,\quad z\in \mathbb{C}-(-\infty,0]$$ belong to the Hilbert space $L^2(-\infty,0)$, i.e., $L^2$-integrable complex-valued functions on the open interval $(-\infty,0)$. I am interested to find out wether the vector subspace of $L^2(-\infty,0)$ spanned by $f_{n,z}$ is dense.

Answer 1

The $f_{1,z}=f_z$ already span a dense subspace: By Stone-Weierstrass, applied to the compactification $X=[-\infty,0]$, any $g\in C(X)$ can be uniformly approximated by functions from the algebra generated by the $f_z$ and $1$. Since $f_w f_z =(f_w-f_z)/(z-w)$, we can in fact approximate by a (finite) linear combination of the $f_z$ and $1$, and if $g(-\infty)=0$, then we don't need $1$.

Now consider a $g\in C(X)$ with $g(x)=0$ in a neighborhood of $x=-\infty$, and approximate $(x+1)(x+2)g(x)$ in this way by $F=\sum \alpha_j f_{z_j}$. Then, by the identity we just observed, $F/(x+1)(x+2)$ is a linear combination of $f_z$'s, too, and this function is close in norm to $g$.