I was wondering if anything was known about the following:

Let $\mathbb{D}^2=\lbrace x^2+y^2< 1 \rbrace \subset \mathbb{R}^2$ be the open unit disk. Consider now the Green's functions $G(z; p)$ of this disk. I.e. here $p\in \mathbb{D}^2$ and $G(z;p)$ is smooth and harmonic in $\bar{\mathbb{D}}^2\backslash \lbrace p \rbrace$, vanishes on the boundary and has the property that $H(z;p) =G(z; p)-\log |z-p|$ is smooth.

Now consider the set of functions: \begin{equation} S=\lbrace f\in C^\infty(\partial \mathbb{D}^2): f= \sum_{i=1}^n \lambda_i \partial_{\nu} G(z; p_i), \lambda_i \in \mathbb{R}, p_i \in \mathbb{D} \rbrace \end{equation} Here $\partial_{\nu} G(z; p_i)$ is the normal derivative on $\partial \mathbb{D}^2$.

My question is what can be said about the set $S$? In particular, is there any hope that it is dense in $L^2(\partial \mathbb{D}^2)$?

Playing around with things all I got was a big mess so any references would be appreciated.