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.



The normal derivative of the Green's function
Yes indeed, $S$ is dense in $L^2(\partial \mathbb{D})$.


This is because any $g\in L^2(\partial \mathbb{D})$ has an $L^2$ harmonic extension $h$ to $\mathbb{D}$, and $$h(p)=\frac{1}{2\pi}\int\_{\partial \mathbb{D}} g(z) \partial\_{\nu} G(z; p) dz\;\;\;(\*)$$ by Green's formula. Hence if $g$ is orthogonal to $S$, $h(p)=0$ for all $p$, and $g=0$. 


Implicit are the non obvious results that there is a continuous trace from $L^2$ harmonic function in $\mathbb{D}$ to $L^2$ functions on the boundary and that the trace of $h$ given by $(\*)$ is indeed $g$. This should be found in many books treating harmonic analysis or one variable complex analysis, like the references [here](http://en.wikipedia.org/wiki/Poisson_kernel), or Rudin's *Real and complex analysis*.


