Recovery of values of analytic functions on the unit disk Is it possible to find constants $\{a_n\}_1^\infty$ and complex points in the unit disk $\{z\}_1^\infty$ ($z_n \neq 0$) that for any analytic function $f\in H_2$ in the unit disk we could find the value $f(0)$ in the form:
$$
f(0) = \sum_{n=1}^\infty c_n f(z_n) \quad (1)
$$
So far, I have found the following result in the article by V.Totik "Recovery of $H^p$-functions":

If $\{\alpha_{nk}\}$ is a point system such that $\alpha_{nk} \neq \alpha_{nj}$ if $k \neq j$ and $\sum_{k=1}^n (1-|\alpha_{nk}|)$ tends to infinity as $n\to\infty$, then to every $\alpha\in U$ there are constants $c_{nk}$ such that for every $f\in H^1$
  $$
\lim_{n\to \infty}\sum_{k=1}^n c_{nk} f(\alpha_{nk}) = f(\alpha)
$$

My questions: 
0) Does the representation $(1)$ exist? It seems that it's the point of the abovementioned article and the representation exists.
1) Are $c_{nk}$ and $\alpha_{nk}$ (as $n\to\infty$) the same for different points $\alpha$ (we are looking for $f(\alpha)$)?
2) Does it work for all $H^p(1\leq p\leq\infty)$ spaces?
3) Why $\sum_{k=1}^n (1-|\alpha_{nk}|)$  as $n\to\infty$ is such a crucial condition?
Thank you for your help in advance!
 A: Lemma 1: Suppose $U\subset \mathbb C$ is open. Then there exists a pairwise disjoint countable collection $D_1,D_2, \dots $ of closed discs of positive radius contained in $U$ such that 
$$A(U\setminus (\cup_n D_n))=0.$$
Here $A$ is Lebesgue area measure on $\mathbb C.$ Perhaps you'd like to try your hand at this.
Lemma 2: If $U\subset \mathbb C$ is open, $\overline {D(z,r)}\subset U,$ and $f$ is analytic in $U,$ then
$$\frac{1}{\pi r^2}\int_{D(z,r)} f\, dA = f(z).$$
This is the area mean value property, which holds for any harmonic function.
Answer to question (0): We can prove this for any $f$ analytic on $D(0,1).$ Let $U= D(0,1/2)\setminus \{0\}.$  Choose closed discs $\overline {D(z_n,r_n)}$ contained in $U$ as in lemma 1. Using both lemmas, we get
$$f(0) = \frac{1}{\pi (1/2)^2}\int_U f\, dA = \frac{1}{\pi (1/2)^2}\sum_{n=1}^{\infty}\int_{D(z_n,r_n)} f\, dA = \frac{1}{\pi (1/2)^2}\sum_{n=1}^{\infty}\pi r_n^2 f(z_n).$$
So there we have it, with the $z_n$'s being the centers of those discs, and $c_n = 4r_n^2$ for each $n.$
I haven't thought very much about the other questions yet.
