I am studying Banach Spaces of Analytic Function by Hoffman. Hoffman proves the following theorem:
The Blaschke product whose zeroes are \begin{align*} \alpha_{1} , \alpha_2 , \ldots \end{align*} converges at all points $z \in \mathbb C$ except those in the compact set $K$ consisting of
- the points $z=1/\bar{\alpha}_n$;
- the points $z$ on the unit circle which are accumulation points of the sequence $(\alpha _n)$.
The convergence is uniform on any closed set $K$ in the plane which is disjoint from $K$, and the Blaschke product is thus analytic off $K$.
The author further remarks that the Blaschke product has a pole $1/\bar{\alpha}_n$ for each $n\in \mathbb N$ and an essential singularity at the accumulation point of the $\alpha _n$.
I have not been able to prove this, however, I wonder if this follows directly from the the theorem that was mentioned. For the zeroes, it directly follows from the convergence theorem, as mentioned in Ash's Complex Variables Theorem 6.1.7.. How does one go about proving that the Blaschke product has a pole of $1/\bar{\alpha}_n$?
The Blaschke product $$ B(z) = \prod_{k =1}^\infty \frac{|\alpha_k|}{\alpha_k} \frac{\alpha_k-z}{1-\bar \alpha_k z} $$ is holomorphic in the exterior of the unit disk except at the points $z=1/\bar \alpha_k$.
Now fix an index $n$ and choose $N \ge n$ so large that $a_k \ne a_n$ for $k > N$. This is always possible because the “Blaschke condition” $\sum_{k=1}^\infty (1-|a_k|) < \infty $ implies that $\lim_{k \to \infty} \lvert a_k \rvert = 1$. Then $$ B(z) = R(z) \cdot \tilde B(z) $$ where $$ R(z) = \prod_{k =1}^N \frac{|\alpha_k|}{\alpha_k} \frac{\alpha_k-z}{1-\bar \alpha_k z} $$ is a rational function, and $$ \tilde B(z) = \prod_{k =N+1}^\infty \frac{|\alpha_k|}{\alpha_k} \frac{\alpha_k-z}{1-\bar \alpha_k z} $$ is again a Blaschke product. $\tilde B$ is holomorphic in the exterior of the unit disk except at the points $1/\bar \alpha_k$, $k > N$. In particular, $\tilde B$ is holomorphic in a neighborhood of $1/\bar \alpha_n$.
So $B$ is the product of a rational function $R$ with a pole at $1/\bar \alpha_n$ and a function $\tilde B$ which is holomorphic in a neighborhood of $1/\bar \alpha_n$. It follows that $B$ has a pole at $1/\bar \alpha_n$.
More precisely, if $\alpha_n$ occurs $p$ times in the sequence $(\alpha_k)$ then $B$ has a $p$-fold pole at $1/\bar \alpha_n$.