On radial limits of Blaschke Products

A Blaschke product is a function of the form $$B(z):=z^k\prod_{n=1}^{\infty}\frac{a_n-z}{1-\overline{a_n}z}\frac{|a_n|}{a_n}$$ where the $a_n$ are the non-zero zeros of $B$, and satisfie $\sum_{n=1}^{\infty}(1-|a_n|) < \infty$.

Blashke products are holomorphic and bounded by 1 on the unit disk. A well known theorem asserts that $B$ has radial limits almost everywhere on the unit circle, i.e. that the limit $$\lim_{r \rightarrow 1} B(re^{i \theta})$$ exist for almost every $\theta$. I'm looking for an example of Blashke product such that the radial limit does not exist at a certain point, say $1$ for example. In particular, a Blaschke product with zeros in $(0,1)$ such that $$\limsup_{r \rightarrow 1}|B(r)| =1$$ would work.

Does anyone have a construction or reference?

Thank you, Malik

Solution 1:

There is an exercise in Rudin's Real and complex analysis whose solution would answer your question, #14 in Chapter 15:

Prove that there is a sequence $\{\alpha_n\}$ with $0\lt\alpha_n\lt1$, which tends to $1$ so rapidly that the Blaschke product with zeros at the points $\alpha_n$ satisfies the condition $$\limsup_{r\to1}|B(r)|=1.$$ Hence this $B$ has no radial limit at $z=1$.

(The previous exercise says that the limit is $0$ if $\alpha_n=1-n^{-2}$.)

Instead of trying to solve it (with the guess of something like $\alpha_n=1-4^{-n}$), I found the article "On functions with Weierstrass boundary points everywhere" by Campbell and Hwang, which says on page 510 (page 4 of the pdf):

Let $B(z)$ be the Blaschke product with zeros at $z=1-\exp(-n)$, for $n=1,2,\ldots$. Then $B(z)$ has no radial limit at $z=1$....

The authors cite page 12 of "Sequential and continuous limits of meromorphic functions" by Bagemihl and Seidel for this fact, but I do not currently have access to that article. Hopefully you can track it down to get your question answered, or perhaps someone will take up the challenge of solving Rudin's problem.

Solution 2:

Let $c_n$ be a sequence dense in $S^1$ and define $a_n = \left ( 1 + \frac{1}{n^2} \right ) c_n$. Then $1 - |z_n| = \frac{1}{n^2}$ so this are the zeros of a Blaschke product.

So, let $c_n = r e^{i \phi_n}$ and fill this in in the Blaschke product. Then we if $B_n(z,r)$ is the term inside the Blaschke product, then we can try to evaluate if $\sum 1 - B_n(z,r)$ converges. We can evaluate the convergence of this with the integral test and then we see the limit does not exist as $r \to 1^{-}$.