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

  • $\begingroup$ Have you checked with the book "$H^p$-spaces" by P. Duren? $\endgroup$ – AD. Sep 23 '10 at 20:35
  • $\begingroup$ Yes, I have, but I couldn't find anything about this. $\endgroup$ – Malik Younsi Sep 24 '10 at 0:19

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.

  • $\begingroup$ Another relevant reference is "Radial limits and star invariant subspaces of bounded mean oscillation" by William S. Cohn, which contains necessary and sufficient conditions for radial limits to exist at a particular point for a Blaschke product and all of its subproducts. However, I do not see any conditions there to guarantee that a particular Blaschke product does not have a radial limit at a particular point. $\endgroup$ – Jonas Meyer Jan 13 '11 at 22:25
  • $\begingroup$ Thank you very much, I will take a look at those very interesting references. $\endgroup$ – Malik Younsi Jan 21 '11 at 1:15
  • $\begingroup$ @Malik: You're welcome. If you find a more complete answer, e.g., in the Bagemihl and Seidel reference, please share. $\endgroup$ – Jonas Meyer Jan 21 '11 at 5:52
  • $\begingroup$ Has this problem by any chance been solved ? $\endgroup$ – dezdichado Apr 22 '16 at 13:56

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 $r \to 1^{-}$.


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