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An infinite Blaschke product whose zeros accumulate at the boundary

Example

Let $a_{n} : = 1 - \frac{1}{n^{2}}$ for $n \geq 2$ . Then $\sum_{n \geq 2} (1 - ∣ a_{n} ∣) = \sum_{n \geq 2} \frac{1}{n^{2}} < \infty$ , so $(a_{n})$ is a Blaschke sequence and the Blaschke product $B = \prod_{n \geq 2} b_{a_{n}}$ converges normally on $D$ to a holomorphic function with $∣ B ∣ \leq 1$ , whose zeros are exactly the points $1 - 1 / n^{2}$ , each simple, and whose boundary function is unimodular $m$ -almost everywhere. The value at the origin is $B (0) = \prod_{n \geq 2} (1 - \frac{1}{n^{2}}) = \lim_{N \to \infty} \frac{N + 1}{2 N} = \frac{1}{2} .$ The zeros accumulate at $1 \in T$ , so $B$ has no continuous extension to $\overline{D}$ and is not a finite product. The associated $H^{2}$ norm is $∥ B ∥_{H^{2}} = 1$ , while the boundary function has modulus $1$ almost everywhere.

Facts & Assumptions

Given: The sequence $a_{n} = 1 - 1 / n^{2}$ ( $n \geq 2$ ), its Blaschke product $B = \prod_{n \geq 2} b_{a_{n}}$ , and the partial products $B_{N} = \prod_{n = 2}^{N} b_{a_{n}}$ .

[F1]

A Blaschke sequence $\sum_{n} (1 - ∣ a_{n} ∣) < + \infty$ has a normally convergent Blaschke product $B$ , holomorphic with $∣ B ∣ \leq 1$ , whose zeros are exactly the $a_{n}$ with multiplicity and whose boundary function satisfies $∣ B^{*} ∣ = 1$ $m$ -almost everywhere; each $b_{a}$ has $b_{a} (0) = ∣ a ∣$ (Blaschke factors and Blaschke products, Boundary values and zeros of a Blaschke product).

[F2]

For rational $p > 0$ the $p$ -series $\sum_{k \geq 1} 1 / k^{p}$ converges if and only if $p > 1$ ; in particular $\sum_{k \geq 1} 1 / k^{2}$ converges at $p = 2$ , and deleting the first term preserves convergence (For rational $p > 0$ , $\sum 1 / k^{p}$ converges iff $p > 1$ ).

[F3]

If $∣ g_{r} ∣ \leq 1$ for all $r$ and $g_{r} \to g_{*}$ $m$ -almost everywhere as $r ↑ 1$ , then $\int_{T} g_{r} d m \to \int_{T} g_{*} d m$ ; the radial means $\int_{T} ∣ g (r ζ) ∣^{2} d m$ are nondecreasing in $r$ and their supremum is $∥ g ∥_{H^{2}}^{2}$ (Dominated convergence, Radial p-means of a holomorphic function are nondecreasing, Analytic Hardy spaces on the unit disc, The one-dimensional torus and its normalized Haar integral).

Verification

1.1givenF1F2algebra

The sequence is Blaschke. Since $∣ a_{n} ∣ = 1 - 1 / n^{2}$ , one has $1 - ∣ a_{n} ∣ = 1 / n^{2}$ and $\sum_{n \geq 2} 1 / n^{2} < \infty$ by [F2]; by [F1] the product $B$ converges normally, is holomorphic with $∣ B ∣ \leq 1$ , has exactly the simple zeros $1 - 1 / n^{2}$ , and has $∣ B^{*} ∣ = 1$ $m$ -almost everywhere.

2.1step 1.1F1algebra

Value at the origin. By [F1], $B (0) = \prod_{n \geq 2} ∣ a_{n} ∣ = \prod_{n \geq 2} (1 - \frac{1}{n^{2}})$ , and the finite products telescope: using $1 - \frac{1}{n^{2}} = \frac{(n - 1) (n + 1)}{n^{2}}$ , $\prod_{n = 2}^{N} (1 - \frac{1}{n^{2}}) = \frac{(N - 1)! \cdot (N + 1)! / 2}{(N!)^{2}} = \frac{N + 1}{2 N} ⟶ \frac{1}{2} .$

2.2step 1.1F1algebra

No continuous extension. The zeros $a_{n} = 1 - 1 / n^{2}$ converge to $1 \in T$ . If $B$ had a continuous extension to $\overline{D}$ , then along $a_{n} \to 1$ one would get $∣ B (1) ∣ = \lim_{n} ∣ B (a_{n}) ∣ = 0$ , while on the other hand the boundary values of the extension agree $m$ -almost everywhere with $B^{*}$ , so the continuous function $∣ B ∣$ restricted to $T$ equals $1$ on a set of full measure, hence equals $1$ everywhere by continuity; at $ζ = 1$ this gives $∣ B (1) ∣ = 1$ , a contradiction. Thus $B$ has no continuous extension to $\overline{D}$ ; in particular $B$ is not a finite Blaschke product, since a finite product would extend continuously.

3.1step 1.1F3algebra∎

The $H^{2}$ norm. Since $∣ B ∣ \leq 1$ and $B (r ζ) \to B^{*} (ζ)$ for $m$ -almost every $ζ$ (the nontangential limits of [F1] in particular give radial limits almost everywhere), [F3] gives $\int_{T} ∣ B (r ζ) ∣^{2} d m (ζ) \to \int_{T} ∣ B^{*} ∣^{2} d m = 1$ as $r ↑ 1$ . The radial means are nondecreasing in $r$ by [F3], so their supremum is the limit, that is, $∥ B ∥_{H^{2}}^{2} = 1$ and hence $∥ B ∥_{H^{2}} = 1$ .

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Sources

R. K. Srivastava, Lecture Notes on Hardy Spaces (MA650, IIT Guwahati), §5.8 (standard reference, not scraped)
J. B. Garnett, Bounded Analytic Functions, revised first edition, Chapter II §2 (standard reference, not scraped)