## Blaschke products and Nevanlinna-Pick interpolation.(English)Zbl 1337.30049

Let $$H(\mathbb{D})$$ be the set of all holomorphic functions $$f$$ in the unit disk $$\mathbb{D}$$ with $$|f(z)| \leq 1$$ for all $$z \in \mathbb{D}$$. Assume that the Nevanlinna-Pick interpolation problem $f(z_n)=w_n\,, \quad n=1,2,3,\dotsc\,, \quad f \in H(\mathbb{D}) \tag{$$*$$}$ has at least two different solutions. If $$E$$ denotes the set of all solutions of $$(*)$$, then R. Nevanlinna found functions $$P$$, $$Q$$, $$R$$, $$S \in H(\mathbb{D})$$ such that $$E$$ can be parametrized by $$H(\mathbb{D})$$ in the form $E = \left\{\, \frac{P-Qw}{R-Sw} : w \in H(\mathbb{D}) \,\right\}\,.$ Furthermore, if $$w \equiv e^{i\alpha}$$, $$\alpha \in [0,2\pi)$$, then the corresponding solution $$I_\alpha$$ is an inner function, and the range $\Delta(z) = \{\, f(z) : f \in H(\mathbb{D})\,, \text{$$f$$ solves $$(*)$$} \,\}$ is a disk and the boundary of $$\Delta(z)$$ is in a one-to-one correspondence to $$\{\, I_\alpha(z) : \alpha \in [0,2\pi) \,\}$$. For this reason the set $$\{\, I_\alpha : \alpha \in [0,2\pi) \,\}$$ is also called the set of extremal solutions of $$(*)$$.
In [J. Lond. Math. Soc., II. Ser. 32, 488–496 (1985; Zbl 0595.30048)] and [Bull. Lond. Math. Soc. 20, No. 4, 329–332 (1988; Zbl 0643.30029)] the author proved that $$I_\alpha$$ is a Blaschke product for almost all $$\alpha$$, and the exceptional set of $$\alpha$$-values has zero logarithmic capacity. If the interpolation problem $$(*)$$ has only finitely many solutions, then every $$I_\alpha$$ is a Blaschke product. In the paper under review the author continues this work and proves the following results. He gives a rather precise condition on the sequence $$(z_n)$$ such that for any indeterminate problem $$(*)$$ all $$I_\alpha$$ are Blaschke products. Furthermore, he shows that for any closed set $$K$$ of zero logarithmic capacity on the unit circle $$\mathbb{T}$$, there exists a problem $$(*)$$ such that $$I_\alpha$$ is a Blaschke product if and only if $$e^{i\alpha} \in \mathbb{T} \setminus K$$.

### MSC:

 30E05 Moment problems and interpolation problems in the complex plane 30J10 Blaschke products

### Citations:

Zbl 0595.30048; Zbl 0643.30029
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