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 \(e^{i\alpha} \in \mathbb{T} \setminus K\).


30E05 Moment problems and interpolation problems in the complex plane
30J10 Blaschke products
Full Text: DOI Euclid