Boundary interpolation with Blaschke products of minimal degree
The classical Nevanlinna-Pick interpolation problem is to find functions \(f\), analytic in the unit disc \(\mathbb D\), which are bounded by \(1\) and satisfying the interpolation conditions \(f(z_i)=w_i\) (\(i=1,\dots,n\)) for given points \(z_1,\dots,z_n\) in the open unit disk \({\mathbb D}\) and complex numbers \(w_1,\dots,w_n\). Nevanlinna and Pick provided necessary and sufficient conditions for the solvability of the problem [see \textit{J. B. Garnett}, ``Bounded analytic functions.'' Grad. Texts Math. 236 (2006; Zbl 1106.30001)]. We remark that when the problem is solvable it has a solution which is a finite Blaschke product of degree at most \(n\). In this paper the author considers a natural modification of the Nevanlinna-Pick problem for Blaschke products. Namely, he places all the interpolation points \(z_k\) in the unit circle \(\mathbb T\). Since Blaschke products are unimodular on \(\mathbb T\), the interpolations values \(w_k\) must also lie on \(\mathbb T\). That is, the paper deals with the following boundary interpolation problem: Given \(n\) distinct points \(z_ 1,\dots,z_ n\in\mathbf C\) and \(n\) points \(w_ 1,\dots,w_ n\in\mathbf C\) with \(| z_ j| =| w_ j| =1, j=1,\dots, n\), find a finite Blaschke product \(B\) satisfying the interpolation condition \(B(z_ j)=w_ j, j=1,\dots,n\). \textit{D. G. Cantor} and \textit{R. R. Phelps} [Proc. Am. Math. Soc. 16, 523--525 (1965; Zbl 0146.09604)] proved that the problem is always solvable, \textit{R. Younis} [Proc. Am. Math. Soc. 78, No. 3, 451--452 (1980; Zbl 0439.30020)] proved that there is always a solution of degree at most \(n^2\) and \textit{W. B. Jones} and \textit{S. Ruscheweyh} [Constr. Approx. 3, No. 4, 405--409 (1987; Zbl 0634.41007)] improved this showing that there is always a solution of degree at most \(n-1\). The authors split the boundary interpolation problems into the following three classes depending on how the minimal possible degree \(m\) of a solution is related to the number \(n\) of interpolation conditions: \[ (\text{a})\,\, 2m<n-1;\quad (\text{b})\,\, 2m=n-1;\quad (\text{c})\,\, 2m>n-1. \] They study the properties of these classes. In particular, they prove that the boundary interpolation problem has a unique minimal degree solution if and only if \(2m\leq n-1\). They show also that in case (b), the unique solution depends continuously on the interpolation. Furthermore, the authors present an algorithm for determining to which class a given problem belongs. This algorithm also gives the value of \(m\) in many cases.
scientific article; zbMATH DE number 5152630
scientific article