Test functions in constrained interpolation (Q2844721)

In this paper, the authors determine a set \(\Psi\) of (complex-valued) test functions in the unit disk \(\mathbb D\) in the complex plane for \(H_1^{\infty}:=\{g\in H^{\infty}(\mathbb D): g'(0)=0\}\) in constrained interpolation problems, via a Herglotz-type representation for \(H_1^{\infty}\), which means that (1) \(\Psi\) is in \(H_1^{\infty}\) such that sup\(\{|\psi(x)|:\psi \in \Psi\}<1\) for each \(x\in \mathbb D\), and (2) whenever \(F\subset\mathbb D\) is finite set with \(n\) elements, the unital algebra generated by \(\Psi|_F\) (the restriction of \(\Psi\) to \(F\)) is \(n\)-dimensional. These test functions turn out to be rational functions and can be parametrized by the sphere. The authors show (see Theorem 9) that the set of test functions is minimal in the sense that there is no proper closed subset \(C \subset \Psi\) so that \(C\) is a set of test functions for \(H_1^{\infty}\).