Boundary interpolation and rigidity for generalized Nevanlinna functions

Abstract

We solve a boundary interpolation problem at a real point for generalized Nevanlinna functions, and use the result to prove uniqueness theorems for generalized Nevanlinna functions.

Truncated moment problems in the class of generalized Nevanlinna functions

“…Now, let \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\psi \in \mathbf {N}_{\kappa }$\end{document} be such that Then φ and ψ both are solutions of the problem IP κ ( s , 2 n ) and hence ψ(λ) ≡ φ(λ). This proves the rigidity result for generalized Nevanlinna functions obtained in 5 and proved originally by Burns and Krantz for functions in the Schur class in 8.…”

“…In the odd case a similar description of the sets \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal M}_\kappa ({\bf s},\ell )$\end{document} and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal I}_\kappa ({\bf s},\ell )$\end{document} is given in Theorem 5.1, with the parameter τ ranging over the classes \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {N}_{\kappa -\nu _-(S_n),1}$\end{document} and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {N}_{\kappa -\nu _-(S_n)}$\end{document}, respectively, and satisfying the condition (O). It should be mentioned, that this result for \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal I}_\kappa ({\bf s},\ell )$\end{document} can be derived also from the recent paper 5 on boundary interpolation in generalized Nevanlinna classes.…”

Indefinite Inner Product Spaces, Schur Analysis, and Differential Equations

“…Heinz proposed that Aad, Henk, their student Piet Bruinsma and this editor consider the interpolation problem using Krein's formula and the theory of resolvent matrices for the description of the self-adjoint extensions of a given Hermitian operator (see [22,33]); this led in particular to the publications [4,5]. A bit later, collaboration between Aad, Heinz and DA began (mainly on the Schur algorithm for generalized Schur functions) and lead to seventeen publications, some of them written in collaboration with Thomas Azizov, R. Buursema, Simeon Reich, David Shoikhet, Yuri Shondin, Dan Volok, and Gerald Wanjala; see for instance [2,3,8,9,10]. The encounter, and the subsequent collaboration with Heinz was fascinating on numerous grounds.…”

Convex cones of generalized positive rational functions and the Nevanlinna–Pick interpolation

“…Within GP, variants of this interpolation problem are well studied, see e.g. [3], [10], [5], [18], [29,Section 3] and [4] for generalized Nevanlinna functions and for generalized Schur functions see e.g. [13], [15], [19], and [30] and [2].…”

“…Nevanlinna-Pick interpolation problem of generalized Schur and Nevanlinna functions has been well addressed in the literature: For generalized Nevanlinna functions see e.g. [3], [10], [5], [18], [29,Section 3] and [4]. For generalized Schur functions see e.g.…”