Notes on interpolation in the generalized Schur class. II. Nudel’man’s problem

Abstract

Abstract. An indefinite generalization of Nudel man's problem is used in a systematic approach to interpolation theorems for generalized Schur and Nevanlinna functions with interior and boundary data. Besides known results on existence criteria for Pick-Nevanlinna and Carathéodory-Fejér interpolation, the method yields new results on generalized interpolation in the sense of Sarason and boundary interpolation, including properties of the finite Hilbert transform relative to weights. The main theorem appeals to the Ball and Helton almost-commutant lifting theorem to provide criteria for the existence of a solution to Nudel man's problem.

On Nudel’man’s Problem and Indefinite Interpolation in the Generalized Schur and Nevanlinna Classes

“…Appendix B in this work describes the problems in the proof and includes an example showing what can go wrong. We have no counterexample to the original statement of the Main Theorem in [3], but we feel that its validity is seriously in doubt. Five open questions are identified in the present work, in Problems 3.3, 4.1, 5.1, 5.3, 5.5.…”

“…Arocena, Azizov, Dijksma, and Marcantognini [6] use a theorem of Ball and Helton to prove an indefinite form of the commutant lifting theorem. This raised the possibility of finding an indefinite generalization of Nudel 1 man's problem, and such a generalization was proposed in our paper [3] with T. Constantinescu. However, the proof of the Main Theorem in [3] has gaps, which are identified in the Corrigendum [4].…”

“…This raised the possibility of finding an indefinite generalization of Nudel 1 man's problem, and such a generalization was proposed in our paper [3] with T. Constantinescu. However, the proof of the Main Theorem in [3] has gaps, which are identified in the Corrigendum [4]. Appendix B in this work describes the problems in the proof and includes an example showing what can go wrong.…”

“…Five open questions are identified in the present work, in Problems 3.3, 4.1, 5.1, 5.3, 5.5. Negative answers to any of them would provide a counterexample to the original form of the Main Theorem in [3].…”

“…This paper is a revised version of [3] that repairs the Main Theorem and shows the changes needed in the applications. Briefly, a stronger hypothesis fixes the problems in the proof of the Main Theorem.…”