The Carath\'eodory-Fej\'er interpolation problem for the polydisc

TL;DR

This paper presents an algorithm for solving the Carathéodory-Fejér interpolation problem on the polydisc, providing necessary conditions, a generalization of Nehari's theorem, and a new proof of the Koraný-Pukánszky theorem.

Contribution

It introduces a new algorithm for the interpolation problem on the polydisc and generalizes Nehari's theorem, along with a spectral theorem-based proof of Koraný-Pukánszky.

Findings

Algorithm for the interpolation problem whenever a solution exists

Necessary condition for solution existence derived from the algorithm

Generalization of Nehari's theorem and a spectral proof of Koraný-Pukánszky theorem

Abstract

We give an algorithm for finding a solution to the Carath\'{e}odory-Fej\'{e}r interpolation problem on the polydisc $\mathbb D^n,$ whenever it exists. A necessary condition for the existence of a solution becomes apparent from this algorithm. A generalization of the well-known theorem due to Nehari has been obtained. A proof of the Kor\'{a}nyi--Puk\'{a}nszky theorem is given using the spectral theorem.