The bitangential matrix Nevanlinna-Pick interpolation problem revisited

TL;DR

This paper revisits four different mathematical approaches to the bi-tangential Nevanlinna-Pick interpolation problem on the right half plane, establishing their equivalence and extending results to generalized Schur classes.

Contribution

It provides new proofs of equivalence among four solution criteria and extends the interpolation theory to Kre
28n-Langer generalized Schur classes.

Findings

All four approaches yield equivalent solution criteria.

Alternative direct proofs of the equivalence are provided.

Results are extended to the Kre
28n-Langer generalized Schur class.

Abstract

We revisit four approaches to the BiTangential Operator Argument Nevanlinna-Pick (BTOA-NP) interpolation theorem on the right half plane: (1) the state-space approach of Ball-Gohberg-Rodman, (2) the Fundamental Matrix Inequality approach of the Potapov school, (3) a reproducing kernel space interpretation for the solution criterion, and (4) the Grassmannian/Kre\u{\i}n-space geometry approach of Ball-Helton. These four approaches lead to three distinct solution criteria which therefore must be equivalent to each other. We give alternative concrete direct proofs of each of these latter equivalences. In the final section we show how all the results extend to the case where one seeks to characterize interpolants in the Kre\u{\i}n-Langer generalized Schur class $\cS_{\kappa}$ of meromorphic matrix functions on the right half plane, with the integer $\kappa$ as small as possible.