Optimal solutions to matrix-valued Nehari problems and related limit theorems

TL;DR

This paper extends Helton and Young's scalar Nehari problem approximation results to matrix-valued cases, using relaxed commutant lifting theory to analyze optimal solutions and convergence rates.

Contribution

It introduces a matrix-valued Nehari problem framework and demonstrates how to approximate solutions via restricted problems solved with relaxed commutant lifting theory.

Findings

Extended scalar Nehari approximation to matrix-valued setting.

Established convergence properties depending on initial space choice.

Provided a method for efficient approximation of matrix-valued Hankel operators.

Abstract

In a 1990 paper Helton and Young showed that under certain conditions the optimal solution of the Nehari problem corresponding to a finite rank Hankel operator with scalar entries can be efficiently approximated by certain functions defined in terms of finite dimensional restrictions of the Hankel operator. In this paper it is shown that these approximants appear as optimal solutions to restricted Nehari problems. The latter problems can be solved using relaxed commutant lifting theory. This observation is used to extent the Helton and Young approximation result to a matrix-valued setting. As in the Helton and Young paper the rate of convergence depends on the choice of the initial space in the approximation scheme.