Inverse spectral analysis for a class of finite band symmetric matrices

TL;DR

This paper develops a new inverse spectral analysis method for finite band symmetric matrices, providing conditions and algorithms for reconstructing matrices from spectral data, especially for cases not handled by existing methods.

Contribution

It introduces a novel reconstructive algorithm based on rational interpolation theory for finite band symmetric matrices, expanding the scope of inverse spectral analysis.

Findings

Provides necessary and sufficient conditions for spectral functions

Develops an algorithm applicable to previously intractable matrices

Extends inverse spectral analysis techniques using rational interpolation

Abstract

In this note, we solve an inverse spectral problem for a class of finite band symmetric matrices. We provide necessary and sufficient conditions for a matrix valued function to be a spectral function of the operator corresponding to a matrix in our class and give an algorithm for recovering this matrix from the spectral function. The reconstructive algorithm is applicable to matrices which cannot be treated by known inverse block matrix methods. Our approach to the inverse problem is based on the rational interpolation theory developed in a previous paper.