Persymmetric Jacobi matrices, isospectral deformations and orthogonal polynomials

TL;DR

This paper investigates persymmetric Jacobi matrices, their unique orthogonal polynomials, and develops efficient algorithms for reconstructing these matrices from spectral data, including analysis of their isospectral deformations.

Contribution

It introduces properties of persymmetric Jacobi matrices and their orthogonal polynomials, and presents novel inverse problem algorithms for matrix reconstruction from spectral points.

Findings

Orthogonal polynomials are orthogonal on restricted lattices.

Efficient algorithms for reconstructing persymmetric Jacobi matrices.

Explicit expressions for deformed polynomials and weights.

Abstract

Persymmetric Jacobi matrices are invariant under reflection with respect to the anti-diagonal. The associated orthogonal polynomials have distinctive properties that are discussed. They are found in particular to be also orthogonal on the restrictions either to the odd or to the even points of the complete orthogonality lattice. This is exploited to design very efficient inverse problem algorithms for the reconstruction of persymmetric Jacobi matrices from spectral points. Isospectral deformations of such matrices are also considered. Expressions for the associated polynomials and their weights are obtained in terms of the undeformed entities.