Generalized Sobolev orthogonal polynomials, matrix moment problems and integrable systems

TL;DR

This paper develops a comprehensive theory of Sobolev bi-orthogonal polynomials linked to matrix moment problems, explores their deformations, and connects them to integrable systems through evolution equations.

Contribution

It introduces a broad class of Sobolev bi-orthogonal polynomials, establishes their deformation theory, and relates these to integrable hierarchies via matrix factorizations.

Findings

New class of Sobolev bi-orthogonal polynomials introduced

Deformation theory of Sobolev bilinear forms developed

Integrable hierarchies derived from moment matrix deformations

Abstract

We introduce a large class of Sobolev bi-orthogonal polynomial sequences arising from a $LU$-factorizable moment matrix and associated with a suitable measure matrix that characterizes the Sobolev bilinear form. A theory of deformations of Sobolev bilinear forms is also proposed. We consider both polynomial deformations and a class of transformations related to the action of linear operators on the entries of a given bilinear form. Transformation formulae among new and old polynomial sequences are determined. Finally, integrable hierarchies of evolution equations arising from the factorization of a time deformation of the moment matrix are presented.