On modified kernel polynomials and classical type Sobolev orthogonal polynomials

TL;DR

This paper investigates modified kernel polynomials constructed from orthonormal polynomials, revealing their recurrence relations, Sobolev orthogonality, and differential equations, especially for Jacobi and Laguerre cases.

Contribution

It introduces a new class of modified kernel polynomials with adjustable parameters, establishing their recurrence relations, Sobolev orthogonality, and differential equations.

Findings

All modified kernel polynomials satisfy a 4th order recurrence relation.

Certain parameter choices lead to Sobolev orthogonal polynomials with matrix measures.

Some polynomials satisfy differential equations and generalized eigenvalue problems.

Abstract

In this paper we study modified kernel polynomials: $u_n(x) = \sum_{k=0}^n c_k g_k(x)$, depending on parameters $c_k>0$, where $\{ g_k \}_0^\infty$ are orthonormal polynomials on the real line. Besides kernel polynomials with $c_k = g_k(t_0)>0$, for example, $c_k$ may be chosen to be some other solutions of the corresponding second-order difference equation of $g_k$. It is shown that all these polynomials satisfy a $4$-th order recurrence relation. The cases with $g_k$ being Jacobi or Laguerre polynomials are of a special interest. Suitable choices of parameters $c_k$ imply $u_n$ to be Sobolev orthogonal polynomials with a $(3\times 3)$ matrix measure. Moreover, a further selection of parameters gives differential equations for $u_n$. In the latter case, polynomials $u_n(x)$ are solutions to a generalized eigenvalue problems both in $x$ and in $n$.