# Classical Sobolev orthogonal polynomials: eigenvalue problem

## TL;DR

This paper studies classical Sobolev orthogonal polynomials defined via a discrete inner product involving a classical measure and a derivative term, focusing on their eigenvalue asymptotics.

## Contribution

It derives the asymptotic behavior of eigenvalues for Sobolev orthogonal polynomials associated with classical measures and a discrete derivative term.

## Key findings

- Eigenvalues exhibit specific asymptotic growth rates.
- Orthogonal polynomials are eigenfunctions of a differential operator.
- Results extend understanding of Sobolev orthogonal polynomial spectra.

## Abstract

We consider the discrete Sobolev inner product $$(f,g)_S=\int f(x)g(x)d\mu+Mf^{(j)}(c)g^{(j)}(c), \quad j\in \mathbb{N}\cup\{0\}, \quad c\in\mathbb{R}, \quad M>0, $$ where $\mu$ is a classical continuous measure with support on the real line (Jacobi, Laguerre or Hermite). The orthonormal polynomials with respect to this Sobolev inner product are eigenfunctions of a differential operator and obtaining the asymptotic behavior of the corresponding eigenvalues is the principal goal of this paper.