Classical Sobolev orthogonal polynomials: eigenvalue problem

Abstract

We consider the discrete Sobolev inner product $$(f,g{)}_S=\int f(x)g(x)d\mu +M{f}^{(j)}(c)g^{(j)}(c),j\in \mathbb{N}\cup \{0\},c\in \mathbb{R},M>0,$$ 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.

This is a pre-print of an article published in Results in Mathematics. The final authenticated version is available online at: https://doi.org/10.1007/s00025-019-1069-9