Boundedness of zeros of Sobolev orthogonal polynomials via generalised eigenvalues

Abstract

In this work we study the asymptotic behaviour of generalised eigenvalues between infinite Hermitian definite positive matrices in the context of the location of the zeros of Sobolev orthogonal polynomials. To achieve this, we introduce the matrix Sobolev inner products associated with a set of infinite Hermitian positive-definite matrices that generalise a type of Sobolev inner product. This general framework allows us to study the boundedness of the multiplication operator on the polynomial space, which is a sufficient condition for the boundedness of zeros of orthogonal polynomials via this matrix approach. We also provide a criterion to ensure the boundedness of the zeros of Sobolev orthogonal polynomials in terms of the generalised eigenvalues introduced in Escribano et al. (2023).