Relative asymptotics for polynomials orthogonal with respect to a discrete Sobolev inner product

Abstract

We investigate the asymptotic properties of orthogonal polynomials for a class of inner products including the discrete Sobolev inner products $\langle h,g \rangle = \int hg\, d\mu + \sum_{j=1}^m \sum_{i=0}^{N_j} M_{j,i} h^{(i)}(c_j) g^{(i)}(c_j)$, where $\mu$ is a certain type of complex measure on the real line, and $c_j$ are complex numbers in the complement of $\supp(\mu)$. The Sobolev orthogonal polynomials are compared with the orthogonal polynomials corresponding to the measure $\mu$.