Zeros and critical points of Sobolev orthogonal polynomials
In this important and interesting paper, the authors consider Sobolev orthogonal polynomials \(\{p_n\}\), that is polynomials orthogonal with respect to the inner product \[ (f, g):= \int fgd\mu_0+ \int f'g'd\mu_1, \] where \(\mu_0\) and \(\mu_1\) are compactly supported positive measures on the real line, with finite total mass. The authors place two main hypotheses: (I) The support \(\Sigma_j\) of the measure \(\mu_j\) is compact and regular for the Dirichlet problem in \(\mathbb{C}\setminus \Sigma_j\), \(j= 0,1\); (II) The measures \(\mu_j\) are regular in the sense that for every sequence of polynomials \(\{P_n\}\), where \(P_n\) is of degree \(\leq n\), we have \[ \lim_{n\to \infty} \Biggl({\| P_n\|_{L_\infty(\Sigma_j)}\over \| P_n\|_{L_2(\mu_j)}}\Biggr)^{1/n}= 1. \] Then they show that the sequence of zero counting measures of \(p_n'\) (the derivatives of the orthogonal polynomials with respect to the Sobolev inner product) converges weakly to the equilibrium measure for \(\Sigma_1\cup \Sigma_2\). From this, they deduce the zero distribution of the \(\{p_n\}\) themselves. This paper points the way for all future research in asymptotics of Sobolev orthogonal polynomials.
scientific article; zbMATH DE number 1076156