Zero location and \(n\)th root asymptotics of Sobolev orthogonal polynomials (Q1300142)

The authors consider the sequence of polynomials which are orthogonal with respect to the inner product defined by \[ \langle p,q\rangle_s= \sum^m_{k= 0} \int p^{(k)}(x) q^{(k)}(x) d\mu_k(x). \] Here \(\{\mu_k\}\) denotes a set of finite positive Borel measure such that the support \(\Delta_k\) of \(\mu_k\) \((k= 0,1,\dots, m)\) is a compact subset of the real line. Their main aim is to show that the zeros of these polynomials are all contained in the disk \(\{z:|z|\leq 2\|M\|\}\), where \(\|M\|\) stands for the norm of the multiplication operator \(Mf= xf\) computed according to the above-mentioned inner product. They give also an asymptotic expression for the normalized zero counting measure defined by \(\nu(q)= (1/n) \sum^n_{j=1} \delta_j\), where \(q\) stands for any polynomial of (exact) degree \(n\) with zeros at \(z_j\) \((j= 1,\dots, n)\) while \(\delta_j\) denotes the Dirac measure with mass one at the point \(z_j\).