Asymptotic properties of balanced extremal Sobolev polynomials: Coherent case
Consider a Sobolev inner product of the form \[ (p,q)=\int pq d\mu_0+\int p'q' d\mu_1. \] If \(\{P_n\}\) and \(\{Q_n\}\) denote the sequences of monic orthogonal polynomials with respect to the measures \(d\mu_0\) and \(d\mu_1\), respectively, then \((d\mu_0,d\mu_1)\) is called a coherent pair of measures if there exist non-zero constants \(D_n\) such that \[ Q_n=\frac{P_{n+1}'}{n+1}+D_n\frac{P_{n-1}'}{n-1},\quad n\geq 2. \] In [J. Approximation Theory 89, No. 3, 321-343 (1997; Zbl 0880.42012)] \textit{H. G.~Meijer} classified all coherent pairs of measures. In the present paper this result is used to study the case that \(\text{supp}(d\mu_0)=[-1,1]\). In that case the asymptotic properties of the Sobolev orthogonal polynomials are studied. Also the behaviour of the norms and the zeros of these polynomials is discussed. It is shown that in some case these Sobolev orthogonal polynomials asymptotically behave like the monic polynomial sequence of polynomials which is orthogonal with respect to a new measure being a linear combination of the measures \(d\mu_0\) and \(d\mu_1\). The authors conjecture that this result is still valid in a more general setting.
scientific article; zbMATH DE number 1356330