Asymptotics of Sobolev orthogonal polynomials for coherent pairs of measures
The orthogonal polynomials \(Q_n\) for the Sobolev inner product \[ \langle f,g \rangle_S = \int f(x)g(x) d\mu_1(x) + \int f'(x) g'(x) d\mu_2(x), \] are investigated when \((\mu_1,\mu_2)\) is a coherent pair of measures with compact support. All coherent pairs of measures were obtained by \textit{H.G. Meijer} [J. Approximation Theory 89, No. 3, 321-343 (1997; Zbl 0880.42012)] and those with compact support are all related to Jacobi polynomials. If \(T_n\) are the orthogonal polynomials for the second measure \(\mu_2\), then it is shown that \(\lim_{n \to \infty} Q_n(x)/T_n(x) = 1/\Phi'(x)\), uniformly on compact sets of \({\mathbb{C}} \setminus [-1,1]\), where \(\Phi(x) = (x+\sqrt{x^2-1})/2\) is the conformal mapping that maps the exterior of the interval \([-1,1]\) to the exterior of the unit circle.
scientific article; zbMATH DE number 1146512