Asymptotics of Sobolev orthogonal polynomials for a Jacobi weight
The asymptotic behaviour is obtained for polynomials which are orthogonal with respect to the Sobolev inner product \[ \langle f,g \rangle_S = \int_{-1}^1 f(x)g(x) d\mu(x) + \int_{-1}^1 f'(x)g'(x) (1-x)^\alpha(1+x)^\beta dx, \] where \(\mu\) satisfies some mild conditions. First the authors show how the norm \(\tilde{k}_n\) of the monic Sobolev orthogonal polynomial \(Q_n\) of degree \(n\) is related to the norm of the monic Jacobi polynomial \(P_{n-1}^{(\alpha,\beta)}\) of degree \(n-1\) (Theorem 1), and then, assuming that \(\mu\) is in the Szegő class, the relative asymptotic behaviour of \(Q_n(x)/P_n^{(\alpha,\beta)}(x)\) is given for \(x\) in the complex plane, away from the interval \([-1,1]\) (Theorem 2), which as a corollary gives the asymptotic distribution of the zeros of the Sobolev orthogonal polynomials.
scientific article; zbMATH DE number 1143920