Some asymptotics for Sobolev orthogonal polynomials involving Gegenbauer weights

摘要

We consider the Sobolev inner product 〈f,g〉=∫−11f(x)g(x)(1−x2)α−12dx+∫f′(x)g′(x)dψ(x),α>−12, where dψ is a measure involving a Gegenbauer weight and with mass points outside the interval (−1,1). We study the asymptotic behaviour of the polynomials which are orthogonal with respect to this inner product. We obtain the asymptotics of the largest zeros of these polynomials via a Mehler–Heine type formula. These results are illustrated with some numerical experiments.