Some asymptotics for Sobolev orthogonal polynomials involving Gegenbauer weights

AbstractWe 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

Asymptotics for Jacobi–Sobolev orthogonal polynomials associated with non-coherent pairs of measures

AbstractWe consider the Sobolev inner product 〈f,g〉=∫−11f(x)g(x)dψ(α,β)(x)+∫f′(x)g′(x)dψ(x), where dψ(α,β)(x)=(1−x)α(1+x)βdx with α,β>−1, and ψ is a measure involving a rational modification of a Jacobi weight and with a mass point outside the interval (−1,1). We study the asymptotic behaviour of the polynomials which are orthogonal with respect to this inner product on different regions of the complex plane. In fact, we obtain the outer and inner strong asymptotics for these polynomials as well as the Mehler–Heine asymptotics which allow us to obtain the asymptotics of the largest zeros of these polynomials. We also show that in a certain sense the above inner product is also equilibrated