Asymptotics of Sobolev orthogonal polynomials for symmetrically coherent pairs of measures with compact support

We study the strong asymptotics for the sequence of monic polynomials Qn(x), orthogonal with respect to the inner product (f,g)s=∫ f(x)g(x)dμ1(x)+σ ∫ f′(x)g′(x)dμ2(x), σ > 0, with x outside of the support of the measure μ2. We assume that μ1 and μ2 are symmetric and compactly supported measures on R satisfying a coherence condition. As a consequence, the asymptotic behaviour of (Qn,Qn)s and of the zeros of Qn is obtained.