On asymptotic properties of Freud–Sobolev orthogonal polynomials

AbstractIn this paper we consider a Sobolev inner product (∗)(f,g)S=∫fgdμ+λ∫f′g′dμand we characterize the measures μ for which there exists an algebraic relation between the polynomials, {Pn}, orthogonal with respect to the measure μ and the polynomials, {Qn}, orthogonal with respect to (∗), such that the number of involved terms does not depend on the degree of the polynomials. Thus, we reach in a natural way the measures associated with a Freud weight. In particular, we study the case dμ=e−x4dx supported on the full real axis and we analyze the connection between the so-called Nevai polynomials (associated with the Freud weight e−x4) and the Sobolev orthogonal polynomials Qn. Finally, we obtain some asymptotics for {Qn}. More precisely, we give the relative asymptotics {Qn(x)/Pn(x)} on compact subsets of C⧹R as well as the outer Plancherel–Rotach-type asymptotics {Qn(n4x)/Pn(n4x)} on compact subsets of C⧹[−a,a] being a=4/34