Asymptotics of polynomials orthogonal with respect to a discrete-complex Sobolev inner product

摘要

Let μ be a finite positive Borel measure supported on a compact set of the real line and introduce the discrete Sobolev-type inner product〈f,g〉=∫f(x)g(x)dμ(x)+∑k=1K∑i=0NkMk,if(i)(ck)g(i)(ck),where the mass points ck belong to supp(μ) and Mk,i are complex numbers such that Mk,Nk≠0. In this paper we investigate the asymptotics of the polynomials orthogonal with this product. When the mass points ck belong to C⧹supp(μ), the problem was solved in a paper by G. López, et al. (Constr. Approx. 11 (1995) 107–137) and, for mass points in supp(μ)=[-1,1], the solution was given by I.A. Rocha et al. (J. Approx. Theory, 121 (2003) 336–356) provided that μ′(x)>0 a.e. x∈[-1,1] and Mk,i are nonnegative constants. If μ∈M(0,1), the possibility ck∈supp(μ)⧹[-1,1] must be considered. Here we solve this last case with complex constants Mk,i.