Zeros of Sobolev orthogonal polynomials of Hermite type

摘要

Let {Sn}n denote a set of polynomials orthogonal with respect to the Sobolev inner product〈f,g〉S=∫f(x)g(x)dψ0(x)+λ∫f′(x)g′(x)dψ1,where λ>0 and {dψ0,dψ1} is a so-called symmetrically coherent pair with dψ0 or dψ1 the Hermite measure e−x2dx. If dψ1 is the Hermite measure, then Sn has n different, real zeros. If dψ0 is the Hermite measure, then Sn has at least n−2 different, real zeros. We determine conditions for Sn to have complex zeros.