Sequentially Ordered Sobolev Inner Product and Laguerre–Sobolev Polynomials

Abstract

We study the sequence of polynomials {Sn}n≥0 that are orthogonal with respect to the general discrete Sobolev-type inner product ⟨f,g⟩s=∫f(x)g(x)dμ(x)+∑j=1N∑k=0djλj,kf(k)(cj)g(k)(cj), where μ is a finite Borel measure whose support suppμ is an infinite set of the real line, λj,k≥0, and the mass points ci, i=1,…,N are real values outside the interior of the convex hull of suppμ (ci∈R\Ch(supp(μ))∘). Under some restriction of order in the discrete part of ⟨·,·⟩s, we prove that Sn has at least n−d* zeros on Ch(suppμ)∘, being d* the number of terms in the discrete part of ⟨·,·⟩s. Finally, we obtain the outer relative asymptotic for {Sn} in the case that the measure μ is the classical Laguerre measure, and for each mass point, only one order derivative appears in the discrete part of ⟨·,·⟩s.