H.G. Meijer

An asymptotic result for Laguerre—Sobolev orthogonal polynomials

Abstract Let { S n } denote the sequence of polynomials orthogonal with respect to the Sobolev inner product (f,g)s = ∫ 0 +∞ f(x)g(x)x α e −x d x+λ ∫ 0 +∞ f′(x)g′(x)x α e −x d x where α > − 1, λ > 0 and the leading coefficient of the S n is equal to the leading coefficient of the Laguerre polynomial L n ( α ) . Then, if x ∈C s [0,+∞), lim n→∞ S n (x) L n (α−1) (x) is a constant depending on λ.

On real and complex zeros of orthogonal polynomials in a discrete Sobolev space

Abstract Let {Sn(x; c, N)} denote a set of polynomials orthogonal with respect to the discrete Sobolev inner product 〈f,g〉 = ʃ ∞ -∞ f(x)g(x)dψ(x)+Nf ′ (c)g ′ (c) , where N ⩾ 0, c ϵ R . For N = 0, put Kn(x) = Sn(x; ·, 0). Then Sn(x; c, N) has at least n − 2 different real zeros; their position with respect to the zeros of Kn can be determined using the tangent to the graph of y = Kn(x) in (c, Kn(c)). On the other hand, if n ⩾ 3, then c can be chosen such that Sn(x; c, N) has two complex zeros if N is sufficiently large.

Asymptotics of Sobolev orthogonal polynomials for coherent pairs of Jacobi type

Abstract Let {Sn}n denote a sequence of polynomials orthogonal with respect to the Sobolev inner product (f,g) S =∫f(x)g(x) d ψ 0 (x)+λ∫f′(x)g′(x) d ψ 1 (x) where λ>0 and {dψ0,dψ1} is a so-called coherent pair with at least one of the measures dψ0 or dψ1 a Jacobi measure. We investigate the asymptotic behaviour of Sn(x), for n→+∞ and x fixed, x∈ C ⧹[−1,1] as well as x∈(−1,1).