7 research outputs found
The semiclassical--Sobolev orthogonal polynomials: a general approach
We say that the polynomial sequence is a semiclassical
Sobolev polynomial sequence when it is orthogonal with respect to the inner
product where is a semiclassical linear functional,
is the differential, the difference or the --difference
operator, and is a positive constant. In this paper we get algebraic
and differential/difference properties for such polynomials as well as
algebraic relations between them and the polynomial sequence orthogonal with
respect to the semiclassical functional . The main goal of this article
is to give a general approach to the study of the polynomials orthogonal with
respect to the above nonstandard inner product regardless of the type of
operator considered. Finally, we illustrate our results by
applying them to some known families of Sobolev orthogonal polynomials as well
as to some new ones introduced in this paper for the first time.Comment: 23 pages, special issue dedicated to Professor Guillermo Lopez
lagomasino on the occasion of his 60th birthday, accepted in Journal of
Approximation Theor
Asymptotics and zeros of Sobolev orthogonal polynomials on unbounded supports
In this paper we present a survey about analytic properties of polynomials
orthogonal with respect to a weighted Sobolev inner product such that the
vector of measures has an unbounded support. In particular, we are focused in
the study of the asymptotic behaviour of such polynomials as well as in the
distribution of their zeros. Some open problems as well as some new directions
for a future research are formulated.Comment: Changed content; 34 pages, 41 reference
Asymptotics and zeros of symmetrically coherent pairs of Hermite type
We consider the Sobolev inner product where (Ό0, Ό1) is a symmetrically coherent pair with one of the two measures the Hermite measure. We give a survey of the analytical properties of the corresponding Sobolev orthogonal polynomials and establish a new result about the asymptotic behaviour of these Hermite-Sobolev orthogonal polynomials inside the support of the measures Ό0 and Ό1. Keywords: Sobolev orthogonal polynomials; Hermite polynomials; Asymptotics; Symmetrically coherent pairs; Zero