The semiclassical--Sobolev orthogonal polynomials: a general approach

Details The semiclassical--Sobolev orthogonal polynomials: a general approach The semiclassical--Sobolev orthogonal polynomials: a general approach

2010-03-24

arxiv.org/abs/1003.4612v2

Journal of Approximation Theory Volume 163, Issue 1, 2011, Pages 65-83

Mathematics

Classical Analysis and ODEs

23 pages, special issue dedicated to Professor Guillermo Lopez lagomasino on the occasion of his 60th birthday, accepted in Jo

Scientific paper

10.1016/j.jat.2010.03.001

We say that the polynomial sequence $(Q^{(\lambda)}_n)$ is a semiclassical Sobolev polynomial sequence when it is orthogonal with respect to the inner product $$ _S=<{{\bf u}} ,{p\, r}> +\lambda <{{\bf u}}, {{\mathscr D}p \,{\mathscr D}r}>, $$ where ${\bf u}$ is a semiclassical linear functional, ${\mathscr D}$ is the differential, the difference or the $q$--difference operator, and $\lambda$ 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 $\bf u$. 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 ${\mathscr D}$ 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.

Affiliated with

Also associated with

No associations

Rating

The semiclassical--Sobolev orthogonal polynomials: a general approach does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The semiclassical--Sobolev orthogonal polynomials: a general approach, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The semiclassical--Sobolev orthogonal polynomials: a general approach will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-57006