Talk: The semiclassical-Sobolev polynomials. A general approach

Date: 2008..09..09
Event: IWOPA08. Conference in honor of Guillermo López
Venue: Universidad Carlos III de Madrid. Leganés, Spain

Abstract

We say the polynomial sequence, \((Q_n(x; \lambda))\), is a semiclassical-Sobolev polynomial sequence when it is orthogonal with respect to the inner product \((p, r) = \langle {\bf u}, p\,r\rangle + \lambda \langle {\bf u}, {\mathcal D}\,p {\mathcal D}r\rangle\), where \({\bf u}\) is a semiclassical linear functional, \({\mathcal D}\) is the differential (or the difference or the q-difference, respectively) operator, and \(\lambda\) is a positive constant.

In this talk we present 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 talk is to show a general approach to the study of the polynomials orthogonal with respect to the above nonstandard inner product regardless of the type of operator \({\mathcal D}\) considered. Finally, we illustrate our results 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.

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