On Sobolev orthogonal polynomials

Abstract

Sobolev orthogonal polynomials have been studied extensively in the past 20 years. The research in this field has sprawled into several directions and generates a plethora of publications. This paper contains a survey of the main developments up to now. The goal is to identify main ideas and developments in the field, which hopefully will lend a structure to the mountainous publications and help future research.

Darboux transformations from the Appell-Lauricella operator

“…With the above constructions, we know that the spectral equations (3.22) and (3.25) hold. Finally, we show next that the polynomials Q γ η (x) satisfy a generalized orthogonality relation which resembles the Sobolev inner products studied in the literature, see the review article [22] and the references therein.…”

WHAT IS... a Sobolev Orthogonal Polynomial?

“…We have said nothing here about other prominent and active topics such as Sobolev orthogonal polynomials in several variables or about the theories of Sobolev orthogonality when the derivative operator is changed by other operators. Marcellán and Xu [2] provide an updated survey tackling some of these topics. Our hope is to give readers an initial glance into the world of Sobolev orthogonality.…”

On some classical type Sobolev orthogonal polynomials

“…The theory of Sobolev orthogonal polynomials is nowadays of a considerable interest. A brief account of the theory state can be found in [5] and a more recent survey is given in [10]. Before we shall discuss the examples of Sobolev orthogonal polynomials related to (5), we need the following definition.…”