Orthogonal polynomials and linear functionals. An algebraic approach and applications
type: Item
instance of: scholarly article
title: Orthogonal polynomials and linear functionals. An algebraic approach and applications
zbMATH Open document ID: 1475.33001
DOI: 10.4171/ELM/33
author: Juan Carlos García-Ardila
author: Misael E. Marriaga
author: Francisco Marcellán
published in: EMS Series of Lectures in Mathematics
publication date: 2021-05-05T00:00:00Z
review text: Quoting the abstract of this short book in its entirety: ``This book presents an introduction to orthogonal polynomials, with an algebraic flavor, based on linear functionals defining the orthogonality and the Jacobi matrices associated with them. Basic properties of their zeros as well as quadrature rules are discussed. A key point is the analysis of those functionals satisfying Pearson equations (semiclassical case) and the hierarchy based on their class. The book's structure reflects the fact that its content is based on a set of lectures delivered by one of the authors at the first Orthonet Summer School in Seville, Spain in 2016. The presentation of the material is self-contained and will be valuable to students and researchers interested in a novel approach to the study of orthogonal polynomials, focusing on their analytic properties.'' This reviewer found that abstract to be quite accurate, although the book is very short for the amount of material that it touches on, making the claim of being ``self-contained'' a little iffy. The authors do point out ``Hoping to make up for this lack of exhaustiveness, we have added a list of references that an interested reader can consult.'' Their literature survey is not comprehensive either, but it does contain useful links. Indeed the book, short as it is, contains several ideas that were very surprising to me, and details of proofs that I had not seen elsewhere and which I found to be very interesting and valuable. First, the idea of using linear functionals to unify the treatment of orthogonal polynomials was new to me, although I have used orthogonal polynomials for, well, decades. I found the unity that this idea provides quite remarkable. Second, the book contains a very short but very illuminating chapter on potential theory, a classical subject for orthogonal polynomials that I had previously only seen in approximation theory (see e.g. Nick Trefethen's book Approximation Theory and Approximation Practice). The bo…
Mathematics Subject Classification ID: 33-01
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 33C47
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 65D30
zbMATH DE Number: 7344232
zbMATH Keywords: linear functionals
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: Jacobi matrices
zbMATH Keywords: zeros of orthogonal polynomials
zbMATH Keywords: Gauss quadrature formulas
zbMATH Keywords: special transformations
zbMATH Keywords: matrix factorizations
zbMATH Keywords: semiclassical linear functionals
zbMATH Keywords: Askey scheme
zbMATH Keywords: potential theory
reviewed by: Robert M. Corless
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4171/elm/33
OpenAlex ID: W3181071443
description: scientific article; zbMATH DE number 7344232