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.
Author(s): Juan Carlos García-Ardila, Francisco Marcellán, Misael E. Marriaga
Series: EMS Series of Lectures in Mathematics
Publisher: European Mathematical Society
Year: 2021
Language: English
Pages: 129
City: Berlin
Preface
Contents
1 Introduction
2 Moment functionals on P and orthogonal polynomials
2.1 Existence of orthogonal polynomial sequences
2.2 Three-term recurrence relation
2.3 Christoffel–Darboux kernel polynomials
2.4 Polynomials of the first kind and the Stieltjes function
3 Continued fractions
3.1 Continued fractions and orthogonal polynomials
4 Zeros of orthogonal polynomials
4.1 The interlacing property of zeros
5 Gauss quadrature rules
6 Symmetric functionals
6.1 LU factorization
7 Transformations of moment functionals
7.1 Canonical Christoffel transformation
7.2 Canonical Geronimus transformation
7.3 Uvarov transformation
8 Classical orthogonal polynomials
8.1 The linear differential operator and its solutions
8.2 Weight function and inner product
9 Classical functionals
10 Electrostatic interpretation for the zeros of classical orthogonal polynomials
10.1 Equilibrium points on a bounded interval with charged end points
10.2 Equilibrium points on the complex plane: The Bessel case
10.3 Classical orthogonal polynomials and the inverse problem
11 Semiclassical functionals
12 Examples of semiclassical orthogonal polynomials
13 The Askey scheme
13.1 Hahn polynomials
13.2 Jacobi polynomials
13.3 Meixner polynomials
13.4 Krawtchouk polynomials
13.5 Laguerre polynomials
13.6 Bessel polynomials
13.7 Charlier polynomials
13.8 Hermite polynomials
13.9 Limit relations
References
Index
