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2014 | Book

Orthogonal polynomials on the real line Read chapter Read first chapter

Walter Gautschi, Volume 2

Selected Works with Commentaries

Editors: Claude Brezinski, Ahmed Sameh

Publisher: Springer New York

Book Series : Contemporary Mathematicians

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

Walter Gautschi has written extensively on topics ranging from special functions, quadrature and orthogonal polynomials to difference and differential equations, software implementations, and the history of mathematics. He is world renowned for his pioneering work in numerical analysis and constructive orthogonal polynomials, including a definitive textbook in the former, and a monograph in the latter area.

This three-volume set, Walter Gautschi: Selected Works with Commentaries, is a compilation of Gautschi’s most influential papers and includes commentaries by leading experts. The work begins with a detailed biographical section and ends with a section commemorating Walter’s prematurely deceased twin brother. This title will appeal to graduate students and researchers in numerical analysis, as well as to historians of science.

Selected Works with Commentaries, Vol. 1

Numerical Conditioning

Special Functions

Interpolation and Approximation

Selected Works with Commentaries, Vol. 2

Orthogonal Polynomials on the Real Line

Orthogonal Polynomials on the Semicircle

Chebyshev Quadrature

Kronrod and Other Quadratures

Gauss-type Quadrature

Selected Works with Commentaries, Vol. 3

Linear Difference Equations

Ordinary Differential Equations

Software

History and Biography

Miscellanea

Works of Werner Gautschi

Table of Contents

Frontmatter

Commentaries

Frontmatter
11. Orthogonal polynomials on the real line
Abstract
In about two dozen papers, Walter Gautschi developed the so-called constructive theory of orthogonal polynomials on ℝ, ncluding effective algorithms for numerically generating orthogonal polynomials, a detailed stability analysis of such algorithms as well as several new applications of orthogonal polynomials. Furthermore, he provided software necessary for implementing these algorithms (see Section 23, Vol. 3) and applications.
Gradimir V. Milovanović
12. Polynomials orthogonal on the semicircle
Abstract
In two papers, jointly with Henry J. Landau and Gradimir V. Milovanović, Walter Gautschi investigates polynomials that are orthogonal with respect to a non-Hermitian inner product defined on the upper half of the unit circle in the complex plane. For special choices of the weight function, these polynomials are related to Jacobi polynomials. Their recurrence relation and properties of their zeros are investigated, and applications to Gauss quadrature are explored. We first discuss the importance of orthogonal polynomials that satisfy recurrence relations with few terms, and then focus on the special properties of orthogonal polynomials on the semicircle.
Lothar Reichel
13. Chebyshev quadrature
Abstract
Here we review Gautschi’s work on Chebyshev quadrature, first his 1975 survey paper, and then original work by him and his colaborators.
Jaap Korevaar
14. Kronrod and other quadratures
Abstract
This section is devoted to Gautschi’s work on Kronrod quadratures and other special quadrature rules. For Gauss-type quadrature rules, see Section 15.
Giovanni Monegato
15. Gauss–type quadrature
Abstract
Walter Gautschi’s work in this area has had a profound impact, especially on the computational and practical aspects of Gauss quadrature. I have heard people refer to it as Gautschian quadrature, just to emphasize Walter’s many contributions to the theory and computation.
Walter Van Assche

Reprints

Frontmatter
16. Papers on Orthogonal Polynomials on the Real Line
Walter Gautschi
17. Papers on Orthogonal Polynomials on the Semicircle
Walter Gautschi
18. Papers on Chebyshev Quadrature
Walter Gautschi
19. Papers on Kronrod and Other Quadratures
Walter Gautschi
20. Papers on Gauss-type Quadrature
Walter Gautschi
Metadata
Title
Walter Gautschi, Volume 2
Editors
Claude Brezinski
Ahmed Sameh
Copyright Year
2014
Publisher
Springer New York
Electronic ISBN
978-1-4614-7049-6
Print ISBN
978-1-4614-7048-9
DOI
https://doi.org/10.1007/978-1-4614-7049-6

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