zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Orthogonal polynomials. Computation and approximation. (English) Zbl 1130.42300
Numerical Mathematics and Scientific Computation. Oxford: Oxford University Press (ISBN 0-19-850672-4/hbk). viii, 301 p. £ 55.00 (2004).

The book by Walter Gautschi, the world renowned expert in computational methods and orthogonal polynomials, came out in the series “Numerical Mathematics and Scientific Computation”. The choice of topics reflects the author’s own research interests and involvement in the area, but it is hoped that the exposition will be acceptable and useful to a wide audience of readers.

The computational methods in the theory of orthogonal polynomials on the real line, which constitute a backbone of the whole book, are treated in Chapter 2. The fundamental problem is to compute the first n recursion coefficients α k (dλ), β k (dλ), k=0,1,,n-1, where n is a typically large integer and dλ a positive measure given either implicitly via moment information or explicitly. There is a simple algorithm due to Chebyshev that produces the desired coefficients in the former case but its effectiveness depends critically on the conditioning of the underlying problem. In the latter case discretization of the measure and subsequent approximation of the desired recursion coefficients by those relative to a discrete measure are applicable. Other problems calling for numerical methods are the evaluation of Cauchy integrals and the problem of passing from the recursion coefficients of a measure to those of a modified measure—the original measure multiplied by a rational function.

In Chapter 1 a brief, but essentially self-contained account of the theory of orthogonal polynomials is presented with the focus on the parts of the theory most relevant to computation. The exposition combines nicely the standard topics such as three-term recurrence relations, Christoffel–Darboux formulae, quadrature rules and classical orthogonal polynomials as well as not quite traditional issues (kernel polynomials, Sobolev orthogonal polynomials and orthogonal polynomials on the semicircle). A number of applications, specifically numerical quadrature, discrete least squares approximation, moment-preserving spline approximation, and the summation of slowly convergent series is given in Chapter 3. Many tables throughout the book report on numerical results of various algorithms.

42-02Research monographs (Fourier analysis)
33-02Research monographs (special functions)
42C05General theory of orthogonal functions and polynomials
33C45Orthogonal polynomials and functions of hypergeometric type
33F05Numerical approximation and evaluation of special functions