*(English)*Zbl 1130.42300

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 ${\alpha}_{k}\left(d\lambda \right)$, ${\beta}_{k}\left(d\lambda \right)$, $k=0,1,\cdots ,n-1$, where $n$ is a typically large integer and $d\lambda $ 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.

##### MSC:

42-02 | Research monographs (Fourier analysis) |

33-02 | Research monographs (special functions) |

42C05 | General theory of orthogonal functions and polynomials |

33C45 | Orthogonal polynomials and functions of hypergeometric type |

33F05 | Numerical approximation and evaluation of special functions |