Interpolation and approximation by polynomials.

*(English)*Zbl 1023.41002
CMS Books in Mathematics/Ouvrages de MathÃ©matiques de la SMC. 14. New York, NY: Springer. xiv, 312 p. (2003).

This is a very nicely written textbook to learn about polynomial interpolation and approximation. As the author states in the Introduction, the discourse is restricted to the use of polynomials as the full generality (he quotes the famous book “Interpolation and Approximation” by Philip J. Davis, published in 1963 by Blaisdell and reprinted by Dover in 1975, that in a certain way served as a role model) would not be appropriate for the use intended.

After the introduction of univariate interpolation (Lagrange interpolation using divided differences, the Neville-Aitken algorithm, forward-, backward- and central-differences and \(q\)-differences) the author turns to best approximation on bounded intervals (using several norms), featuring Legendre- and Chebyshev polynomials, general minimax approximations, the Lebesgue function and the use of a modulus of continuity.

This is followed by a short chapter on numerical integration, and the Euler-MacLaurin formula, after which a short introduction into Peano’s kernel and its applications is given.

As a sort of ‘interlude’ the next chapter looks into multivariate interpolation using rectangular and triangular regions, including interpolation on so-called \(q\)-integers (using coordinates of the form \([i]=1+q+q^2+\cdots+q^{i-1}\) with \(q>0\); these are studied in more detail in the final chapter of the book). Also splines and B-splines are considered with evenly spaced nodes or nodes at \(q\)-integers.

Finally (although it precedes the treatment of the \(q\)-integers) a quite broad treatment of Bernstein polynomials and the monotone operator theorem is given, touching upon things as total positivity.

Throughout the book nice sets of “Problems” are placed: exercises that tax the reader to show his mastering of the theory given. It is to be hoped that this book will appear in a friendly priced paperback shortly.

After the introduction of univariate interpolation (Lagrange interpolation using divided differences, the Neville-Aitken algorithm, forward-, backward- and central-differences and \(q\)-differences) the author turns to best approximation on bounded intervals (using several norms), featuring Legendre- and Chebyshev polynomials, general minimax approximations, the Lebesgue function and the use of a modulus of continuity.

This is followed by a short chapter on numerical integration, and the Euler-MacLaurin formula, after which a short introduction into Peano’s kernel and its applications is given.

As a sort of ‘interlude’ the next chapter looks into multivariate interpolation using rectangular and triangular regions, including interpolation on so-called \(q\)-integers (using coordinates of the form \([i]=1+q+q^2+\cdots+q^{i-1}\) with \(q>0\); these are studied in more detail in the final chapter of the book). Also splines and B-splines are considered with evenly spaced nodes or nodes at \(q\)-integers.

Finally (although it precedes the treatment of the \(q\)-integers) a quite broad treatment of Bernstein polynomials and the monotone operator theorem is given, touching upon things as total positivity.

Throughout the book nice sets of “Problems” are placed: exercises that tax the reader to show his mastering of the theory given. It is to be hoped that this book will appear in a friendly priced paperback shortly.

Reviewer: Marcel G.de Bruin (Delft)

##### MSC:

41-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to approximations and expansions |

41A10 | Approximation by polynomials |

41A05 | Interpolation in approximation theory |

42-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces |