Spline functions: basic theory.

*(English)*Zbl 1123.41008
Cambridge Mathematical Library. Cambridge: Cambridge University Press (ISBN 978-0-521-70512-7/pbk). xv, 582 p. (2007).

This book is a third edition of the famous and highly useful book on splines by one of the most well-known approximation theorists. In his work, Larry Schumaker describes mostly univariate spline in much detail, and in particular different kinds of splines as they were known in 1981. (Some multivariate material is also included at the end of the book, especially tensor-product splines, and also there is a supplement catching up with a large number of later developments). All the basic material on polynomial splines, their properties, especially their dependence on knots and multiple knots, and the B-spline basis is contained, including the important dual bases, L-splines, perfect splines and other generalised splines. Approximation powers for uniform approximation to sufficiently smooth approximands were then and are now one of the most interesting and relevant aspects of piecewise polynomial interpolation, and as such are also included in this comprehensive volume. This addresses both the standard work on splines with fixed knots as well as the non-linear part of splines with free knots. The new edition, published by Cambridge University Press, becomes even more useful by a new bibliography and details on web-sites where large bibliographies on splines and related approximation tools are kept up-to-date by Larry Schumaker and Carl de Boor. Larry Schumaker’s book on splines is a standard reference book, equally useful today as it was when it was first published.

Reviewer: Martin D. Buhmann (Gießen)

##### MSC:

41A15 | Spline approximation |

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

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

41A46 | Approximation by arbitrary nonlinear expressions; widths and entropy |

41A10 | Approximation by polynomials |

41A36 | Approximation by positive operators |