The subject of Spline functions has its roots in research conducted during the Second World War by a variety of premier computer scientists including I. J. Schoenberg, D. Greenspan, C. de Boor, G. Fasshauer, J. Jerome, D. Kincaid, W. Cheney, G. Nurnberger, S. Sard, M. H. Schultz, L. Schumaker, G. Wahba, and others. Much of this earlier work was developed at the US Army Mathematics Research Center at the University of Wisconsin, involving work in the 1950s and 1960s at the beginnings of computer and computational sciences as they developed.

This book, along with MATLAB software, provides a thorough treatment mathematically and computationally of methods in the use of spline functions and the understanding and methods for studying these techniques. In my role as an editor, I generally do not encourage a straightforward list of chapters in a book to illustrate the content of a book such as this. However, in this case, I find little justification for eliminating any of the chapters since the content is so thorough and concise that the 11 chapters are of central importance; in addition, this book not only has a bibliography, but also indices of scripts, functions, and subjects.

This book provides a thorough treatment of the state of the art in the theory and applications of spline functions. Its focus is on the mathematics and theory along with applications for computational scientists and engineers. Schumaker provides a thorough treatment that is clearly focused on the history as well as the current methodology, including the state of the art.

The chapters begin (as you might expect) with univariate splines and move on to interpolation and approximation along with tensor-product splines. Further details are given in the preface of the book.

The book begins with the classical approach of univariate splines and has as its foundation much of the basic theory I recall from undergraduate courses early in my career. However, this opening chapter deals with B-splines and Hermite interpolation, and introduces the topics of shape control, noise control, and optimal knots locations. In this regard, even the first chapter presents the up-to-date approach to least-squares splines and concludes with two-point boundary-value problems and error bounds. Chapter 1 includes remarks and historical notes, which the author consistently provides as each chapter develops.

Chapter 2 moves into tensor-product B-splines, includes Hermite interpolation and Lagrange interpolation on a grid, extends the treatment of the previous chapter involving noise and penalized least-squares approximation, and contains more advanced treatment of error bounds.

Chapter 3 introduces triangulations and more advanced topics such as storage and plotting, barycentric coordinates, and grid generation including Euler relations.

Chapter 4 deals with computing and splines as the presentation becomes more advanced. This chapter begins with Bernstein basis polynomials and moves on to integrals of inner products and general functions over triangles. More advanced topics include plotting and splines, as well as derivatives.

Chapter 5 deals with macro-element interpolation methods and several C1 Powell–Sabin interpolants and other interpolants including the C2 Wang interpolant. Chapter 5 also includes a discussion of rates of convergence for less smooth functions, scale invariance, as well as (of course) remarks and historical notes.

Chapter 6 deals with scattered data interpolation including minimal energy interpolating splines, local C0 cubic splines, and a C1 Clough–Tocher method. This chapter moves into more advanced topics including derivative estimation of derivatives from scattered data, scale invariance, nonrectangular domains with holes, non-Delaunay triangulations, and the use of radial basis functions and interpolation with tensor-product splines.

Chapter 7 deals with scattered data fitting and least-squares fitting with splines, scale invariance, and penalized least-squares fitting including macro-element spline spaces.

Chapter 8 is focused on shape control. This chapter deals with interpolation, least squares, and monotone and convex methods.

Chapter 9 begins with an example of boundary-value problems, but includes the Ritz–Galerkin method and solution of problems including the biharmonic equations involving C1 quintic splines.

Chapter 10 deals with spherical splines and begins with spherical triangulations and polynomials.

Chapter 11 deals with applications of spherical splines, beginning with Hermite interpolation with macro-element spaces, and going on to scattered data interpolation of the sphere, least-squares fitting with spherical splines and (again) noise effects, spherical penalized least squares, and partial differential equations on the sphere.

The book concludes with a bibliography and three indices for scripts, functions, and subjects.