This book was originally published by Wiley-Interscience in 1981. A second edition was published in 1993 by Krieger. The two differ only in that a number of misprints were corrected. Both editions are now out of print. However, spline functions remain an active research area with important applications in a wide variety of fields, including some, such as Computer-Aided Geometric Design (CAGD) and Wavelets, which did not exist in 1981. This continued interest in the basic theory of splines was the motivation for preparing this third edition of the book.

There have been many developments in the theory of splines over the past twentyfive years. While it was not my intention of rewrite this book to cover all of these developments, David Tranah of Cambridge University Press convinced me that it would be useful to prepare a supplement to the book which gives an overview of the main developments with pointers to the literature. Tracking down this literature was a major undertaking, and more than 250 new references are included here. However, this is still far from a complete list. For an extended list, see the online bibliography at www.math.vanderbilt.edu/∼schumake/splinebib.html. I include links there to a similar bibliography for splines on triangulations, and to the much larger spline bibliography in TEX form maintained by Carl de Boor and I.

Interpolation, approximation, and the numerous other applications of splines are not treated in this book due to lack of space. Consequently, I have elected not to discuss them in the supplement either, and the newlist of references does not include any applied papers or books.

The theory of spline functions and their applications is a relatively recent development. As late as 1960, there were no more than a handful of papers mentioning spline functions by name. Today, less than 20 years later, there are well over 1000 research papers on the subject, and it remains an active research area.

The rapid development of spline functions is due primarily to their great usefulness in applications. Classes of spline functions possess many nice structural properties as well as excellent approximation powers. Since they are easy to store, evaluate, and manipulate on a digital computer, a myriad of applications in the numerical solution of a variety of problems in applied mathematics have been found. These include, for example, data fitting, function approximation, numerical quadrature, and the numerical solution of operator equations such as those associated with ordinary and partial differential equations, integral equations, optimal control problems, and so on. Programs based on spline functions have found their way into virtually every computing library.

It appears that the most turbulent years in the development of splines are over, and it is now generally agreed that they will become a firmly entrenched part of approximation theory and numerical analysis. Thus my aim here is to present a fairly complete and unified treatment of spline functions, which, I hope, will prove to be a useful source of information for approximation theorists, numerical analysts, scientists, and engineers.

This book developed out of a set of lecture notes which I began preparing in the fall of 1970 for a course on spline functions at the University of Texas at Austin.