Splines play an important role in applied mathematics since they possess high flexibility to approximate efficiently, even nonsmooth functions which are given explicitly or only implicitly, e.g. by differential equations. The aim of this book is to analyse in a unified approach basic theoretical and numerical aspects of interpolation and best approximation by splines in one variable. The first part on spaces of polynomials serves as a basis for investigating the more complex structure of spline spaces. Given in the appendix are brief introductions to the theory of splines with free knots (an algorithm is described in the main part), to splines in two variables and to spline collocation for differential equations.A large number of new results presented here cannot be found in earlier books on splines. Researchers will find several references to recent developments. The book is an indispensable aid for graduate courses on splines or approximation theory. Students with a basic knowledge of analysis and linear algebra will be able to read the text. Engineers will find various pactical interpolation and approximation methods.

Spline functions entered Approximation Theory as solutions of natural extremal problems. A typical example is the problem of drawing a function curve through given n + k points that has a minimal norm of its k-th derivative. Isolated facts about the functions, now called splines, can be found in the papers of L. Euler, A. Lebesgue, G. Birkhoff, J. Favard, L. Tschakaloff. However, the Theory of Spline Functions has developed in the last 30 years by the effort of dozens of mathematicians. Recent fundamental results on multivariate polynomial interpolation and multivari ate splines have initiated a new wave of theoretical investigations and variety of applications. The purpose of this book is to introduce the reader to the theory of spline functions. The emphasis is given to some new developments, such as the general Birkoff's type interpolation, the extremal properties of the splines and their prominant role in the optimal recovery of functions, multivariate interpolation by polynomials and splines. The material presented is based on the lectures of the authors, given to the students at the University of Sofia and Yerevan University during the last 10 years. Some more elementary results are left as excercises and detailed hints are given.

Splines play an important role in applied mathematics since they possess high flexibility to approximate efficiently, even nonsmooth functions which are given explicitly or only implicitly, e.g. by differential equations. The aim of this book is to analyse in a unified approach basic theoretical and numerical aspects of interpolation and best approximation by splines in one variable. The first part on spaces of polynomials serves as a basis for investigating the more complex structure of spline spaces. Given in the appendix are brief introductions to the theory of splines with free knots (an algorithm is described in the main part), to splines in two variables and to spline collocation for differential equations.A large number of new results presented here cannot be found in earlier books on splines. Researchers will find several references to recent developments. The book is an indispensable aid for graduate courses on splines or approximation theory. Students with a basic knowledge of analysis and linear algebra will be able to read the text. Engineers will find various pactical interpolation and approximation methods.

This classic work continues to offer a comprehensive treatment of the theory of univariate and tensor-product splines. It will be of interest to researchers and students working in applied analysis, numerical analysis, computer science, and engineering. The material covered provides the reader with the necessary tools for understanding the many applications of splines in such diverse areas as approximation theory, computer-aided geometric design, curve and surface design and fitting, image processing, numerical solution of differential equations, and increasingly in business and the biosciences. This new edition includes a supplement outlining some of the major advances in the theory since 1981, and some 250 new references. It can be used as the main or supplementary text for courses in splines, approximation theory or numerical analysis.

A NATO Advanced Study Institute on Approximation Theory and Spline Functions was held at Memorial University of Newfoundland during August 22-September 2, 1983. This volume consists of the Proceedings of that Institute. These Proceedings include the main invited talks and contributed papers given during the Institute. The aim of these lectures was to bring together Mathematicians, Physicists and Engineers working in the field. The lectures covered a wide range including ~1ultivariate Approximation, Spline Functions, Rational Approximation, Applications of Elliptic Integrals and Functions in the Theory of Approximation, and Pade Approximation. We express our sincere thanks to Professors E. W. Cheney, J. Meinguet, J. M. Phillips and H. Werner, members of the International Advisory Committee. We also extend our thanks to the main speakers and the invi ted speakers, whose contri butions made these Proceedings complete. The Advanced Study Institute was financed by the NATO Scientific Affairs Division. We express our thanks for the generous support. We wish to thank members of the Department of Mathematics and Statistics at MeMorial University who willingly helped with the planning and organizing of the Institute. Special thanks go to Mrs. Mary Pike who helped immensely in the planning and organizing of the Institute, and to Miss Rosalind Genge for her careful and excellent typing of the manuscript of these Proceedings.

This volume reports a series of research investigations concerned with spline functions and approximation theory. The common thread of the studies derives from the facts that (1) the subject matter of the individual articles relate and significantly complement each other; 92) part of the genesis and certainly the main developments of these studies occurred at the Weizmann Institute of Science, Rehovot, Israel, commencing about September 1970 through June 1974. The contributions cover aspects of the theory of best approximation and quadratures, the solution of certain extremal problems embracing generalized Landau and Markov-type inequalities for derivative functionals, and a hierarchy of interpolation and convergence properties of classes of spline functions.