Theodore J. Rivlin An Introduction to the Approximation of Functions

The needs of automatic digital computation have spurred an enormous revival of interest in methods of approximating continuous functions by functions that depend only on a finite number of parameters. This concise but wide-ranging text provides an introduction to some of the most significant of these methods, with particular emphasis on approximation by polynomials. Because approximation theory is an area of mathematics with important practical applications in computation, the author takes pains to discuss not only the theoretical underpinnings of many common algorithms, but to demonstrate the practical applications of the procedure. Thus, for each method of approximation studied, at least one algorithm leading to actual numerical approximation is described and traced to its present formulation. Apart from purely practical considerations, approximation theory is also a lively branch of mathematical analysis. Material in this book will be especially useful as additional reading in introductory courses in both mathematical and numerical analysis. It is particularly helpful in its coverage of a variety of approximation methods, including interpolation methods, and an introduction to splines, currently an area of great interest in approximation theory. Written for upper-level graduate students, this book presupposes a knowledge of advanced calculus and liner algebra, but the author has made a special effort to avoid some of the more sophisticated prerequisites in order to keep the topic within the grasp of the uninitiated. Abundant exercise material at the end of each chapter and an excellent bibliography contribute to the uniqueness and value of Dr. Rivlin's book. As mathematical research continues to increase in complexity, the need to understand methods of approximation becomes even greater. This valuable text, one of the most accessible of its kind, represents an excellent introduction to this significant area of mathematical study.