This is a very erudite exposition of the mathematics that is needed to work on interpolation and approximation problems. It’s fun just to browse through and look at the convergence theorems, even if you’re not specifically interested in approximation. The focus of the book is on mathematical background more than specific techniques, and on breadth rather than depth. The book was published in 1963 by Blaisdell, then in a “minor corrections” reprint by Dover in 1975 (that adds a preface with many references to new works); it was reprinted by Dover again in 2014.

By “interpolation” the book means creating a function that passes through specific points, not simply a method for finding a value at an intermediate point. By “approximation” the book nearly always means approximation of whole functions, not calculation of approximate values of numbers such as roots, definite integrals, or infinite series (although there is some of that). On p. 24 the book says, “This whole book can be regarded as a theme and variation on two theorems: an interpolation theorem of great antiquity and Weierstrass’ approximation theorem of 1885.” Consequently the book has a great deal about orthogonal polynomials and functions, many kinds of approximation theorems, and a moderate amount of inner product spaces and functional analysis.

Despite its age the book holds up very well; the subjects covered are still in the mainstream of approximation theory. Happily computers were already in heavy use in this field when the book was written, and it does cover the methods that work well with computers but not by hand. One (today) conspicuous omission is splines; they were just becoming popular in the late 1960s and are not covered at all in the present book.

It’s hard to say exactly what the level of the book is; the prerequisites are light and it explains everything from the beginning, but it covers an enormous amount of ground and the examples and exercises are skimpy (but intriguing and well-chosen). By present-day standards it is probably either a graduate text or a reference.

A few somewhat-similar books include Theodore J. Rivlin’s *An Introduction to the Approximation of Functions* (Dover reprint, 1981), which is much more elementary but does give a good introduction and does cover splines; and Mastroianni & Milovanovic’s *Interpolation Processes*, that is very modern and has a lot of overlap with the present book but is not as comprehensive. A good introductory book, but not a very broad one (it is mostly about Chebyshev polynomials) is Trefethen’s *Approximation Theory and Approximation Practice*.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.