This book is a precariously-positioned and choppy introduction to function spaces and orthogonal expansions, aimed at students of applied mathematics and engineering. Its position is precarious because it describes a large number of tools without showing any of them in action. Because of this omission, it reads like the first book (“Theory”) of a two-volume set, for which we don’t have the “Applications” volume. It is choppy because each (short) chapter is largely independent of the others; there’s no real thread connecting them.

In the book’s favor, it is clearly written and it does provide a useful summary of the basic properties of the tools it covers. It does a good job of explaining the difference in the various function spaces. It has detailed coverage of wavelets and the related subjects of B-splines and multiresolution analysis, although still without applications.

Although it’s not a cookbook, it doesn’t give a complete set of proofs either. It tends to prove only the easier theorems, stating the more difficult ones without proof. The book has about 150 exercises, and most of these do not test the student‘s mastery or advance the narrative but merely complete proofs that were sketched in the body. The book undertakes the unenviable task of explaining the L^{p} spaces without explicitly using the Lebesgue integral, even though it does quote most of the key theorems of Lebesgue theory, including the bounded and dominated convergence theorems and the completeness of the L^{p} spaces.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.