Many theories and algorithms in numerical mathematics and scientific computing are built on approximation theory. Many deep theoretical results have been successfully used to construct extremely powerful numerical algorithms. However, to a certain extent, the traditional means of approximation theory--a major part of which is linear theory--have reached the limits of their capabilities. In recent years, though, very interesting, innovative ideas have been raised. Temlyakov’s book is devoted to a detailed study of one of these concepts--so-called greedy approximation--and its applications.

Essentially, classical approximation looks for a good approximant to a given function from a set of linear combinations of a given finite set of functions. The new aspect introduced in greedy approximation is that only the dimension of this approximation space is prescribed, not the basis elements. Rather, infinitely many basis elements are offered, and one may freely choose the required number of basis functions from this set. It is evident that this approach provides much more flexibility and the freedom to adapt the approximating function to the target; thus, much better results can be obtained. The difficulty is then twofold: one needs to determine which properties of the given basis functions really lead to efficient algorithms, and then the finitely many functions need to be appropriately selected.

In the first half of the book (chapters 1 through 3), Temlyakov provides a very detailed and thorough account of the mathematical theory underlying greedy approximation. While this part of the book is written in the language of mathematical analysis, it is usually rather easy to transfer the theorems into actual algorithms. Chapters 4 and 5 are devoted to two important applications of the theory: learning theory and compressed sensing. Both of these important topics are of current interest and have important implications in areas such as signal and image processing and scientific computing; Temlyakov clearly demonstrates how greedy approximation can help here. The last chapter picks up where the theoretical part leaves off and generalizes some of the basic results even further.

The book is written in a mathematically rigorous but clear and readable style. It is completely self-contained; Temlyakov includes all of the necessary background information. Therefore, it is certainly a well-suited textbook for graduate courses. Moreover, graduate students and professional researchers who are prepared to work their way through the classical mathematical definition-theorem-proof style should read the book on their own--they will greatly benefit from its excellent introduction to a highly interesting and novel area of applied mathematics.