Volker Michel’s new book Lectures on Constructive Approximation Theory is concerned with methods of approximating functions on the real line, and the sphere and ball in three dimensional Euclidean space. At first glance, the third space may seem out of place. The ball is just a nice solid chunk of Euclidean space, so if you could approximate functions on R3 then you could approximate functions on the ball. While that is true, it misses an important point. Understanding why the ball is treated specially helps understand the purpose of the book.
The motivation of approximation is to replace something that is difficult to use with something easier to use, and “easier to use” depends on context. For example, we might want to approximate a function by a polynomial for numeric computation because polynomials are easy to evaluate in a computer. But in integration theory, you usually want to approximate functions by linear combinations of set indicator functions because indicator functions are easy to work with in that context. Functions that are convenient in one context might not be in other contexts.
The author is interested in constructing approximations that are useful for a class of problems that take advantage of the symmetry of the sphere and the ball. For these problems the sphere is not simply a 2-manifold sitting in R3 nor is the ball simply an open set in R3. For example, one of the motivating problems is the inverse gravimetric problem, determining the mass density distribution inside the Earth from the gravitational potential on its surface. (The book mentions applications in tomography, but the emphasis is on geoscience.) For such applications, it is important for functions to have nice properties when expressed in spherical, not rectangular, coordinates.
In each domain, the book examines Fourier, spline, and wavelet approximations. Polynomial approximation, such as spherical harmonics, is included not so much for its own sake but for its utility in constructing other kinds of approximation.
This is a book of hard analysis. It constructs explicit functions and computes quantitative estimates rather than making existential statements about this space being dense in that space. This explains in part why the book focuses on “the sphere” and “the ball,” i.e. in R3, rather than looking at spheres and balls in general Rn. Another reason is the emphasis on geoscience applications.
Lectures on Constructive Approximation has a pleasant balance of exposition and derivation. Although there are some extensive calculations, it is not difficult to maintain one’s bearings. The book also has quite a few colorful graphics. After reading about abstract topics such Sobolev spaces on spheres etc., the graphics provide a down-to-earth contrast, especially since many of these images are literally graphs of functions on the Earth!
One nice feature of the book is the “Questions for Understanding’ at the end of each chapter. Some of these questions are simple indications of which concepts were most important. Others are more specific exercises that an instructor could assign as homework.
John D. Cook is an independent consultant and blogs at The Endeavour.