This is the first volume of a new series Mathematical Textbooks for Science and Engineering. It builds up standard approximation theory from scratch (requiring only some advanced calculus and linear algebra) up to a reasonably advanced level. No complex analysis or advanced functional analysis is needed. As a textbook it includes all the proofs and has many exercises following each chapter.
The idea of the new series Mathematical Textbooks for Science and Engineering (MSTE) is to publish self contained textbooks on applied or applicable mathematics at undergraduate and graduate level of mathematics science and engineering. It should be easy reading, even for self study. This book is volume 1 and it is devoted to the standard basics of approximation in the broad sense.
The subjects are not a surprise and line up as one would expect: Polynomial interpolation, best approximation (in $L_2$ and $L_\infty$-sense by polynomials), elementary orthogonal polynomials (Chebyshev, Legendre), numerical quadrature, Fourier series, and spline approximation. A more detailed table of contents can be found on the publisher's website. Note that it involves only approximation by polynomials, trigonometric polynomials or piecewise polynomials (no rationals or other function systems) and only real approximation (complex analysis is avoided completely).
All the material is standard and fully proved in the book, and no historical notes are given about the origin of these results. Hence no references to research literature are needed and thus there is no list of references. As with most textbooks, it is an endeavour resulting from many years of experience while teaching the course to many students and it is not different in this case. Such a ripening process is needed to make all the things explicit that a teacher considers obvious but may actually be a source of wild and unexpected speculations by students. So it is a remarkable achievement to introduce the right amount of rigour to avoid any misunderstanding from the reader's side and yet not to overload the text and keep it very readable also for self studying. In that respect de Villiers has made some specific choices that allowed to deliver a thoroughly thought over product that serves these characteristics very well.
Some of these choices have their consequences. Although the series aims at an audience from `science and engineering', this book is in my opinion more oriented towards science than towards engineering. It is practical in the sense that a lot of attention goes to error estimates and that it handles approximation techniques that are important and actually used in practical applications, but it is not so practical that proper attention is paid to the pure numerical and algorithmic aspects. Many of the subjects discussed will also appear in a textbook on numerical analysis, but there much more attention will be given to rounding errors and efficient implementation. This particular choice also shows by a total absence of graphs. A simple graph of an orthogonal polynomial, the location of the Chebyshev points, a spline, or whatever could relax the reader a bit from the rigorous and sometimes complicated formulas to explain a relatively simple idea. Another exponent of this choice of approach is a lack of numerical examples. There are some examples included, but mostly quite simple and of an academic nature. So there are also no graphs or tables that illustrate the error or the convergence behaviour of some of these methods. Of course a particular phenomenon might require a numerical explanation that has been deliberately avoided in this book. Another example is the incidental mentioning of FFT in the chapter on Fourier series: `[...] indeed (9.3.73)-(9.3.76) form the basis of the widely used Fast Fourier Transform (FFT), a detailed presentation of which is beyond the scope of this book.'. In many computational issues, the linear algebra aspect takes an important role when it comes down to implementation (e.g. the computation of Gaussian quadrature formulas by solving an eigenvalue problem). This is another aspect that has not been included in this book.
But even, if the contents has a strong theoretical component, the scrupulous attention paid to error estimates, estimates of Lebesgue constants, and the different ways in which a solution can be represented and computed have a definite `applicable' component as well. So, I am somewhat surprised that, besides Lagrange and Newton forms of interpolation polynomials, there is no mentioning of the barycentric representation which allows some nice theory and has definitely great practical importance. Another nice piece of theory could have been devoted to wavelets, which are absent as well (except for a brief note in an exercise about splines). Let me make it clear that I do not consider my enumeration of what is not in this book as a critique. After all, any course that is actually taught has to transfer knowledge in a finite number of teaching hours, and there is enough material left to teach a major course on approximation. My only intention is to inform potential buyers of what is and what is not to be expected.
A completely different approach is for example taken in the matlab based package Chebfun developed by Nick Trefethen and his team. This is less broad, less theoretical, and obviously much more hands-on. Learn by experience, not by formulas. A book on Chebfun is announced as a SIAM publication for early 2013. That book will be in many aspects different from the present book under review: probably not the opposite but certainly a complement.