A book on classical topics like linear algebra or Fourier analysis can be written in one of two ways: either as a purely theoretical treatise, aimed mainly at mathematics students or theoretical researchers in a given field, or as a guide focused on applications of the theory with appropriate background information. The book by Chui and Jiang pursues the latter avenue.

The book covers linear algebra and various methods for spectrum analysis. Linear algebra is dealt with in the first two chapters, “Linear Spaces” and “Linear Analysis.” The first presents problems like vector, function, and inner-product spaces, while the second deals with topics including matrix algebra, linear transformations, and eigenvalue theory. The book continues by addressing ideas related to spectrum analysis. The third chapter, “Spectral Methods and Applications,” focuses on two basic methods of analysis: singular value decomposition (SVD) and principal component analysis (PCA). It addresses these in the context of many practical applications, including signal processing. Here, the related theoretical basis is introduced. The next chapter, “Frequency-Domain Methods,” elaborates on the commonly used transforms: discrete Fourier transform (DFT), discrete cosine transform (DCT), and fast Fourier transform (FFT). The subsequent chapter on data compression is the most application oriented of all of the topics in the book; it covers entropy introduction, binary source coding, and relationships between lossless/lossy compression and DCT. Chapter 6 elaborates on Fourier series, constituting the introduction to chapter 7, which presents Fourier transforms and other time-frequency methods. Chapter 8 introduces wavelet transformations and finite impulse response (FIR) filter theory. These issues are extended in the last two chapters: “Compactly Supported Wavelets” and “Wavelet Analysis.”

Even when the book deals with applications like coding or filtering, it is highly formal and focuses on the theoretical results. While the presentation starts with the basic notions and then goes to very advanced ideas, the material as a whole is very demanding. Therefore, I would not use it as a main textbook, except for seminars in theoretical computer science or lectures on applied mathematics for mathematics students only. Nevertheless, the book could be used as a supplement for a graduate- or doctoral-level course: the presentation is very clear and logical, and there are proofs of many theorems, an uncommon feature of books on applications of mathematics. I like the way the book is edited to improve reading. In addition, the main ideas appear in boxes, enabling the reader to find information very quickly and easily. Each chapter has examples and exercises (unfortunately, they are mainly theoretical rather than practical), which helps with repeating the material.

The book is a second volume of the series of textbooks for science and engineering by Atlantis Press/Springer. In future editions, I would propose extending the material with examples and exercises illustrated with the use of software well known by practitioners (like MATLAB or Mathematica). Students and engineers like to see concrete examples, and this would make it easier for them to understand the material and imagine how to use it to solve problems. Such an approach has been successfully applied in many recent books, such as Singh’s [1], which presents much easier topics in linear algebra in a comprehensible way that other books should follow.