Charles Byrne begins his book by mentioning briefly his own experience as a pure mathematician wishing to get involved with signal processing. He candidly remarks that this book contains all the math that he wishes he had known during his transition to becoming an applied mathematician working in this area. A very promising premise; one expects a most appealing book, one that will be not only satisfactory to a pure mathematician in its rigor and depth but also convincing and relevant to the applied mathematician.

The book consists of 11 main parts, and a total of 63 chapters in all. Each chapter is short enough to be an easy and quick read by itself. However even though earlier chapters are relatively self contained, as the story unfolds requirements and expectations from the reader increase considerably. At the beginning, the student is introduced to complex numbers and the complex exponential function. At this point, the book seems to be written for a student coming out of a serious calculus sequence, and that this background should suffice for being able to follow the discussion. However, as we move further into the more advanced material, we see that that calculus sequence has to have been digested really well, to say the least. Just when the reader is comforted with finding basic definitions, like that of inner products, a casual reference is made to topics like calculus of variations or to Bessel functions, and one comes to realize that to understand all the details in the text requires a more serious mathematical foundation than just calculus. (Still, a good engineering or math junior should be able to get a lot out of it).

Throughout the text, every notion is explained in very intuitive language. Every concept developed is very well-motivated. For instance this reviewer found the idea of the Ferris wheels very useful for describing the notion of hidden periodicities. The discussion of a fictional radio broadcast on the West coast is used most profitably while describing the ideas of directionality and signal concentration. The Fast Fourier Transform algorithm is described in some detail, which this reviewer admits provided her with the first real understanding of this famous and most useful method for computing Fourier transforms. Overall, the motivating discussions provide excellent launching pads for the following developments of new concepts and methods.

This is a perfect book for those engineering students who may be interested in exploring some serious mathematical topics in order to understand the theoretical underpinnings of signal processing. One thing is certain: It will undoubtedly whet their appetite for more math. As a former engineering undergraduate who took signal processing courses, I for one would have loved to have come across this text then.

However, as a mathematician, I can say that I am left not fully satisfied. The pure mathematician in me now expects some more rigor, or at least some more precise definitions of the terms used regularly throughout. For instance, the discussion of time-invariant linear systems or that of the Hilbert transform can get a bit frustrating because the general ideas are not really introduced at all or they are postponed until the very end of the discussion, at which point the reader is already somewhat turned off. The inner-outer factorization for functions in the Hardy space is introduced and used, but without sufficient explanation. This reader often felt that the story was not being told in a straightforward manner, and many times, a reference for more mathematical developments was lacking. (Perhaps, Byrne was not only talking about convergence issues of infinite sums when on page 34 he wrote: "Our discussion here will be more formal and less rigorous.")

Overall, the book still successfully delivers most of what it sets out to do: A mathematician interested in learning about some applications of classical mathematics like Fourier analysis and more generally harmonic and functional analysis will still find this text a good read. The well-motivated discussion of wavelet analysis and the brief discussion on frames and Riesz bases are some of the parts of the text that make it most stimulating and informative. This reviewer could not help but dream about a fun teaching project for a second course in analysis, but she would certainly want to put in a lot more theory; after all she is still a pure mathematician!

Gizem Karaali teaches at the University of California in Santa Barbara.