One reviewer characterized this book as little more than class notes. It is much better than that. It is, however, concise; the authors cover a lot of ground in few pages. As usual for graduate courses, a lot is required of the student. A good foundation in real analysis, or at least calculus, is assumed. For example, the author say that Lebesgue's Dominated Convergence Theorem is beyond the scope of the book. This suggests that it is a more advanced topic, while really it is a less advanced topic that ought to have been covered in a prior course. In general, the book could be amplified to two or three times its length without taking up additional topics. I would not say that there are too few exercises, but it is true that a complete working out of the exercises in the back of the book would have been good.
P.S. Different writers use different definitions of the Fourier transform. The authors mention this but do not say why they chose the one that they did. He calls the differences "purely technical," but of course the whole subject is technical. I would have appreciated a discussion of his choice. Also, I would warn the reader that the exercises do not seem given in increasing order of difficulty as is usual. Nevertheless, this is a good book if you are prepared to work through it.
It appears that well-written advanced mathematics texts are few and far in between. "Fourier Series and Integral Transforms" is no exception. The authors belie their goal in the preface, stating that the "aim of this book is to provide ... important examples of useful series of functions." They admit that this text was a bundling of class notes from a course of the same name, but do not specify who actually gave the course. After reading the first few chapters, I'm forced to conclude that either the authors have little experience in writing comprehensive texts or they do not know the material. I'll say one nice thing before I rip into this: the price. When all of my other science books cost a minimum of $100 each, it is nice to only have to fork out $... for a book. One axiom that holds true in all mathematics courses is that the way to quickly master the material is to work problems. It goes without saying that rote problem working without feedback serves no purpose but to burn trees and waste time. This is the crux of the problem that I have with this book: inadequate number of problems, no problem key and no examples in the book. The number of problems in the book is sparse at best, and most aspects of the ideas in the text seem underrepresented in the problems. The lack of a problem key means that when the student is working the problems she has no idea if she got it correct. Doing a problem wrong serves no purpose whatsoever, except to demoralize the student. Providing a key does not allow students to "cheat" as it were, since one needs to demonstrate the steps toward getting the answer. Finally, there are "examples" in the book, but the majority of them end abruptly with the sentence "it is not difficult to show~" or "it is easy to verify~." This is fine and dandy if there were enough complete examples to get you going. Unfortunately, there are not. I would expect this sort of thing from someone who slept through the actual class, then-faced with a publishing deadline-was forced to "finish" his book quickly. My recommendation: if you have the choice-don't bother. If you are forced to purchase this book (i.e., required reading for signal analysis), then buy it used or better yet get it from the library. Sell it back as quickly as possible, because once someone else realizes that there is money to be made in writing (and selling) well-written signal analysis texts, this one will end up with no resale value whatsoever.
This is by far one of the best mathematics texts I've read. It's clearly written and provides substantial background behind the development of Fourier series with some excellent examples and exercises. I used this text for an independent study course and needed only some help from my professor. Undergraduate courses in Real Analysis and Vector Calculus is recommended to help speed up the learning process. I highly recommend this text for your personal library.
I think this is an excellent book on Fourier series and Fourier transforms. I think their style is clear, concise and informative. For example their treatment of pointwise conergence is explained in far simpler way then in other text. The book is well done.