Katznelson's An Introduction to Harmonic Analysis is, of course, a classic. Since it won the 2002 Steele Prize for Mathematical Exposition, that hardly needs to be said. So the first thing to say is "thank you," to Cambridge for doing this new edition, and to Prof. Katznelson for undertaking the task of updating his book.
This Introduction was first published in 1968. It is an ambitious book, moving all the way from Fourier series to Banach algebras and analysis on locally compact abelian groups. It is densely but clearly written, with the occasional flash of wit. For example, from the introduction: "When I was writing Chapter III of this book, I was very pleased to produce the simple elegant proof of Theorem III.1.6 there. I could swear I did it myself until I remembered two days later that six months earlier, 'over a cup of coffee,' Lennart Carlson indicated to me this same proof." Or, from the chapter on Banach algebras, "There is no claim, of course, that every problem in harmonic analysis has to be considered in this setting; however, if a space under study happens to be either a Banach algebra, or the dual space of one, keeping this fact in mind usually pays dividends."
In his introduction to the new edition (dated 2003), Katznelson says that he has added some material and revised some of the wording. However, he says, "The added material does not reflect the progress in the field in the past thirty or forty years. Much of it could and, in retrospect, should have been included in the first edition of this book." So this remains an introduction, and students wanting to go deeper will need to go on to other books. But this is a great place to start.
Fernando Gouvêa is the editor of MAA Reviews.