If there was ever a “gift that keeps on giving” to the most esoteric of mathematico-physical theories or to the most pragmatic of engineering practice, it is Fourier analysis. (On my bucket list is someday to discover, in principle from Georg Cantor’s original papers, exactly how Cantor’s seminal and arguably still-controversial set theory emerged from his study of 19th-century Fourier analysis.)

Just as the principle of least action [1] retains a mystery (that is not born of mysticism) regarding its effectiveness in both classical and quantum physics, so Fourier analysis realizes to an astounding degree the assertion, variously attributed to James Clerk Maxwell or David Hilbert, that “there is nothing more practical than a good theory.”

This book is a smooth continuation, in content and philosophy, of its excellent predecessor volume (subtitled “Fundamentals” [2]). As regards the two volumes’ philosophy and pedagogy, these are well captured by the author’s statement in the preface to the current volume:

Engineers who wish to play important roles in developing future technologies must do more than just deal with [the] black boxes [that current Fourier-analysis tools provide]. Engineers must understand the basis of the present Fourier technology in order to create and build up new technologies based on it.

This second volume succeeds very well in supplying the “knowledge . . . required for in-depth understanding of . . . the *application* of Fourier analysis to a wide area” [emphasis mine].

My take on the nature of Fourier analysis compels me to continue, as briefly as I can, in the above philosophical vein, so as to support my conclusion that this book, along with its first volume, strikes the exact theory-practice balance that imparts that “no-frills-or-side-streets” understanding, which is required for the development of new applications.

The book’s main effect on me was to help me emerge from a learning rut, which I’m sure was self-imposed via the imaginary Scylla (mathematical rigor) and Charybdis (applications) that my stubbornness and non-commitment constructed. Two (blameless) pillars of this rock-and-a-hard-place dilemma were the now-classic Körner book [3] and, say, Jackson’s excellent, ultra-pragmatic electrodynamics book [4] or (more extremely) the seminal book of P. A. M. Dirac [5]. The latter’s immortal statement to his student, the prominent mathematician Harish-Chandra, was: “I am not interested in proofs; I am only interested in how nature works.” And then there was Nobel laureate I. I. Rabi’s remark upon introducing Werner Heisenberg’s final public lecture, which I had the privilege to attend, at Columbia University: “Heisenberg’s great discovery was based on a misunderstanding of Fourier’s theorem.” (Rabi acknowledged, from the pulpit, Heisenberg’s help in Rabi’s decision to “go into physics,” instead of opting for a “career in real estate.”)

I am far from being alone in having had experiences similar to this with respect to Fourier analysis, and I endorse this book as an excellent preventative of perpetual bouncing between the above two extremes. Its six chapters begin where volume 1 left off: “Convolution,” “Correlation,” “Cross-Spectrum Method,” “Cepstrum Analysis,” “Hilbert Transform,” and “Two-Dimensional Transform.” The short appendices are nevertheless thorough and deal with derivations and rewriting of formulas. The figures are well rendered, not overly busy, and multi-colored when it enhances clarity. The author gives the answers to odd and even-numbered exercises, and this is a good thing in a book such as this; it certainly does not constitute spoon-feeding. The index entries are now in exact lexical order, thus correcting a minor flaw of the first volume.

This excellent book is best read and worked in linear order, beginning with the first volume. However, in my first scan, two things caught my eye, both having to do with my personal “world line:” The first was the word “causality,” and the section thereon. Here, the recollection of dispersion relations, the complex index of (light) refraction, and the work of the great Eugene Wigner [6] came to mind. Suffice it to say that the Fourier-analysis-related theory is very tricky, but this book is a help, not a confusing hindrance. The second word (phrase) was “Hilbert transform.” It reminded me of having walked into a lab (decades ago, in an aerospace venue) and having spotted an engineer at a computer who pushed the proverbial button, thus performing a Hilbert transform. I thought that David Hilbert, one of those mathematical personalities who was the purest of the pure (and who asked in public, “Excuse me, what is a Hilbert space?”), should have witnessed *that* push of a button! Be that as it may, the book does much to demystify transforms in general, and that named Hilbert in particular.

The claim on the book’s back cover, that it is “a thorough, accessible introduction to advanced digital Fourier analysis for undergraduate and graduate students,” is in my view fully justified. Seasoned professionals will certainly find it very useful.