Harmonic Analysis: From Fourier to Wavelets (Student Mathematical Library) (Student Mathematical Library - IAS/Park City Mathematical Subseries, 63)

I have worked with HA for decades in array processing and MR Imaging. I always enjoy authors who take a "specialty" area, then go really deep and broad to explore both the body and frontiers. The problem with handling HA this way is that the field, and Fourier transforms in general, has become SO broad and diverse that the math itself is all over the board in complexity, from "relatively" simple partial differential equations for continuous signal processing waveforms, to extremely difficult Hilbert space translations between pure and general functional analysis and HA, including rotational invariance of the Fourier tools and decompositions.Far from just thermodynamics and Newtonian physics, the dynamical systems and advanced Fourier applications have broadened HA to Neurology, Electronics, Quantum Physics, sound and musical eigenvalue applications, and many others. These don't even begin to scratch the surface of the REALLY advanced research in pure math such as topology, duality and other very abstract research areas. Note also that this book is about the Fourier aspects of non-musical applications (though they are covered and mentioned as examples), NOT the strictly "other" form of HA in music. I'm sure you know this, but I have many music oriented buddies who frown at my (and this book's) broad view of HA!The bottom line is that, even though the problems are very well presented and the book is brilliantly written, I'd take exception to the publisher's "undergraduate" statement. If you're a high level mathematician and perhaps a Senior at a really good technical school, maybe, but engineers like me, or even physics majors, will find much of this material challenging, and the author does not write it at a "slow general explain it from many angles" pedagogic level-- the pace is pure graduate style, with many assumptions that you will already know why that sine or radial/ spherical component popped up. I'd even venture to say that a previous Fourier course would probably be a requisite. When you start breaking functions into trig components, you better really know your trig both in calculus and linear algebra.Underneath it all, a transform is a transform, and the inverse, always the tougher little beast, isn't really very intuitive until some advanced analysis courses show you that a MAJORITY of problems are not amenable to the "relatively" easy solutions presented in mosts texts. Once you see how nasty many of the nonlinear, discontinuous functions can be in the advanced areas, like me, you might say... "Oh, yeah, good old heat dynamics and the superpositioning of nice tame waves..." Might even make you look longingly AT the music side...It might not be all that helpful to point out how diverse Fourier topics have become, but HA now even includes spherical harmonics, Bessel functions, Laplacian eigenvectors and even graph theory! The authors do a great job of covering the entire field, with very current leading edge ideas as well as centuries old applications and in between but still very important ideas like oscillatory integrals. If you're going into either applied engineering in areas like signal processing or imaging, or math research at the graduate level, there is no book more recent and complete than this fine work. I just don't want you to pick it up as an undergrad and find many of the author's relatively fast paced decompositions leave you digging back through 5 other texts! If you're already a pro using FT's in any capacity in your work, this is a must have for your library, as ALL the most recent "named" algorithms are covered (you know, Paley-Weiner, Peter-Weyl, etc.).Library Picks reviews only for the benefit of Amazon shoppers and has nothing to do with Amazon, the authors, manufacturers or publishers of the items we review. We always buy the items we review for the sake of objectivity, and although we search for gems, are not shy about trashing an item if it's a waste of time or money for Amazon shoppers. If the reviewer identifies herself, her job or her field, it is only as a point of reference to help you gauge the background and any biases.

I had spent a lot of time looking for a harmonic analysis book at the right level -- it seemed like the market was mostly split into very theoretical treatments like Rudin's Fourier Analysis on Groups, or books that were just trying to present the computational tools to engineers. I thought this book bridged the divide very well, and is well suited for self-study for students who have some advanced calculus or analysis under their belt.