I've revised my question a bit in response to the (very helpful) advice so far–
I have an engineering background but am interested in learning abstract harmonic analysis. My interest is rather unstructured; the Fourier transform sparked my interest in mathematics, and I've been following since I first learned of it as a sort of guide through mathematics. I've decided it's time I delved into the abstract, and from my readings so far it's become clear to me that this will require a sojourn into a swathe of unfamiliar mathematics. My background in classical analysis, linear algebra, and probability is rather good (probably the rough equivalent of a 1st year american grad student), and so I'm familiar as well with point-set topology. I'm also familiar with measure theory (via probability, and some real analysis). I have essentially no abstract algebra.
I would be interested in three bits of advice:
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What algebra is essential for the canon of harmonic analysis? Perhaps this is a vacuous question (i.e., that an answer is as simple "just google" or "open a book")? It strikes me that techniques or background might work their way in unnamed. I'm hoping to prevent the sinking feeling that I don't know something I should.
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Specific examples (theorems, objects, counterexamples) which illustrate important applications of harmonic analysis, especially in relation to algebra. I'm not looking for explanations, just statements of what's thought to be important. I'm hoping that this information will help me sift through the different branches of harmonic analysis, especially where I am most unfamiliar with the language.
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Canonical/important texts which I might find useful. I'm especially interested in those good for self study (well-edited with many examples and end-of-chapter exercises, and ideally a conversational tone with lengthy remarks and some historical content).
Best Answer
Let me suggest that you look at the book "An introduction to topology and modern analysis", by Simmons. It covers concepts of point set topology that you presumably already know, but also gives a fairly concise, but quite readable, introduction to abstract algebra. It then brings these together in its final sections into a discussion of Banach algebras and related topics, culminating in a proof of the Gelfand--Naimark theorem. [See below for a brief remark about this theorem, and the other theorems mentioned in the subsequent paragraphs.]
Since you say that you are interested in abstract harmonic analysis, the Gelfand--Naimark theorem is a good place to start. (For example, it is not so far to go from there to the abstact form of Wiener's Tauberian theorem.) Note: I am interpreting abstract harmonic analysis to mean something like harmonic analysis on locally compact abelian groups (and related topics).
Simmons also has exercises, I think.
When I was studying this stuff, the next place I went to after Simmons was Naimark's tome Normed rings. (There are various editions, and some of the later ones might be called Normed algebras instead, if I'm not misremembering. They are translated from Russian, so the slighly unusual, and changing, name may be an artefact of this; I'm not sure. In any case, they are basically about the theory of Banach algebras and its applications to abstract harmonic analysis.) This is a place where one can read about various group rings of topological groups, Haar measure, the general form of Wiener's Tauberian theorem, and other concepts of abstract harmonic analysis. It is essentially too long to read from start to finish, but in my experience one can dip into it in bits and pieces, and having a firm understanding of the material from Simmons helps a lot.
Naimark's book is a monograph, not a textbook as such, and although it has many historical comments and illustrative examples (although the examples are often at a theoretically fairly high level), I don't remember it as having exercises. But in any case, I am not suggesting it as a first point of call, but as somewhere to go after you have some basics under your belt.
There is also a book by Loomis, An introduction to abstract harmonic analysis, which also treats Haar measure, various group rings, and so on. If I remember correctly it is less condensed than Naimark and also less comprehensive. My memory is that I preferred Naimark, but probably for idiosyncratic reasons. I don't remember whether Loomis's book has exercises.
All the books I'm mentioning are probably out of print, so I'm also assuming that you have access to a university library or something similar. (Any decent such library should have them.)
Finally, some fundamental results that I would recommend aiming for, which combine algebra and analysis nicely:
Gelfand's generalization of Wiener's theorem (that if $f$ is a nowhere zero continuous periodic function whose Fourier series is absolutely convergent, than the Fourier series of $1/f$ is also absolutely convergent).
the Gelfand--Naimark theorem (identifying certain commutative Banach algebras with extra structure as being algebras of continuous functions on compact topological space; it is a beautiful generalization of the classical spectral theorems for matrices).
The generalization of Wiener's Tauberian theorem to arbitrary commutative locally compact groups. (Wiener's original theorem says that if $f$ is an $L^1$-function on the real line whose Fourier transform is nowhere zero, then the translates of $f$ span a dense subspace of $L^1$.)
More abstract, but basic to the previous example and lots of other things, is the existence of Haar measure for any locally compact group.
As already mentioned, the first two results are in Simmons, and are easier, but already involve a very nice interplay between analysis and algebra. (The general framework is that of Banach algebras, which combine the analysis of Banach spaces with the algebra of rings, ideals, and so on.)
The second two results are in Naimark, and Haar measure is also in Loomis (and many other places) (and the general form of Wiener may be in Loomis too; I forget now).