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I'm new here, and I hope this is within the scope of the website. I'll try to ask few advisory type questions in the future... I'm a college junior, and I was wondering if you guys could offer any course guidance on independent studies I could try to take my senior year. I have some ideas, but I was wondering whether you guys could give me any recommendations, especially textbook recommendations.

My background: The professors I am closest with here (and whom I will probably ask to write my recommendations for graduate school) are both specialists in Harmonic analysis, so I'm thinking of going deeper into more advanced analysis coursework. I didn't come into college wanting to go into mathematics, so keep in mind that I only started taking mathematics courses last year. Nonetheless, by the end of my junior year I'll have taken:

Calc I, II (AP BC calc)

Multivariable Calculus

Linear Algebra

Analysis I, II (Wade...)

Ordinary Differential Equations

Algebra I

Algebra II

Probability Theory

Differential Geometry (Barrett O'Neil)

Complex Analysis (Ruel & Churchill, though prof's notes gave a more rigorous treatment, though still very much at an undergraduate level.)

I've gotten A's without too much difficulty in all of my classes, and I have currently worked through these books through self study: Hardy & Wright's Intro to Theory of Numbers (no exercises, tried to work out proofs of theorems myself before reading them in the book). GF Simmon's intro to topology and modern analysis (did all of the problems. Did not get up to the last few chapters, though.)

and I am currently reading through Munkres' Topology on my own (and working through the problems).

Therefore, I have experience in analysis to the degree of finishing Wade, and I have developed quite a bit of topological knowledge through Simmons, Munkres.

EDIT: So I cut out quite a lot of what I was thinking because apparently I should really work through baby Rudin and learn Lebesgue Integration earlier. I have winter break (in which i usually work extremely hard on maths), next semester, and all summer (minus possible internships/research time) to go through baby rudin, and learn as much measure theory/lebesgue integration as possible.

Given this new addition to my background, what would be the best suggestions for real analysis/fourier analysis texts?

Any suggestions are welcome! I just want to best position myself for applying to a specific group when I apply to grad school.

Also, feel free to recommend other classes I really should take, but not in place of answering my questions. I have a lot of free space my senior year, so I can take these independent studies while still filling in any other holes in my learning.

Thanks so much!

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Do baby Rudin. Even though much of it is topical overlap, the proofs and problems are generally a cut above other textbooks. It's not a waste of effort. Don't build too high without strengthening your foundation. –  Arkamis Nov 16 '12 at 18:34
    
I have a year until fall 2013, and I have a copy of Baby Rudin (began working through it). Since this wouldn't be my first encounter with analysis, even the topological concepts, do you think I would be able to work through the proofs/problems myself before senior year (I've had really good retention with the other books I've self studied)? Or should I really get a professor to walk me through it? I understand that I want as strong a foundation as possible but at the same time I started getting into mathematics a bit later than others applying to graduate school and I need to play catch up. –  Dave Nov 16 '12 at 18:49
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If you've seen analysis stuff before, the first four chapters should go easily; nevertheless, there are many good problems contained in these chapters. Even if you have seen the topology stuff before, the material on integration and differentiation is presented differently from most analysis books that I've seen. By presenting topology early, Rudin doesn't jump "back and forth" like some other books. So there are new ideas to be had there, and again, some very good problems. –  Arkamis Nov 16 '12 at 18:53
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Simmons especially covers a lot on metric spaces, banach spaces, and hilbert spaces. You may want to look at Neal L. Carother's text Real Analysis (2000). Although his book is probably a little on the easy side for you at this time, I think it would prove useful to have this book on hand as a reference to fill in any possible gaps you might later find yourself having. The writing is clear and the book has many exercises and extensive (worthwhile) references. –  Dave L. Renfro Nov 16 '12 at 20:40
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I'd definitely recommend baby Rudin for general introductory analysis, his followup textbook is also my favourite analysis book. Fourier analysis is generally very reliant on Lebesgue integration. A book which uses only the Riemann integral (if I recall correctly) is Dietmar's first.

The route I took into harmonic analysis was starting with A (Terse) Introduction to Lebesgue Integration by Franks, which has an online draft here, which introduces the Lebesgue measure/integral in a very rigorous manner before establishing the basic $L^{2}$ treatment of Fourier series on $\Bbb{T}$. After that, the best recommendation that most people interested in harmonic analysis should read is katznelson's book, which covers the standard Fourier transform on $\Bbb{R}$ material very nicely, as well as sketching the locally compact abelian group stuff. From there, there seems to be less of a general consensus. I found Rudin's Fourier Analysis on Groups excellent for the locally compact abelian case, giving nice proofs of several theorems for which I wasn't happy with the proofs given in other books. I also enjoyed Classical Harmonic Analysis and Locally Compact Groups by Reiter and Stegeman as more of a broad introduction to abelian harmonic analysis, although it does ommit quite a few key proofs. I can't offer many references beyond the abelian case, and certainly not beyond the compact case, but the second of Deitmar's books was my favourite general reference for introductory nonabelian harmonic analysis. I've not read much of it, but my favourite treatment of the compact case is that found in Folland's book, which is online here; in particular, I found that its description of the representation theory was much more natural than other treatments.

I'm not American, so I can't relate what I've said to the courses you've listed, but hopefully this will help somewhat. I'd certainly recommend starting with Franks and Katznelson.

As a general aside related to your comments, I'd recommend trying to read as much as possible without asking for help from your professors - even if you ultimately have to ask for some help understanding something, you'll get much, much more from it if you only ask for help once you've beaten your brains out trying to understand it.

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So do you think it would be a good idea to work through any holes in my analysis study with Baby Rudin by myself? Since a lot of it would conceptually be review, it'd mainly just be looking at the proofs/doing problems. Do you think that within a semester + one summer I could teach myself enough Real Analysis and Lebesgue Integration to jump right into something like Rudin's second book or something like katznelson's book? I'll edit my question to reflect my new plans based on your suggestions. –  Dave Nov 16 '12 at 20:30
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This book doesn't get the recognition it deserves:

Pugh's "Real Math. Analysis."

It gives a very intuitive approach and quit thorough in its presentation.There are many examples, many, many problems including some from Berkeley pre-lim exams.

Especially good for self-study.

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I have all 4 Stein and Shakarchi books, and I've found them to be very useful in my first couple years of grad school. My graduate real analysis course used book III, so I bought the others to have the complete set. They have pros and cons, of course. Pros: they cover a ton of material, have more good exercises than you could possibly finish in a year (if you can, props to you), and approach everything in as rigorous a manner as possible. In particular Book I is possibly the most digestible rigorous introduction to Fourier analysis that I know of, at least at the advanced undergraduate level. The first chapter reviews motivations from partial differential equations, but from there on out you're developing the basic Fourier series convergence results and the Fourier transform on $\mathbb{R}$ and $\mathbb{R}^d$. There are tons of 'applications' which is good to stay grounded. The last two chapters are a nice short introduction to finite Fourier analysis and some analytic number theory, a nice contrast to most of the rest of the book.

Book III is again, in my opinion, a very digestible introduction to Lebesgue theory. Whereas many books on real analysis (like Papa Rudin) tend to start with the definition of a measure space, S&S stick with Lebesgue measure and integration on $\mathbb{R}^d$ for most of the book. I would definitely recommend the first 3 chapters. You can probably do better for the abstract measure theory (Papa Rudin is a good choice). Cons of the S&S books: some of the proofs are a bit hasty, so make sure you flush out all the details for yourself. Also, some of the exercises are a worded bit confusing at first. The other problem I have with the books is that each book isn't really self contained - they make a lot of references to the other books in the series.

Another good, albeit lighter, introduction to measure theory is "Measure, integral and probability" by Capinski and Kopp.

As far as getting in to harmonic analysis, I'm not sure if there is a perfect book at your level. I did an independent study course last year and we worked out of "real variable methods in harmonic analysis" by Torchinsky. It was dense, but I was able to get through the basic chapters (Weak Lebesgue spaces, Interpolation of Lebesgue spaces, the Hilbert transform) in less than 10 weeks. Plus it's a Dover book, so it's cheap.

Some might disagree with me on this, but I've found that a great way to motivate what I need to learn is to keep a "goal" book or paper in mind that I'd eventually like to understand. I've had a few of these shelf-dwellers staring me down for the past year or two, including the harmonic analysis Bible (Stein's "Harmonic Analysis"), Grafako's "Modern Fourier Analysis," and so on. Occasionally you might head to the library and just thumb through a book to see where you're at (if you don't recognize most of the words, you've got a lot more work to do!)

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