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!
Best Answer
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.