# Road map to teach undergrads a first course in real analysis that concludes with convergence of fourier series

I am planning to teach (unofficially, I am a Grad student) a course in real analysis.

Aim of the course is to understand the convergence of Fourier series.

I want to start with the notion of sequence of real numbers, their convergence, metric space, convergence of sequence in metric space and so on...

Has any one tried such a course with the time bound of 20 lectures, 1.5 hours each? Any suggestions are welcome.

I want to teach derivatives and integration. 30 hours may be less for usual course in real analysis that start with definition of convergence of sequence of real numbers to reach till the definition of Fourier series. What I had in mind is some of the topics can be skipped without breaking the flow. So, I am asking what would be a sequence of topics that starts with notion of limit, continuity and ends with convergence of Fourier series. This is for the benifit of students who take physics as major in their undergraduate. They might not take more courses in analysis in their undergraduate but use the setup of Fourier series without being sure if that makes sense or not.

• Welcome!! A little more detail would be good; who is the audience? Are you looking for a textbook? Oct 29, 2019 at 18:01
• I checked out the MIT OCW catalog to see how your class would line up. ocw.mit.edu/courses/mathematics It seems like you plan to teach the first half of 18.100 and then go to the second half of 18.103. Based on what (little) I know of Fourier series, this might be achievable. The biggest question in my mind is whether your school would want you to teach a class called "Real Analysis" that jumped over to Fourier Analysis without teaching derivatives and integrals. But if you've got buy-in for that, then go for it. Oct 29, 2019 at 20:25
• The closest fit I found on my bookshelves is Introduction to Real Analysis by Burton Randol (1969) --- Chapter 1: Numbers, Sequences, and Series (5 sections, pp. 1-25); Chapter 2: Functions (4 sections, pp. 26-38); Chapter 3: Power Series (7 sections, pp. 39-57); Chapter 4: The Weierstrass Approximation Theorem (2 sections, pp. 58-69); Chapter 5: Fourier Series (5 sections, pp. 70-94); Chapter 6: The Lebesgue Integral (2 sections, pp. 95-108). See also my comments about it here. Oct 29, 2019 at 22:15
• I am not looking for a book but a reasonable sequence that starts with limit, continuity and ends with Fourier series convergence... @Chris Oct 30, 2019 at 1:28
• Thanks for the links. it is useful. @Matthew Daly Oct 30, 2019 at 1:34

I can think of two books which might provide a roadmap, though you would have to flesh it out a lot depending on your prerequisites and the timeline you propose.

• Frank Morgan's Real Analysis has a minimum of prereqs and a maximum of topology and series. Now, it doesn't construct the real numbers (they are just infinite series of decimals, if I recall correctly), and saying it is terse in exposition is an understatement. But if you know what you are doing and choose sections carefully, you can definitely get to not just Fourier series convergence, but also Lebesgue integration.
• David Bressoud's A Radical Approach to Real Analysis instead takes a historical view of the subject, both starting and ending with Fourier series. You may find this review helpful.

In any case, whether or not you adopt a particular text, getting them from the library should give you a few sense of how to approach this goal. If I had to do it, I'd try the historical approach, but the direct one also has a charm.

• Thanks for the links. I will see. Oct 31, 2019 at 4:26

This is just an idea, not a complete answer. And may be wrongheaded, but doing what I can.

You might look at the approach in courses called "sequences and series". For instance, see here:

https://www.usna.edu/MathDept//_files/documents/courses/fall-2019-2020/SM333_Fall20.pdf

If there are some things missing, you may need to add them. Or to find things to prune. But at least it gives you a skeleton to start with. Also, note this looks like 60 hours of instruction, not 30.

FYI: USNA has three tracks for real analysis. A standard one (with a regular approachable real analysis text), an honors one with Baby Rudin, and then this one (which looks easiest and most applied). The honors track also allows for a followon second semester.

In addition, I recommend to look at the textbook Advanced Engineering Mathematics by KReyszig and the three chapters entitled Fourier Series; Sequences and Series; and Power Series, Taylor Series and Laurent Series. I generally find Kreyszig to be very time efficient in conveying useful content and not making it rely on things when it doesn't need to. (Maybe still not all you need, but I suspect those three chapters could be covered in 30 hours.)

P.s. Kudos for the specificity of the time duration. But would also be good to give some other factors. What level students (in terms of prereqs and brains). Also, what is driving this course description, is it some sort of specialized math track or service to another department? (May be insights from that.)

• Thank you for the link... It looks like what I want... I will add more details about the students prerequisites Oct 31, 2019 at 4:10
• If they are physicsers, definitely go with the sequence/series approach (doing convergence tests) and skip some of the less applied stuff (real number construction, L-thingie integral). Oct 31, 2019 at 18:26
• I especially like the K approach in that they introduce FS BEFORE PDEs, thus independent of it. And give lots of practice in actually using them to model functions. Other books often address FS as an outgrowth of PDEs and never give any independent practice in using the FS. The approach of looking at FS first means they are familiar when getting to PDEs and encountering them there. Oct 31, 2019 at 18:28
• I really like the Kreyszig book for physicists. It's much more approachable and more of a normal mathbook than Arfken Weber (kind of a mess of hodgepodge topics, without good explication even of what it selects). There's a lot of other stuff in there that helps the physicsers also. P.s. They will end up buying AW also for grad school...but if both books address a topic, K does it better, so you actually learn the math topic (not just cite a formula to plug into a problem). Oct 31, 2019 at 18:30