Fourier Analysis Notes

This semester I’ve been teaching a course on Fourier Analysis, which has, like just about everything, been seriously disrupted by the COVID-19 situation. Several class sessions have been canceled, and future ones are supposed to resume online next week. To improve matters a bit, I’ve been writing up lecture notes for the material since in-person lectures were canceled, and we’ll see how long I have the energy to keep this up.

The website for the course is here, giving detailed information about what it covers. In terms of level of mathematical rigor, the concept is to use the course as an opportunity to give students some motivation for a conventional real analysis course. The only prerequisite for the course is our usual Calculus sequence, which is not proof-based. In this class students are expected to try and follow proofs given in the book and in class, but not expected to be very good at coming up with their own proofs in the assignments, which mostly are computational. The textbook (Stein-Shakarchi) is based on a Princeton course with a somewhat similar philosophy of providing an introduction to analysis, but it is very challenging for the students to follow. I’ve looked around, but not found a better alternative. Other books on the subject tend to be either books for mathematics students that are even more abstract and challenging, or books for engineers that focus on either signal analysis or PDEs. Since the math department already has a PDE class I want to emphasize other things you can do with the subject.

The first set of lecture notes I wrote up were only loosely connected to Fourier analysis, through the Poisson summation formula. They dealt with theta functions and the zeta function, giving the standard proof of the functional equation for the zeta function that uses Poisson summation. I confess that one reason for covering this material is that I’ve always been fascinated by the connection between theta functions, quantization, representation theory (through the Heisenberg and metaplectic groups), and number theory. This subject contains a wealth of ideas that bring together fundamental physics and deep mathematics. On the mathematics side, this story was generalized by Tate in his thesis, where he developed what is essentially the GL(1) case of the modern theory of automorphic forms that underpins the Langlands program. On the physics side, one can think of what is going on as the standard canonical quantization of a finite-dim phase space, but with a lattice in the phase space giving a discrete subgroup of the usual Heisenberg group, and lots of new structure. For the details of this, one place to look is volume III of Mumford’s books on theta functions.

The second set of lecture notes, which I’ve just started on, are intended as an introduction to the theory of distributions, a topic that isn’t in Stein-Shakarchi. I highly recommend the book by Strichartz referred to in the notes for more details, with the notes maybe best used just as an introduction to that book.

I don’t want to turn this blog into yet one more place for discussion of the COVID-19 situation that just about all of us are obsessed with at the moment. If you’re interested in my personal experience, I’m doing fine. Almost all of us in New York are now pretty much confined to our apartments (it helps that the weather outside today is terrible), other than for short ventures out to get food or some exercise. I’d like to optimistically think that New York started taking action to stop the virus spread early enough to avoid disaster at the local hospitals. The best place I’ve seen to try and follow what is happening there is this web page. We’ll see in the next couple days if the problem has started to peak, or has much further to go.

I’m in much better shape than most people, having left town early for a spring break vacation. I had been planning a trip to Paris, at the very last minute instead rented a car and started out on a road trip in the general direction of New Orleans, consulting coronavirus report maps for where to avoid. Ended up in Memphis and then the Mississippi Delta (it had become clear New Orleans was a bad idea) before finally deciding that the situation was getting serious everywhere and it was time to head home. So, ended up back here in New York in relatively good mental shape for the confinement to come. Good luck to all of us in dealing with the coming challenges…

Update: The semester is now over, and I’ve put together all the notes I wrote up in one document, available here. This fixes mistakes/typos/etc. in earlier versions of the notes, so I’m changing the links to point to the final version.

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9 Responses to Fourier Analysis Notes

  1. Akhil says:

    Thanks Peter! We hope more genuine content on the Fundamental physics-Mathematics connections side is coming, which is hard to find elsewhere.

  2. Roderic C Deyo says:

    Perhaps it’s not fully appropriate for the course since it uses complex analysis, but I noticed that Richard Bellman’s wonderful short classic “A Brief Introduction to Theta Functions” was missing from the references.

  3. Hugh Osborn says:

    You are missing a – in the heat kernel in lecture 2

  4. Peter Woit says:

    Hugh Osborn,
    Thanks! Fixed.

  5. Jackiw Teitelboim says:

    Dear Peter,

    It’s interesting that Vol.I of Stein & Shakarchi gets very very close to introducing the notion of Schwartz’s distributions in their discussion of Poisson, Dirichlet, heat kernels &., but always avoids introducing the notion of continuous linear functional on the space of test functions.

    There’s a short, rigorous and quite elementary book by Friedlander, “Introduction to the theory of distributions,” which as a student at the time I found a valuable complement to Stein & Shakarchi.

    Regards.

  6. David Brown says:

    In the second set of notes, on page 1 in the last paragraph, shouldn’t “Schwarz” be “Schwartz” for Laurent Schwartz? No tee Schwarz is either the string theorist Schwarz or Hermann Amandus Schwarz.

  7. Peter Woit says:

    David Brown,
    Thanks, yes it’s Laurent Schwartz. Fixed.

    Jackiw Teitelboim,
    Thanks, the Friedlander/Joshi book is a good more advanced reference for distribution theory. I still would recommend the Strichartz book as less heavy on the theory and having more about applications and motivation.

    I can see why Stein-Shakarchi decide not to introduce distributions, but then you miss a huge part of the subject. The simplest way to introduce distributions, especially in the context of Fourier analysis, seems to me to be to stick to tempered distributions (i.e. Schwartz functions as test functions), but I don’t think I’ve seen any text that does this.

  8. Babak says:

    Dear Peter,

    As you know, there’s also a group theory point of view that can be found for instance in a beautiful book by Audrey Terras

    https://www.amazon.ca/Fourier-Analysis-Finite-Groups-Applications/dp/0521457181

    I think that is perhaps the true reality of Fourier analysis.

  9. Babak says:

    David: There’s also Albert Schwarz. 😊☺️

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