Our semester here at Columbia is finally over, and I’ve put the lecture notes on Fourier analysis that I wrote up in one document here. A previous blog posting explained the origin of the notes: they cover the second half of this semester’s course, from the point at which the course became an online course due to the COVID-19 situation.
Not much blogging going on here, mainly since everyone staying home seems to have kept news of much interest to a minimum.
Great notes and thank you for making them public.
Peter,
this upcoming event might be pretty interesting.
https://conference.ippp.dur.ac.uk/event/906/
Correct me if this is way off:
> The Fourier transform is possible because the 1-sphere is parallelizable;
> What is known about generalizing concept to the 3- and 7-spheres?
I thought you could use a good laugh during these troubled times. Unfortunately, the article doesn’t seem to be in jest.
https://www.google.com/amp/s/whdh.com/news/nasa-uncovers-evidence-of-bizarre-parallel-universe-where-physics-time-operate-in-reverse/amp/
Justin B. Glick,
See
https://www.math.columbia.edu/~woit/wordpress/?p=11747#comment-236286
Geoffrey Dixon,
It’s not $S^1$ being parallelizable (all 1-manifolds are parallelizable…) that makes Fourier analysis work, it’s that $S^1$ is a group (rotations of the plane). $S^3$ is also a group (SU(2) or Sp(1), unit quaternions if you prefer). There is an analogy of Fourier analysis in this case (like for any compact Lie group). Abstractly
$$Functions(G)=\sum_{irreps\ V} V\otimes V^* =\sum_{irreps\ V} End(V)$$
Or, in the case of SU(2):
$$Functions(S^3)=\sum_{n=0}^{\infty}(\text{matrix elements of}(n+1)\times(n+1) \text{matrices})$$
This says that any function on $G$ can be expanded in matrix elements of irreducible representations, e.g. any function on $S^3$ can be expanded in matrix elements of the spin 0,1/2,1, etc. representations (these matrix elements depend on the group element, so are functions on the group).
If your group is commutative, all irreducible representations are one-dimensional, and you get an expansion in these (as character functions on the group). For instance, the $e^{in\theta}$ for $S^1$.
$S^7$ isn’t a group, so this won’t work.
Hi Peter,
But the group property is not necessary for orthogonal expansions to exist…
Spherical harmonics on $S^2$, expansions in terms of orthogonal polynomials, Bessel functions. Orthogonality and completeness are all proved (despite there not being a Peter-Weyl/Plancherel theorem, which you have for groups). Granted that some (but I think not all) of these are obtained by reducing from a group to a lower-dimensional manifold.
Not trying to be snarky here, just mentioning something that appears relevant. Maybe calling an expansion into orthogonal functions “Fourier” depends on the group property (since it is an expansion into group characters) of the manifold. That would make the term purely a matter of semantics. Beyond that, I’m not sure how it is important.
For example, the symmetry group of $S^7$ is SO(7) or $spin(7)$. There are certainly spherical harmonics, respecting this group symmetry, on $S^7$.
Hi Peter,
Yes, there are various other generalizations. There are bases of orthonormal functions of various kinds without an obvious group theory origin. For a pair of Lie groups G, H, you can think of functions on G/H as the subspace of functions on G that are right invariant under H. This is how you get spherical harmonics: think of $S^2$ as $SU(2)/U(1)$, functions on $S^2$ as functions on $S^3$ right invariant under the $U(1)$ action. In terms of the decomposition I wrote above into a sum over spins of $V\otimes V^*$, the $U(1)$ acts on one of these, with invariant piece one-dimensional for integral spin, zero for non-integral spin. So, the sum is over all integral spin representations, and this is exactly the decomposition of functions on $S^2$ into spherical harmonics.
While $S^7$ is not a Lie group (and $G/H$ in general is not parallelizable), $S^7$ does have an amazing (“unparalleled” by any other example) variety of different geometries, thought of as different identifications with a $G/H$.
$$S^7=Spin (5)/Spin(3)=Spin(6)/SU(3)=Spin(7)/G_2=Spin(8)/Spin(7)$$
(you might want to think of $Spin(5)$ as $Sp(2)$, $Spin(6)$ as $SU(4)$).
This might be entertaining for just over 40 mins.
The first part has some interesting aspects.
There are a couple of questions at the end, one re the landscape…
https://www.youtube.com/watch?v=GC2mqMmfW6o
Your comments on S^7 are fascinating (I knew about S^2), but I guess I was making the point that there also exist complete sets of orthogonal functions on manifolds in all sorts of situations; even with no symmetry (by which I mean no Killing vector at almost every point).
I suppose I could have made my earlier point clearer by not citing examples of manifolds with symmetry.
…but what seems clear is that when there IS symmetry, constructing orthogonal families of functions becomes an easier algebraic problem (with all kinds of beautiful aspects), rather than a hard and ugly analytic problem. Maybe that is the real point.
I found an 8 year old introductory text on Spherical Harmonics in q Dimensions (Frye and Efthimiou). It claims there is scant literature on this topic, so something at this level is probably great for someone who has taught advanced analysis, but not by preference, and decades ago. I name no names.
The group property is needed for convolution.
Spherical harmonics in any dimension are quite adequately discussed in the book by Hochstadt on Mathematical Physics.
Peter O: Hochstadt is mentioned in the text of Frye & Efthimiou. They say it is more advanced and definitive. I wonder if I own the thing – down in the Bat Cave. For the time being, given how unlikely it is at his point that it will help me crystallize my intuition, I’ll continue perusing F&E in a desultory way.