Let $\mathbb{R}^n/\mathbb{Z}^n$ be equipped with the quotient topology with respect to the standard topology on $\mathbb{R}^n$, and denote this space by $\mathbb{T}^n$.

Is there a text introducing $\mathbb{T}^n$ rigorously with a view towards harmonic analysis?

For example, I'm curious how a natural measure on this space can be defined to talk about $L^2(\mathbb{T}^n)$, and here is what I have guessed. I can see the map $\phi:\mathbb{R}^n/\mathbb{Z}^n \rightarrow S^n$ defined by $\phi(x_1+\mathbb{Z}^n,...,x_n+\mathbb{Z}^n)=(e^{2\pi i x_1},...,e^{2\pi i x_n})$ is a homeomorphism. Moreover $\psi:[-\frac{1}{2},\frac{1}{2}]^n \rightarrow S^n:(x_1,...,x_n)\mapsto (e^{2\pi i x_1},...,e^{2\pi i x_n})$ is a continuous surjection. Hence a natural measure on $\mathbb{T}^n$ would be the pushforward measure of the $n$-dimensional Lebesgue measure with respect to $\phi^{-1}\circ \psi$. Am I correct? I'm curious if there is a text introducing $\mathbb{T}^n$ just as much of rigor as I explained above (I like formal and a bit dry texts, not focuing heavily on intuitions and examples). Thank you in advance.

  • $\begingroup$ Maybe Folland's textbook on abstract harmonic analysis? IIRC it starts with Haar measures based on the structure of $\mathbb{T}^n$ as a topological group, which is exactly how the general theory proceeds on other spaces. $\endgroup$ – user296602 Aug 9 '16 at 7:48
  • $\begingroup$ @T.Bongers I am skimming the text but I cannot find anything about $\mathbb{T}^n$ (maybe $(Z_2)^\omega$ is relevant to this? I am not sure). Btw this text seems very nice! $\endgroup$ – Rubertos Aug 9 '16 at 8:11

A text that contains an entire chapter entitled "Fourier Analysis on the Torus" is the following:

  • Grafakos' Classical Fourier Analysis, chapter 3.

At the beginning of the chapter it is said that "Haar measure on the $n$-torus is the restriction of $n$-dimensional Lebesgue measure to the set $\mathbb{T}^n = [0,1]^n$."

The treatment may not be as formal or as rigorous as you would like, though, but I find this text and its sequel to have a rare pedagogical quality for material that gets all the way to open problems in harmonic analysis.


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