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If $H \subseteq G$ are abelian groups, how does $L^2(H)$ embed into $L^2(G)$? In particular, I am trying to find the Pontriyagin dual of the p-adic numbers.

Example

Here is a rather stunning visualization of the $2$-adic case. However, I could not translate this into Fourier series.

I am trying to understand how fourier series works on the $p$-adic numbers $\mathbb{Z}_p$. These numbers are a direct limit: $$ \mathbb{Z}_p = \lim_{\to} \mathbb{Z}/p^k \mathbb{Z}$$ An element of this ring is a sequence of numbers $a_m$ such that if $m \leq n$ then $a_m \equiv a_n \mod p^m$, looking at the smaller exponent.

For any finite $k$ there is a meaninigful discrete fourier transform and therefore it could be possible to extend this to the p-adic numbers. We can define a sequence of functions: $$ f_m: \mathbb{Z}/p^m \mathbb{Z} \to \mathbb{C} $$ and it will have an expansion as a linear combination of waves $e_m(x) = e^{2\pi i \,\frac{x}{p^m}}$. This does not seem right.

cactus314
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  • I think my mistake is to try to embed $\mathbb{Z}_p$ inside the unit circle $S^1$ certainly not with the topology inherited from $\mathbb{R}$. My goal is to write down explicit characters so I can evaluate a Fourier series in $p$-adic numbers (in fact in the adeles, $\mathbb{A}$). – cactus314 Nov 18 '16 at 17:05
  • Your question does not really have much of an answer... If H is $\mathbb Z$ and G is $\mathbb R$ there is no sensible embedding. – Mariano Suárez-Álvarez Nov 18 '16 at 17:11
  • @MarianoSuárez-Álvarez well... there are several. You can place $a \mathbb{Z} + b \subset \mathbb{R}$ for any real numbers $a,b \in \mathbb{R}$ usually $b = 0$. Is this the canonical choice? No, but we can always do something. – cactus314 Nov 18 '16 at 17:56
  • I don't understand in what way that is an embedding of $L^2$s. – Mariano Suárez-Álvarez Nov 18 '16 at 18:02
  • @MarianoSuárez-Álvarez Hmm... $L^2(\mathbb{Z}) \not \subseteq L^2(\mathbb{R})$ I think I left out the word "finite" in my question. However, consider the Nyquist-Shannon sampling theorem used to store real-time signals using discrete-time Fourier analysis. – cactus314 Nov 18 '16 at 18:11

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