Rellich's theorem for Sobolev space on the torus

Question

From John Roe: Elliptic operators, topology and asymptotic methods, page 73:

Let $H^{k}$ be the Soblev space defined on the torus $\mathbb{T}^{n}$ with the discrete $k$-norm: $$ \langle f_{1}, f_{2}\rangle_{k}=(2\pi)^{k}\sum_{v\in \mathbb{Z}^{n}}\tilde{f}_{1}(v)\overline{\tilde{f}_{2}}(v)(1+|v|^{2})^{k} $$

John Roe claimed that there is a Rellich type compact embedding theorem available. If $k_{1}<k_{2}$, then the inclusion operator $H^{k_{2}}\rightarrow H^{k_{1}}$ is a compact linear operator. The proof goes with the following steps:

1. Let $B=\{x:|x|=1,x\in H^{k_{2}}\}$.
2. Let $\epsilon>0$, choose subspace $Z\subset H^{k_{2}}$ such that $\dim (H^{k_{2}}/Z)<\infty$, and for all $f\in B\cap Z$, $|f|_{k_{1}}<\epsilon$.
3. The unit ball of $H^{k_{2}}/Z$ is compact, so can be covered by finitely many balls of radius $\epsilon$.
4. Hence $B$ can be covered by finitely many balls of radius $2\epsilon$ in $H^{k_{1}}$ norm. Since $\epsilon$ is arbitrary, $B$ is totally bounded and compact in $H^{k_{1}}$. Therefore the inclusion map is compact.

Here $Z$ can be explicitly constructed by taking it to be the space $$ \{f:\tilde{f}(v)=0,\forall v>N \} $$ where $N$ is some large enough constant.

I am fine with the strategy, but I am a little disturbed by $Z$'s construction at here. It is not clear to me that give $N$ large enough, I would be able to force all $f\in B\cap Z$ to have small enough norm. Can someone give me a hint? Thinking this in terms of Fourier series in the circle, it seems the terms $\tilde{f}(v)$ for $v>N$ can be arbitrarily close to $1$ and $|f|$ would also be quite large. For example if $k_{2}=3, k_{1}=2$, then there seem to be no reason $\tilde{f}$'s $H^{2}$ norm should be really small if the first $N-1$ terms are zero. I do not really know otherwise how to construct $Z$.

Answer 1

In the definition of $Z$, you probably want $|v|>N$ instead of $v>N$. Also, in item 1, the definition of $B$, you have the Sobolev norm of $x$, so it's better to use norm notation for that. Let's also not use subscripts in superscripts... say, $p<q$ and the embedding is $H^q\to H^p$. The $H^q$ norm is given by $$\|f\|_{H^q}^2 = (2\pi)^{k}\sum_{v\in \mathbb{Z}^{n}}|f(v)|^2(1+|v|^{2})^{q}$$ (oops, now I'm using superscripts in subscripts...) Suppose $f\in Z$, then $$\begin{split}\|f\|_{H^p}^2 &= (2\pi)^{k}\sum_{|v|>N }|f(v)|^2(1+|v|^{2})^{p}\\ &\le (2\pi)^{k} (1+N^2)^{p-q} \sum_{|v|>N }|f(v)|^2(1+|v|^{2})^{q} \\ &\le (1+N^2)^{p-q} \|f\|_{H^q}^2 \end{split}$$ The factor $(1+N^2)^{p-q} $ is small when $N$ is large. And when $f\in B\cap Z$, we get $$\|f\|_{H^p}^2 \le (1+N^2)^{p-q}$$