Im reading about fourier transform $\widetilde{f:}L^{2}(T)\rightarrow l^{2}(\mathbb{Z})$. The periodic space of Sobolev $H^{s}$ are the functions f in $L^{2}(T)$ such that $\sum_{k\epsilon \mathbb{Z}}(1+\left | k \right |^{2})^{s}\left | \widetilde{f(k)} \right |^{2}< \infty $.
how can i show that the next three norms are equivalents in $H^{m}$ for $m \epsilon \mathbb{N}$?
$\sum_{j=0}^{m}\left \| f^{(j)} \right \|_{L_{2}}$ with $f'(x)=\sum_{k\in \mathbb{Z}}ik\widetilde{f(k)}e^{ikx}$, $(\sum_{k\epsilon \mathbb{Z}}(1+\left | k \right |^{2s})\left | \widetilde{f(k)} \right |^{2})^{1/2}$ and $(\sum_{k\epsilon \mathbb{Z}}(1+\left | k \right |^{2})^{s}\widetilde{f(k)}\overline{\widetilde{f(k)}})^{1/2}$
A function $f \in L^2$ is absolutely continuous and periodic on $\mathbb{R}$ with period $2\pi$ and $f' \in L^2$ iff $\sum_{k}k^2|\widetilde{f(k)}|^2 < \infty$. In such a case, $\|f'\|_{L^2}^2=\sum_{k}k^2|\widetilde{f(k)}|^2$.
Likewise $f$ is $n$-times absolutely continuous and periodic on $\mathbb{R}$ with $f^{(n)}\in L^2$ iff $\|f^{(n)}\|_{L^2}^2=\sum_{k}k^{2n}|\widetilde{f^{(n)}(k)}|^2 < \infty$.
It is clear that $(1+k^{2n})\le (1+k^2)^{n}$ holds for all integers $k$. So the only thing you have to do to prove equivalence of norms is to show that there is a constant $C_n$ such that $(1+k^{2})^n \le C_n(1+k^{2n})$ for a given $n$ and for all integers $k$. It's an easy estimate that $k^{2j} \le k^{2n}$ holds for $1 \le j \le n$ for all integers $k$, and that's enou.gh