Let us denote by $L^2([0,2\pi])$ the space of all periodic functions that are square integrable. Usually one defines the $H^s$-space for $s>0$ by \begin{align} H^s([0,2\pi]) = \left\{ u \in L^2([0,2\pi]) \, \bigg| \, \int_{(0,2\pi)} \int_{(0,2\pi)} \frac{(u(x)-u(y))^2}{|x-y|^{1+2s}}dx\, dy<\infty\right\}. \end{align}
I've come across several references where also the definition \begin{align} \left\{ u \in L^2([0,2\pi]) \, \bigg| \, \sum_{m\in \mathbb{Z}}(1+m^2)^s |\hat{u}(m)|^2<\infty \right\} \end{align} is used. Here, $\hat{u}$ is the Fourier transform of $u$. Now I'm wondering why both definitions give the same space. I know the equivalent definition via the Fourier transform in the case $H^s(\mathbb{R})$. But somehow, the known techniques do not work if I want to show that both norms are equivalent in the case of periodic functions. Is there an easy trick?
Maybe someone also knows some nice references where fractional Sobolev spaces for periodic functions are treated. I only found references for the case when $s$ is an integer.
If a function is periodic, then we can use Fourier Series instead of the Fourier Transform to represent it. Then the above result is intuitively similar to Parseval's Theorem.