How big is $ H^{\frac{1}{2}}_{per} (0,2\pi) $?

Question

Define $2\pi-$ periodic Sobolev function space as \begin{equation*} H^{r}_{per} (0,2\pi): = \{ \sum_{k \in \mathbb{Z}} a_k e^{ikt}: \sum_{k \in \mathbb{Z}} (1+k^2)^r\vert a_k \vert^2 < \infty \} \end{equation*} Let $ r =\frac{1}{2} $, how big is $ H^{\frac{1}{2}}_{per} (0,2\pi)$?

To ask more specifically,

1. Is it equivalent to $ H^{\frac{1}{2}} (0,2\pi) $, the classical Sobolev space of $ \frac{1}{2} $order?

2. Are wavelet (for instance, Haar function), finite element, spline, orthogonal polynomial (for instance, Legendre polynomial) contained in $ H^{\frac{1}{2}}_{per} (0,2\pi)$?

Thank you in advance for your consideration in this question!