In Tsybakov's book(Page 51), Sobolev space (or Ellipsoid) for positive smoothness parameter $s$ is defined with sequential model, i.e. the series of the Fourier coefficients is finite.
On the other hand, Sobolev space is defined in a general sense as follows,
$$W_2^{\beta}(\mathbb{R}):=\left\{ f\in \mathcal{S}': \, \left((1+|\xi|^2)^{\beta/2}\hat{f}\right)^{\vee}\in L^2 \right\}.$$ For subdomain $\Omega \subset \mathbb{R}$, the space is defined by restriction.
Besides, we also have the original definition of Sobolev space with integer-valued smoothness, defined with existence of derivatives in $L^2$ sense.
I am looking for references which discuss or connect the different definitions mentioned above.