I have some confusion about the definition of Sobolev space for a real positive real $r>0$.
I know that for a positive integer $m>0$ and $p\geq 1$ and any subset $\Omega$ of $\mathbb{R^n}$, we define the Sobolev space denoted by $W^{m,p}$ by $$W^{m,p}=\{f\in L^p : \text{the partial derivatives up to order m are in } L^p \}.$$ I am struggling with the definition of a Sobolev space by the summation with the Fourier coefficients. And I want also to know the general definition of Sobolev space for any real positive $r >0.$
As explained here https://en.wikipedia.org/wiki/Sobolev_space and here Is $B^s_{p,p} = W^{s,p}$?, there are several extensions of Sobolev spaces for non integer spaces: the Sobolev-Bessel spaces $H^{s,p} = F^s_{p,2} = (1-\Delta)^{-s/2} L^p$ and the Sobolev-Slobodeckij spaces $W^{s,p} = F^s_{p,p} = B^s_{p,p}$. More generally, the Besov spaces $B^s_{p,q}$ and the Triebel-Lizorkin spaces $F^s_{p,q}$ are intermediate spaces between Sobolev spaces of integer order.