Consider function $f$ contained in periodic Sobolev space $H^k$, then it has Sobolev norm $\|f\|_{H^k}^2 = \sum_i (1+i^k)^2 f_i^2$, where $\{f_i\}_i$ are Fourier coefficients.
I am wondering if $f\in H^s$ with $ 0<s\notin \mathbb{N}$, do we have similarly $\|f\|_{H^s}^2 = \sum_i (1+i^s)^2 f_i^2$, for any one of the standard fractional Sobolev spaces, i.e. defined by Fourier transform, as Slobodeckij spaces, etc.