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
Does the norm of fractional Sobolev spaces can be expressed in a series manner?
1
$\begingroup$
sobolev-spaces
-
0This is one of the ways to define a norm on $H^s$. The norm is equivalent to the other ones, but not necessarily identical. How would you use the Fourier _transform_ in the setting of the periodic space? – 2017-01-16