1

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.

newbie
  • 3,441
  • This 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? –  Jan 16 '17 at 22:41

0 Answers0