Query: In what reference could I find the definition of the Sobolev space using Fourier series? I mean the following:
The Sobolev space with the Fourier transform is defined by
$$H^2(\mathbb{R})=\left\{u\in L^2(\mathbb{R}: \int_{\mathbb{R}} (1+\xi^2)^2|\widehat{u}(\xi)|^2\,d\xi<\infty\right\}$$ where $\widehat{u}$ is the Fourier transform of $u$.
So the Sobolev space with fourier series should be the following?
$$H^2(\mathbb{T})=\left\{u\in L^2(\mathbb{T}): \sum_{m\in\mathbb{Z}}(1+m^2)^2|\widehat{u}(m)|^2<\infty\right\}$$ where $\widehat{u}$ is the Fourier series of $u$.
thank you