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In nonparametric statistics, the following space is often used $$H_{per}^\beta := \left\{f:[0,1]\to\mathbb{R}:\,D^{\beta-1}f\,\text{absolutely continuous and } D^\beta f\in L^2[0,1], \\D^{k}f(0) = D^{k}f(1),\quad \forall k\leq\beta-1 \right\}. $$ I think it is the classical Sobolev space (on a compact domain) when $\beta$ is nonnegative integers. Furthermore, since the space is a subspace of $L^2[0,1]$, we can find Fourier series for each element in $H_{per}^\beta$ and recharaterize the space as follows, $$ \left\{ f:\,f=\sum_{k\in\mathbb{Z}}c_k e^{-ik\pi x},\, \overline{c_k} = c_{-k},\, \sum_{k\in\mathbb{Z}}(k)^{2\beta}c_k^2 <\infty \right\}. $$ The space is extended by allowing $\beta$ to take nonnegative real value.

On the other hand, fractional Sobolev space is defined (in $\mathbb{R}$) by $$W_2^{\beta}(\mathbb{R}):=\left\{ f\in \mathcal{S}': \, \left((1+|\xi|^2)^{\beta/2}\hat{f}\right)^{\vee}\in L^2 \right\}.$$

I would like to know how these two types of spaces relate to each other if we restrict the second space to periodic distributions. If they do hold a relation, can it be extended to higher dimension, $\mathbb{R}^d$?

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  • $\begingroup$ The only periodic $L^2$-function on $\mathbb{R}$ is the trivial function. However, for $\beta\geq 0$, any function in $H^\beta_{per}$ is the restriction of a function in $H^\beta(\mathbb{R})$. $\endgroup$ – Liviu Nicolaescu Dec 17 '14 at 22:43
  • $\begingroup$ @LiviuNicolaescu Is there a correspondence if we restrict $W_2^\beta(\mathbb{R})$ to $W_{2,per}^\beta([0,1])$? I want to know if $W_{2,per}^\beta([0,1]) = H_{per}^\beta([0,1])$. $\endgroup$ – newbie Dec 19 '14 at 14:52
  • $\begingroup$ What you are actually considering are Sobolev spaces over a compact manifold without boundary, namely the one-dimensional torus. That these two constructions agree is standard, see e.g. the book by Adams-Fournier. $\endgroup$ – Delio Mugnolo Dec 19 '14 at 20:47
  • $\begingroup$ @DelioMugnolo Thanks for the comment, but I didn't find any discussion in book Sobolev Spaces about on manifolds. $\endgroup$ – newbie Jan 5 '15 at 12:09

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