d}^k \varphi(u)|\big\}_{k \in {{\mathbb N}}_0}$. Then ${\ensuremath{{{\mathcal S}}_{{\ensuremath{\operatorname{Dir}}}}}}$ and $ {{\mathcal S}}_{\rm Neu}$ are Fréchet spaces, and we write ${\ensuremath{{{\mathcal S}}_{{\ensuremath{\operatorname{Dir}}}}}}'$ and $ {{\mathcal S}}_{\rm Neu}'$ for their topological duals.

Now, let $-\Delta$ be the closure of the Laplacian operator $-\Delta \colon {{{\mathcal S}}_{{\ensuremath{\operatorname{Dir}}}}} \to L^2([0,1]) $ as an unbounded operator in the Hilbert space $L^2([0,1])$. It is a positive and self-adjoint operator whose eigenvalues and eigenfunctions are given for any integer $m{\geqslant}1$, respectively by $\lambda_m=(m\pi)^2$ and $e_m(u)=\sqrt{2}\sin(m\pi u)$. We note that the set $\{e_m\; ; \;m{\geqslant}1\}$ forms an orthonormal basis of $L^2([0,1])$. We denote by $\{ T_t^{\rm Dir} \; ; \; t {\geqslant}0\}$ the semi-group associated to $\Delta$.

For $k\in\mathbb{N}$, let us define ${\mathcal{H}_{{\ensuremath{\operatorname{Dir}}}}^k}=\mathrm{Dom}((-\Delta)^{k/2})$, endowed with the inner product $$(\varphi,\psi)_k=\int_0^1(-\Delta)^{k/2}\;\varphi(u)\;(-\Delta)^{k/2}\;\psi(u)\, du.$$ By the spectral theorem for self-adjoint operators, ${\mathcal{H}_{{\ensuremath{\operatorname{Dir}}}}^k}$ equals $${\mathcal{H}_{{\ensuremath{\operatorname{Dir}}}}^k} = \Big\{\varphi\in L^2([0,1])\; ;\;
\|\varphi\|_k^2<\infty\Big\} ,$$ where $\|\varphi\|_k^2=(\varphi,\varphi)_k$ and for $\varphi,\psi\in  L^2([0,1])$ we also have $$(\varphi,\psi)_k = \sum_{m=1}^{+\infty}\bigg\{(m\pi)^{2k}\langle\varphi,e_m\rangle_{L^2([0,1])}\langle\psi,e_m\rangle_{L^2([0,1])}\bigg\}.$$ Moreover, if ${\mathcal{H}_{{\ensuremath{\operatorname{Dir}}}}^{-k}}$ denotes the topological dual space of ${\mathcal{H}_{{\ensuremath{\operatorname{Dir}}}}^k}$, then $${\mathcal{H}_{{\ensuremath{\operatorname{Dir}}}}^{-k}}= \Big\{{{\mathcal Y}} \in {\ensuremath{{{\mathcal S}}_{{\ensuremath{\operatorname{Dir}}}}}}' \; ;\;\|{{\mathcal Y}}\|_{-k}^2<\infty\Big\},$$ where $\|{{\mathcal Y}}\|_{-k}^2=({{\mathcal Y}},{{\mathcal Y}})_{-k}$ and the inner product $(\cdot,\cdot)_{-k}$ is defined as $$({{\mathcal Y}},{{\mathcal Z}})_{-k} = \sum_{m=1}^{+\infty}\Big\{(m\pi)^{-2k} {{\mathcal Y}}(e_m){{\mathcal Z}}(e_m)\Big\}, \label{eq:norm-k}$$ with $ {{\mathcal Y}}(e_m)$ denoting the action of 