Let $u\in C^\infty(\mathbb T^k)$, where $\mathbb T^k$ is the $k$-dimensional torus. (Equivalently, $u\in\mathbb R^k$ and $u$ is $2\pi$-periodic with respect of each argument.) We define the semi-norm $$ \|u\|_s=\|(-\Delta)^{s/2}u\|_{L^2(\mathbb T^k)}=\Big(\sum_{\ell_1,\ldots,\ell_k\in\mathbb Z}(\ell_1^2+\cdots+\ell^2_k)^{s/2}\big|\hat{u}_{\ell_1,\ldots,\ell_k}\big|^2\Big)^{1/2}, $$ where $s>0$ and $u(x_1,\ldots,x_k)=\sum_{\ell_1,\ldots,\ell_k\in\mathbb Z}\mathrm{e}^{2\pi i(\ell_1x_1+\cdots\ell_kx_k)}\hat{u}_{\ell_1,\ldots,\ell_k}$.

My question is the following: *Which rate of growth of $\|u\|_s$, as $s\to\infty$, implies that $u$ extends holomorphically to an open neighborhood of $\mathbb R^k$ in $\mathbb C^k$?*

More specifically: *Which rate of growth of $\|u\|_s$, as $s\to\infty$, guarantees that $u$ extends holomorphically to
$$
\Omega_\alpha= \{(x_1+iy_1,\ldots,x_k+iy_k): x_1,y_1,\ldots,x_k,y_k\in\mathbb R\,\&\,|y_1|,\ldots,|y_k|<\alpha\} \subset \mathbb C^k,
$$
for a given $\alpha>0$.*

Non-Homogeneous Boundary Value Problems and Applications– Liviu Nicolaescu Dec 9 '13 at 16:22