Let $u\in L^1(\mathbb T^n)$ on the torus. The Fourier transform of $u\in\mathcal S'(\mathbb R^n)$ (the temperated distribution) is given by $$\hat u=\sum_{k\in\mathbb N}c_k\delta _k,$$ where $c_k$ are the Fourier coefficients and $\delta _k$ the dirac with mass on $\{k\}$.
Sorry but I'm quite new with these distribution.
Q1) First, why $u\in \mathcal S'(\mathbb R^n)$ ? Does $u\in L^1(\mathbb T^n)$ implies $u\in \mathcal S'(\mathbb R^n)$ ?
Q2) The proof of the claim goes as follow : For $\varphi \in \mathcal S(\mathbb R^n)$. $$\left<\hat u, \varphi \right>=\int_{\mathbb R^n}u\hat \varphi \underset{(1)}{=}\sum_{k\in\mathbb Z^n}\int_{\mathbb R^n}u(x)\hat \varphi (x)\boldsymbol 1_{[0,1)^n+k}(x)dx$$ $$=\sum_{k\in\mathbb Z^n}\int_{[0,1)^n}u(x+k)\hat \varphi (x+k)dx=...=\sum_{k\in\mathbb Z^n}\varphi (k)c_k.$$
I don't understand why we can permute the limit and the integral in $(1)$. I thought it was dominated convergence, but since $u$ is not supposed bounded, what could be a function $g$ s.t. $|u\hat \varphi |\leq g\in L^1(\mathbb R^n)$.