Schwartz periodisation over a lattice is smooth with locally uniformly summable derivatives

Convergence of every derivative series. Fix $h\in \mathcal{S}({\mathbb{R}}^n)$ and a multi-index $\gamma$, and put $g:=({\mathrm{\partial }}^{\gamma }h)\circ A$, which lies in $\mathcal{S}$ by [F1]. For every $x$, writing $\lambda =Ak$ and $y=A^{-1}x$ gives $x+\lambda =A(y+k)$ and hence the termwise identity ${\sum }_{\lambda \in \mathrm{\Lambda }}{\mathrm{\partial }}^{\gamma }h(x+\lambda )={\sum }_{k\in {\mathbb{Z}}^n}g(y+k)$; by [F2] applied to $g$, this series and the derivative series converge locally uniformly in $y$, hence by [F5] in $x$. Absolute convergence at each fixed $x$: for the fixed $y$, the translate $z\mapsto g(y+z)$ is in $\mathcal{S}$ by [F4], so the absolute-convergence clause of [F2] applied to that translate gives ${\sum }_k\mathrm{\mid }g(y+k)\mathrm{\mid }<\mathrm{\infty }$.

Differentiation along coordinate lines. Fix $h\in \mathcal{S}$, $x_0\in {\mathbb{R}}^n$ and a coordinate $j$, and for $t\in [-1\mathrm{/}2,1\mathrm{/}2]$ put $\mathrm{\Phi }(t):={\sum }_{\lambda \in \mathrm{\Lambda }}h(x_0+t{e}_j+\lambda )$, a sum that converges by step 1.1 applied to $h$. The finite partial sums ${\mathrm{\Phi }}_N(t):={\sum }_{\mathrm{\mid }k\mathrm{\mid }\le N}h(x_0+t{e}_j+Ak)$ are continuously differentiable with ${\mathrm{\Phi }}_N^{\mathrm{\prime }}(t)={\sum }_{\mathrm{\mid }k\mathrm{\mid }\le N}{\mathrm{\partial }}_{j}h(x_0+t{e}_j+Ak)$, and ${\mathrm{\Phi }}_N(0)\to {\sum }_{\lambda }h(x_0+\lambda )$ by step 1.1; by step 1.1 applied to ${\mathrm{\partial }}_{j}h$, the derivatives ${\mathrm{\Phi }}_N^{\mathrm{\prime }}$ converge uniformly on the closed interval $[-1\mathrm{/}2,1\mathrm{/}2]$ to ${\sum }_{\lambda }{\mathrm{\partial }}_{j}h(x_0+t{e}_j+\lambda )$. Applying [F3] on this closed interval to the real and imaginary parts gives that $\mathrm{\Phi }$ is differentiable at every interior point, in particular at $t=0$, with ${\mathrm{\Phi }}^{\mathrm{\prime }}(0)={\sum }_{\lambda }{\mathrm{\partial }}_{j}h(x_0+\lambda )$. Since $x_0$ and $j$ were arbitrary, the partial derivative ${\mathrm{\partial }}_j$ of the sum function exists at every point and equals ${\sum }_{\lambda }{\mathrm{\partial }}_{j}h(\cdot +\lambda )$, which is continuous by [F7] as a locally uniform limit of continuous functions.

Apply step 2.1 successively along any ordered word of coordinate differentiations, starting with $h=f$. At each stage its derivative is again Schwartz by [F4], so the next differentiation is licensed and the resulting derivative series is continuous and locally uniformly convergent. Induction on the word length establishes every ordered derivative of $P_{\mathrm{\Lambda }}f$ and its continuity, hence smoothness. By [F6], grouping the derivatives of $f$ by their multi-indices gives ${\mathrm{\partial }}^{\beta }(P_{\mathrm{\Lambda }}f)={\sum }_{\lambda }{\mathrm{\partial }}^{\beta }f(\cdot +\lambda )$ for every $\beta$. Absolute and locally uniform convergence are supplied by step 1.1.

Periodicity. For $\mu \in \mathrm{\Lambda }$, reindexing the absolutely convergent series by the bijection $\lambda \mapsto \lambda +\mu$ of $\mathrm{\Lambda }$ gives $P_{\mathrm{\Lambda }}f(x+\mu )={\sum }_{\lambda }f(x+\mu +\lambda )={\sum }_{\lambda }f(x+\lambda )=P_{\mathrm{\Lambda }}f(x)$. Countable Choice enters only through the published Poisson theorem's convergence clause in [F2], which is quoted for the Schwartz functions $({\mathrm{\partial }}^{\gamma }h)\circ A$; the lattice reindexing, the partial sums and the interval differentiations are explicit.