Fourier series of the periodized Gaussian

Given: Countable Choice and a real parameter $\tau >0$.

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$): Countable Choice, as required by the Schwartz Poisson-summation and Euclidean Gaussian-transform suppliers.

Poisson summation for Schwartz functions: for a Schwartz function $h$ on $\mathbb{R}$, ${\sum }_{n\in \mathbb{Z}}h(x+n)={\sum }_{r\in \mathbb{Z}}\widehat{h}(r)e^{2\pi irx}$, with the left side locally uniform and the right side absolutely uniformly convergent.

Euclidean Gaussian transform with the 2π normalization: under Countable Choice, $\widehat{(\exp (-\pi t{u}^2))}(r)=t^{-1\mathrm{/}2}\exp (-\pi r^2\mathrm{/}t)$ for $t>0$ and $r\in \mathbb{R}$.

Schwartz space and its seminorms: $h\in \mathcal{S}(\mathbb{R})$ when it is smooth and every seminorm ${\sup }_u\mathrm{\mid }{u}^m{h}^{(q)}(u)\mathrm{\mid }$ is finite.

The exponential function is smooth and $(\exp {)}^{\mathrm{\prime }}=\exp$ and Sum, product, quotient, and chain rules from difference quotients: ${\exp }^{\mathrm{\prime }}=\exp$, and derivatives of products and compositions obey the product and chain rules.

The exponential dominates every fixed nonnegative integer power at $+\mathrm{\infty }$: for every fixed nonnegative integer $m$ and $a>0$, $x^m\mathrm{/}\exp (ax)\to 0$ as $x\to +\mathrm{\infty }$.

Local uniform convergence of the periodized Gaussian: the defining kernel series is locally uniformly convergent and equals ${\varphi }_{\tau }$.

Real elementary functions and their domains: $\arctan (1)={\int }_0^1(1+t^2{)}^{-1}dt\ge 1\mathrm{/}2$, so $\pi =4\arctan (1)\ge 2>0$; the real exponential is defined on all of $\mathbb{R}$.