Poisson summation for a full-rank lattice

Given: Countable Choice, $f\in \mathcal{S}({\mathbb{R}}^n)$, the full-rank lattice $\mathrm{\Lambda }=A{\mathbb{Z}}^n$ with dual ${\mathrm{\Lambda }}^{\ast }=A^{-T}{\mathbb{Z}}^n$ and covolume $c=\mathrm{\mid }\det A\mathrm{\mid }$, the fundamental parallelotope $F=A((0,1{]}^n)$, the periodisation $P=P_{\mathrm{\Lambda }}f$, and the $L^1$ transform $\widehat{f}$ of Fourier transform on complex L1 classes.

$P$ is smooth, $\mathrm{\Lambda }$-periodic and absolutely convergent at every point: $P(x)={\sum }_{\lambda \in \mathrm{\Lambda }}f(x+\lambda )$ with locally uniformly summable derivative series (Schwartz periodisation over a lattice is smooth with locally uniformly summable derivatives).

$\widehat{f}\in \mathcal{S}({\mathbb{R}}^n)$ (Fourier transform acts continuously on Schwartz space), and Schwartz functions satisfy the product-weight bound $\mathrm{\mid }{\mathrm{\partial }}^{\beta }g(y)\mathrm{\mid }\le A_{\beta }{\prod }_j(1+y_j^2{)}^{-1}$ for a finite constant built from finitely many seminorms; in particular $\mathrm{\mid }\widehat{f}(\xi )\mathrm{\mid }\le A(1+\mathrm{\mid }\xi \mathrm{\mid }{)}^{-N}$ for every prescribed $N$ (Schwartz derivatives are integrable).

Lattice count: there is a constant $C$ with $\mathrm{\#}\{{\lambda }^{\ast }\in {\mathrm{\Lambda }}^{\ast }:\mathrm{\mid }{\lambda }^{\ast }\mathrm{\mid }\le r\}\le C(1+r{)}^n$ for all $r\ge 0$. Indeed ${\lambda }^{\ast }=A^{-T}k$, so $\mathrm{\mid }{\lambda }^{\ast }\mathrm{\mid }\le r$ implies $\mathrm{\mid }k\mathrm{\mid }=\mathrm{\mid }{A}^T{\lambda }^{\ast }\mathrm{\mid }\le Kr$ for a constant $K$ (Every Euclidean linear map has a unique matrix and satisfies $\mathrm{\parallel }Lh{\mathrm{\parallel }}_2\le K\mathrm{\parallel }h{\mathrm{\parallel }}_2$ for some $K\ge 0$, Full-rank lattices, covolume, and the dual lattice), and the number of integer points with $\mathrm{\mid }k\mathrm{\mid }\le Kr$ is at most $(2Kr+1{)}^n$, as the shell count of Dirac comb shows.

${\sum }_{m\ge 1}{m}^{n-N}<\mathrm{\infty }$ for real $N>n+1$ (The p-series for a real exponent p converges exactly when p is greater than one).

Character orthogonality on the fundamental domain: $(1\mathrm{/}c){\int }_F{e}^{2\pi i({\lambda }^{\ast }-{\eta }^{\ast })\cdot x}dx=1$ if ${\lambda }^{\ast }={\eta }^{\ast }$ and $0$ otherwise (Orthogonality of the lattice characters over a fundamental domain).

The periodisation has $P$'s coefficients: ${\int }_{F}P(x)e^{-2\pi i{\eta }^{\ast }\cdot x}dx=\widehat{f}({\eta }^{\ast })$ for every ${\eta }^{\ast }\in {\mathrm{\Lambda }}^{\ast }$ (Fourier coefficients of a lattice periodisation).

A continuous $\mathrm{\Lambda }$-periodic function with all lattice coefficients zero vanishes identically (Continuous lattice-periodic functions are determined by their lattice Fourier coefficients).

Termwise integration of a uniformly convergent series over a finite-measure set and interchange of an absolutely summable integral are justified by the dominated-convergence and Fubini interfaces (Dominated convergence, Fubini's theorem for L^1 functions on a sigma-finite product).

Uniform limits of continuous real-valued functions are continuous (If for every $\epsilon >0$ some continuous $g:X\to \mathbb{R}$ satisfies $\mid f(x)-g(x)\mid <\epsilon$ for all $x$, then $f$ is continuous; in particular a uniformly convergent series of continuous real functions has a continuous sum), and a complex-valued function is continuous when its real and imaginary parts are continuous (A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions).