Let \(\Lambda \subseteq \mathbb{R}^d\) be a lattice. Let \(\Lambda^{*}\) be the dual lattice.

Let \(\mathcal{F}\) denote the Fourier transform on \(\mathbb{R}^{d}\). Let \(\hat{f}= \mathcal{F}(f)\) for some \(f \in \mathcal{S}(\mathbb{R}^{d})\).

Then, we have the following formula

\[\begin{equation} \sum_{v \in \Lambda} f(v) = \frac{1}{\mathop{\mathrm{vol}}(\mathbb{R}^{d}/\Lambda )} \sum_{\lambda \in \Lambda^{*}} \hat{f}(v). \end{equation}\]

Adelic version

Let \(K\) be a number field and let \(\mathbb{A}_{K}\) be the ring of adeles over \(K\). Let \(f :\mathbb{A}_{K} \rightarrow \mathbb{C}\) be a Schwartz-Bruhat function. Then for an appropriate Haar measure and an appropriate Fourier tranform \(\widehat{f}\) of \(f\), one has \[\begin{equation} \sum_{x \in K}^{} f(x) = \sum_{y \in K} \widehat{f}(y). \end{equation}\] This is used to prove analytic continuity of Hecke \(L\)-functions.

Applications

One example is linear programming bound for sphere packings