 Poisson summation formula

For a sufficiently rapidly decreasing function $f$, the Poisson summation formula relates its samples to those of its Fourier transform:under the convention $\hat{f}(\xi )={\int }_{\mathbb{R}}f(x)e^{-ix\xi }dx$.

The Poisson summation formula is a powerful and essential result in analytic number theory and Fourier analysis, connecting sums of a function at integer points to sums of its Fourier transform. Specifically, it relates a sum over a lattice (for example, the integers) to a sum over the dual lattice.