Poisson summation formula

In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function's Fourier transform. And conversely, the periodic summation of a function's Fourier transform is completely defined by discrete samples of the original function. The Poisson summation formula was discovered by Siméon Denis Poisson and is sometimes called Poisson resummation.

For a smooth, complex valued function ${\displaystyle s(x)}$ on ${\displaystyle \mathbb {R} }$ which decays at infinity with all derivatives (Schwartz function), the Poisson summation formula states that

${\displaystyle \sum _{n=-\infty }^{\infty }s(n)=\sum _{k=-\infty }^{\infty }S(k).}$

Eq.1

where ${\displaystyle S}$ is the Fourier transform of ${\displaystyle s}$, i.e., ${\textstyle S(f)\triangleq \int _{-\infty }^{\infty }s(x)\ e^{-i2\pi fx}\,dx.}$ The summation formula can be restated in many equivalent ways, but a simple one is the following.^[1] Suppose that ${\displaystyle f\in L^{1}(\mathbb {R} ^{n})}$ and ${\displaystyle \Lambda }$ is a unimodular lattice in ${\displaystyle \mathbb {R} ^{n}}$. Then the periodization of ${\displaystyle f}$, which is defined as the sum $${\displaystyle f_{\Lambda }(x)=\sum _{\lambda \in \Lambda }f(x+\lambda ),}$$ converges in the ${\displaystyle L^{1}}$ norm of ${\displaystyle \mathbb {R} ^{n}/\Lambda }$ to an ${\displaystyle L^{1}(\mathbb {R} ^{n}/\Lambda )}$ function having Fourier series $${\displaystyle f_{\Lambda }(x)\sim \sum _{\lambda '\in \Lambda '}{\hat {f}}(\lambda ')e^{2\pi i\lambda 'x}}$$ where ${\displaystyle \Lambda '}$ is the dual lattice to ${\displaystyle \Lambda }$. (Note that the Fourier series on the right-hand side need not converge in ${\displaystyle L^{1}}$ or otherwise.)