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 appropriate functions ${\displaystyle f,\,}$  the Poisson summation formula may be stated as:

${\displaystyle \sum _{n=-\infty }^{\infty }f(n)=\sum _{k=-\infty }^{\infty }{\hat {f}}\left(k\right),}$     where ${\displaystyle {\hat {f}}}$  is the Fourier transform of ${\displaystyle f\,}$;  that is   ${\displaystyle {\hat {f}}(\nu )={\mathcal {F}}\{f(x)\}.}$

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