Poisson summation formula

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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 s ( x ) {\displaystyle s(x)} on R {\displaystyle \mathbb {R} } which decays at infinity with all derivatives (Schwartz function), the simplest version of the Poisson summation formula states that where S...

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