Editing Poisson summation formula (section)

{{Short description|Equation in Fourier analysis}}
{{Use American English|date=January 2019}}In [[mathematics]], the '''Poisson summation formula''' is an equation that relates the [[Fourier series]] coefficients of the [[periodic summation]] of a [[function (mathematics)|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 <math>s(x)</math> on <math>\mathbb R</math> which decays at infinity with all derivatives ([[Schwartz function]]), the simplest version of the Poisson summation formula states that
{{Equation box 1
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|equation={{NumBlk||<math>\sum_{n=-\infty}^\infty s(n)=\sum_{k=-\infty}^\infty S(k).</math> &nbsp; &nbsp;|{{EquationRef|Eq.1}}}}
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where <math>S</math> is the [[Fourier transform]] of <math>s</math>, i.e., <math display="inline">S(f) \triangleq \int_{-\infty}^{\infty} s(x)\ e^{-i2\pi fx}\, dx.</math> The summation formula can be restated in many equivalent ways, but a simple one is the following.<ref>Stein and Weiss, p 251</ref> Suppose that <math>f\in L^1(\mathbb R^n)</math> (''L''<sup>1</sup> for [[Lp space|''L''<sup>1</sup> space]]) and <math>\Lambda</math> is a [[unimodular lattice]] in <math>\mathbb R^n</math>. Then the periodization of <math>f</math>, which is defined as the sum <math display="inline">f_\Lambda(x) = \sum_{\lambda\in\Lambda} f(x+\lambda),</math> converges in the <math>L^1</math> norm of <math>\mathbb R^n/\Lambda</math> to an <math>L^1(\mathbb R^n/\Lambda)</math> function having Fourier series <math display="block">f_\Lambda(x) \sim \sum_{\lambda'\in\Lambda'} \hat f(\lambda') e^{2\pi i \lambda' x}</math> where <math>\Lambda'</math> is the dual lattice to <math>\Lambda</math>. (Note that the Fourier series on the right-hand side need not converge in <math>L^1</math> or otherwise.)