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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.

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rdfs:label	Poissonsche Summenformel (de) Formule sommatoire de Poisson (fr) Formula di sommazione di Poisson (it) ポアソン和公式 (ja) Poisson summation formula (en) Fórmula do somatório de Poisson (pt) 泊松求和公式 (zh)
rdfs:comment	Die poissonsche Summenformel ist ein Hilfsmittel der Fourier-Analysis und Signalverarbeitung. Sie dient unter anderem zur Analyse der Eigenschaften von Abtastmethoden. (de) 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. (en) 数学においてポアソン和公式（ポアソンわこうしき、英語: Poisson summation formula）とは、ある関数列の無限和とその関数列をフーリエ変換したものの無限和が等しいことを主張する公式である。シメオン・ドニ・ポアソン(Siméon Denis Poisson)によって発見された。 (ja) 泊松求和公式（英文：Poisson Summation Formula）由法國數學家泊松所發現，它陳述了一個連續時間的信號，做無限多次的週期複製後，其傅立葉級數與其傅立葉轉換之間數值的關係，亦可用來求周期信號的傅立葉轉換。 (zh) La formula di sommazione di Poisson, anche detta risommazione di Poisson, è un'identità tra due somme infinite, di cui la prima è costruita con una funzione e la seconda con la sua trasformata di Fourier . La funzione è definita sull'asse reale o nello spazio euclideo a dimensioni. La formula è stata scoperta da Siméon Denis Poisson. (it) La formule sommatoire de Poisson (parfois appelée resommation de Poisson) est une identité entre deux sommes infinies, la première construite avec une fonction , la seconde avec sa transformée de Fourier . Ici, f est une fonction sur la droite réelle ou plus généralement sur un espace euclidien. La formule a été découverte par Siméon Denis Poisson. (fr) A fórmula de soma de Poisson (às vezes chamada de retomada de Poisson ) é uma identidade entre duas somas infinitas, a primeira construída com uma função f, a segunda com sua transformada de Fourier . Aqui, f é uma função na linha real ou mais geralmente em um espaço euclidiano . A fórmula foi descoberta por Siméon Denis Poisson .Ela, e suas generalizações, são importantes em várias áreas da matemática, incluindo teoria dos números, análise harmônica e geometria Riemanniana . Uma das maneiras de interpretar a fórmula unidimensional é ver uma relação entre o espectro do operador Laplace-Beltrami no círculo e os comprimentos da geodésica periódica nessa curva. A , na interface de todos os domínios mencionados acima e também da análise funcional, estabelece uma relação do mesmo tipo, mas com (pt)
rdfs:seeAlso	Convolution theorem
dcterms:subject	Fourier analysis Generalized functions Lattice points Theorems in analysis Summability methods
Wikipage page ID	502042 (xsd:integer)
Wikipage revision ID	1124896840 (xsd:integer)
Link from a Wikipage to another Wikipage	Sampling (signal processing) Method of images Fourier analysis Generalized functions Riemann zeta function Robert Langlands Dominated convergence theorem Compact support Convolution theorem Mathematical analysis Mathematics Geometry of numbers Function (mathematics) Fundamental solution Lattice points Modular form Theta function Lp space Short exact sequence Siméon Denis Poisson Harmonic analysis Bandlimiting Cesàro summation Dual lattice Locally compact abelian group Euclidean space Explicit formulae for L-functions Fourier series Fourier transform Number theory Partial differential equations Dirac comb Pointwise convergence Dirac delta Heat equation Heat kernel Atle Selberg Theorems in analysis Summability methods Dirac delta function Discrete-time Fourier transform Distribution (mathematics) Pontryagin duality Schwartz function Indicator function Integration by substitution Nyquist–Shannon sampling theorem Continuous Fourier transform Voronoi formula Ewald summation Periodic summation Sides of an equation Post's inversion formula Uniformly continuous Sphere packing Lattice (group theory)
Link from a Wikipage to an external page	https://www.uni-hohenheim.de/~gzim/Publications/sm.html https://web.archive.org/web/20110524085311/https:/www.uni-hohenheim.de/~gzim/Publications/sm.html http://projecteuclid.org/euclid.bams/1183552334
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