PlanetMath: Poisson summation formula

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Poisson summation formula

(Theorem)

Let $f:\mathbb{R}\to\mathbb{R}$ be an integrable function and let

$\displaystyle \hat{f}(\xi)= \int_{\mathbb{R}} e^{-2\pi i\xi x} f(x)dx,\quad \xi\in\mathbb{R}.$

be its Fourier transform. The Poisson summation formula is the assertion that

$\displaystyle \sum_{n\in\mathbb{Z}} f(n) = \sum_{n\in\mathbb{Z}} \hat{f}(n).$

(1)

whenever

is such that both of the above infinite sums are absolutely convergent.

Equation (1) is useful because it establishes a correspondence between Fourier series and Fourier integrals. To see the connection, let

$\displaystyle g(x)=\sum_{n\in\mathbb{Z}} f(x+n),\quad x\in\mathbb{R},$

be the periodic function obtained by pseudo-averaging ¹

relative to $\mathbb{Z}$ acting as the discrete group of translations on $\mathbb{R}$ . Since

was assumed to be integrable,

is defined almost everywhere, and is integrable over

with

$\displaystyle \Vert g \Vert_{L^{\!1}[0,1]}\leq \Vert f\Vert_{L^{\!1}(\mathbb{R})}.$

Since

is integrable, we may interchange integration and summation to obtain

$\displaystyle \hat{f}(k) = \sum_{n\in\mathbb{Z}} \int_0^1 f(x+n)e^{-2\pi ik x} dx = \int_0^1 e^{-2\pi i k x} g(x) dx$

for every $k\in\mathbb{Z}$ . In other words, the restriction of the Fourier transform of

to the integers gives the Fourier coefficients of the averaged, periodic function

. Since we have assumed that the $\hat{f}(k)$ form an absolutely convergent series, we have that

$\displaystyle g(x) = \sum_{k\in\mathbb{Z}} \hat{f}(k) e^{2\pi ik x}$

in the sense of uniform convergence. Evaluating the above equation at

, we obtain the Poisson summation formula (1).

Footnotes

... pseudo-averaging ¹: This terminology is at best a metaphor. The operation in question is not a genuine mean, in the technical sense of that word.

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"Poisson summation formula" is owned by rmilson. [ full author list (3) | owner history (2) ]

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Poisson summation

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Cross-references: uniform convergence, series, Fourier coefficients, integers, restriction, almost everywhere, translations, group, discrete, mean, operation, periodic function, connection, Fourier integrals, Fourier series, equation, absolutely convergent, sums, infinite, Fourier transform, integrable function
There are 3 references to this entry.

This is version 13 of Poisson summation formula, born on 2003-02-11, modified 2006-10-03.
Object id is 4022, canonical name is PoissionSummationFormula.
Accessed 10581 times total.

Classification:

AMS MSC:	42A38 (Fourier analysis :: Fourier analysis in one variable :: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type)
	42A16 (Fourier analysis :: Fourier analysis in one variable :: Fourier coefficients, Fourier series of functions with special properties, special Fourier series)

Pending Errata and Addenda

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