Poisson summation formula

Let f: be an integrable function and let


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

nf(n)=nf^(n). (1)

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

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


be the periodic function obtained by pseudo-averaging11This terminology is at best a metaphor. The operation in question is not a genuine mean, in the technical sense of that word. f relative to acting as the discrete group of translations on . Since f was assumed to be integrable, g is defined almost everywhere, and is integrable over [0,1] with


Since f is integrable, we may interchange integration and summation to obtain


for every k. In other words, the restriction of the Fourier transform of f to the integers gives the Fourier coefficients of the averaged, periodic function g. Since we have assumed that the f^(k) form an absolutely convergent series, we have that


in the sense of uniform convergenceMathworldPlanetmath. Evaluating the above equation at x=0, we obtain the Poisson summation formula (1).

Title Poisson summation formula
Canonical name PoissonSummationFormula
Date of creation 2013-03-22 13:27:25
Last modified on 2013-03-22 13:27:25
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 16
Author rmilson (146)
Entry type Theorem
Classification msc 42A16
Classification msc 42A38
Synonym Poisson summation