Poisson summation formula
Let be an integrable function and let
be its Fourier transform. The Poisson summation formula is the assertion that
(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
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. relative to acting as the discrete group of translations on . Since was assumed to be integrable, is defined almost everywhere, and is integrable over with
Since is integrable, we may interchange integration and summation to obtain
for every . 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 form an absolutely convergent series, we have that
in the sense of uniform convergence. Evaluating the above equation at , we obtain the Poisson summation formula (1).
Title | Poisson summation formula |
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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 |