Poisson Summation Formula


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Poisson Summation Formula

 

a formula used to calculate sums of series of the form

If

is the Fourier transform (in a form normalized somewhat differently than is usually the case) of the function F(x), then

where m and n are integers. This is the Poisson summation formula. It can be written in a more general form: if λ > 0, μ > 0, λ μ = l, and 0 ≤ t < l, then

This formula holds if F(x) has bounded variation in every finite interval and if, for x → + ∞ and x → – ∞, either (1) F(x) is monotone and ǀF(x) ǀ is integrable or (2) F(x) is integrable and has a derivative F’(x) such that ǀF’(x) ǀ is integrable. In some cases, the Poisson summation formula permits the calculation of the sum of a series to be replaced by the calculation of the sum of a more rapidly converging series.

References in periodicals archive ?
As is evident from the above citations, the Poisson summation formula is well known and has several variations and interpretations.
Meyer, A generalized Poisson summation formula, Appl.
Zimmermann, Sampling multipliers and the Poisson summation formula, J.
Stens, The Poisson summation formula, Whittaker's cardinal series and approximate integration.
Weinberger, On dualizing a multivariable Poisson summation formula, J.