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.