Poisson Summation Relates a Lattice Sum to Its Dual

Under sufficient decay, summing a function over integers equals summing its Fourier transform over integers. Keeping the exact identity in view prevents the geometric or probabilistic intuition from drifting.

Statement

Fourier analysis expands a function into frequencies. On the circle the coefficients are $\widehat f(n)$; on $\mathbf R$ the transform $\widehat f(\xi)$ is an integral against an oscillatory exponential.

$$ f\in\mathcal S(\mathbf R) $$

The definition determines which expressions are legal; only then does the identity become meaningful. An equality in $\mathcal A$ may change ambient meaning, so I keep $\mathsf D$ separate from $\mathsf C$.

$$ \sum_{n\in\mathbf Z}f(n)=\sum_{k\in\mathbf Z}\widehat f(k) $$

Worked algebra

Now evaluate one representative case. The result should agree with the structural law above, but it is obtained without assuming the conclusion.

$$ \sum_{n\in\mathbf Z}e^{-\pi t n^2}=t^{-1/2}\sum_{k\in\mathbf Z}e^{-\pi k^2/t}\qquad(t>0) $$

Conceptual compression

The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.

$$ \begin{aligned} \mathsf{D}\;&:\quad f\in\mathcal S(\mathbf R),\\[5pt] \mathsf{C}\;&:\quad \sum_{n\in\mathbf Z}f(n)=\sum_{k\in\mathbf Z}\widehat f(k). \end{aligned} $$

Caveat

Normalization conventions move factors of $2\pi$ between the transform, inverse transform, derivative rule, and Gaussian formula.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \sum_{n\in\mathbf Z}f(n)=\sum_{k\in\mathbf Z}\widehat f(k) \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Wed 23 April 2014. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.

Poisson Summation Relates a Lattice Sum to Its Dual by Vicfred

Statement

Worked algebra

Conceptual compression

Caveat