Editing Poisson summation formula (section)

==Applications==
===Method of images===
In [[partial differential equations]], the Poisson summation formula provides a rigorous justification for the [[fundamental solution]] of the [[heat equation]] with absorbing rectangular boundary by the [[method of images]].  Here the [[heat kernel]] on <math>\mathbb{R}^2</math> is known, and that of a rectangle is determined by taking the periodization.  The Poisson summation formula similarly provides a connection between Fourier analysis on Euclidean spaces and on the tori of the corresponding dimensions.<ref name="Grafakos"/>  In one dimension, the resulting solution is called a [[theta function]].

In [[electrodynamics]], the method is also used to accelerate the computation of periodic [[Green's function]]s.<ref name="Kinayman"/> 

===Sampling===
In the statistical study of time-series, if <math>s</math> is a function of time, then looking only at its values at equally spaced points of time is called "sampling."  In applications, typically the function <math>s</math> is [[bandlimiting|band-limited]], meaning that there is some cutoff frequency <math>f_o</math> such that <math>S(f)</math> is zero for frequencies exceeding the cutoff: <math>S(f)=0</math> for <math>|f|>f_o.</math> For band-limited functions, choosing the sampling rate <math>\tfrac{1}{T} > 2 f_o</math> guarantees that no information is lost: since <math>S</math> can be reconstructed from these sampled values. Then, by Fourier inversion, so can <math>s.</math> This leads to the [[Nyquist–Shannon sampling theorem]].<ref name="Pinsky"/>

===Ewald summation===
Computationally, the Poisson summation formula is useful since a slowly converging summation in real space is guaranteed to be converted into a quickly converging equivalent summation in Fourier space.<ref>[[Philip Woodward|Woodward, Philipp M.]] (1953). ''Probability and Information Theory, with Applications to Radar''. Academic Press, p. 36.</ref> (A broad function in real space becomes a narrow function in Fourier space and vice versa.)  This is the essential idea behind [[Ewald summation]].

===Approximations of integrals===
The Poisson summation formula is also useful to bound the errors obtained when an integral is approximated by a (Riemann) sum. Consider an approximation of <math display="inline">S(0)=\int_{-\infty}^\infty dx \, s(x)</math> as <math display="inline">\delta \sum_{n=-\infty}^\infty s(n \delta)</math>, where <math> \delta \ll 1 </math> is the size of the bin. Then, according to {{EquationNote|Eq.2}} this approximation coincides with <math display="inline"> \sum_{k=-\infty}^\infty S(k/ \delta)</math>. The error in the approximation can then be bounded as <math display="inline">\left| \sum_{k \ne 0} S(k/ \delta) \right| \le \sum_{k \ne 0} | S(k/ \delta)|</math>. This is particularly useful when the Fourier transform of <math> s(x) </math> is rapidly decaying if <math>1/\delta \gg 1 </math>.

===Lattice points inside a sphere===
The Poisson summation formula may be used to derive Landau's asymptotic formula for the number of lattice points inside a large Euclidean sphere.  It can also be used to show that if an integrable function, <math>s</math> and <math>S</math> both have [[compact support]] then <math>s = 0.</math><ref name="Pinsky"/>

===Number theory===
In [[number theory]], Poisson summation can also be used to derive a variety of functional equations including the functional equation for the [[Riemann zeta function]].<ref>[[Harold Edwards (mathematician)|H. M. Edwards]] (1974). ''Riemann's Zeta Function''. Academic Press, pp. 209–11. {{ISBN|0-486-41740-9}}.</ref>

One important such use of Poisson summation concerns [[theta function]]s: periodic summations of Gaussians. Put <math> q= e^{i\pi \tau } </math>, for <math> \tau</math> a complex number in the upper half plane, and define the theta function:

<math display="block"> \theta ( \tau) =  \sum_n q^{n^2}. </math>

The relation between <math> \theta (-1/\tau)</math>  and <math> \theta (\tau)</math>  turns out to be important for number theory, since this kind of relation is one of the defining properties of a [[modular form]].  By choosing <math>s(x)= e^{-\pi x^2}</math> and using the fact that <math>S(f) = e^{-\pi f ^2},</math> one can conclude:

<math display="block">\theta \left({-1\over\tau}\right) = \sqrt{\tau \over i} \theta (\tau),</math> by putting <math>{1/\lambda} = \sqrt{\tau/i}.</math>

It follows from this that <math>\theta^8</math> has a simple transformation property under <math>\tau \mapsto {-1/ \tau}</math> and this can be used to prove Jacobi's formula for the number of different ways to express an integer as the sum of eight perfect squares.

===Sphere packings===
Cohn & Elkies<ref name="Cohn"/> proved an upper bound on the density of [[sphere packing]]s using the Poisson summation formula, which subsequently led to a proof of optimal sphere packings in dimension 8 and 24.

===Other===
* Let <math>s(x) = e^{-ax}</math> for <math>0 \leq x</math> and <math>s(x) = 0</math> for <math>x < 0</math> to get <math display="block">\coth(x) = x\sum_{n \in \Z} \frac{1}{x^2+\pi^2n^2} = \frac{1}{x}+ 2x \sum_{n \in \Z_+} \frac{1}{x^2+\pi^2n^2}.</math>
* It can be used to prove the functional equation for the theta function.
* Poisson's summation formula appears in Ramanujan's notebooks and can be used to prove some of his formulas, in particular it can be used to prove one of the formulas in Ramanujan's first letter to Hardy.{{Clarify|reason=|date=May 2019}}
* It can be used to calculate the quadratic Gauss sum.