In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function's Fourier transform. And conversely, the periodic summation of a function's Fourier transform is completely defined by discrete samples of the original function. The Poisson summation formula was discovered by Siméon Denis Poisson and is sometimes called Poisson resummation.
For a smooth, complex valued function <math>s(x)</math> on <math>\mathbb R</math> which decays at infinity with all derivatives (Schwartz function), the simplest version of the Poisson summation formula states that
where <math>S</math> is the Fourier transform of <math>s</math>, i.e., <math display="inline">S(\xi) \triangleq \int_{-\infty}^{\infty} s(x)\ e^{-i2\pi \xi x}\, dx.</math> The summation formula can be restated in many equivalent ways, but a simple one is the following. Suppose that <math>f\in L^1(\mathbb R^n)</math> (L<sup>1</sup> for L<sup>1</sup> space) and <math>\Lambda</math> is a unimodular lattice in <math>\mathbb R^n</math>. Then the periodization of <math>f</math>, which is defined as the sum <math display="inline">f_\Lambda(x) = \sum_{\lambda\in\Lambda} f(x+\lambda),</math> converges in the <math>L^1</math> norm of <math>\mathbb R^n/\Lambda</math> to an <math>L^1(\mathbb R^n/\Lambda)</math> function having Fourier series <math display="block">f_\Lambda(x) \sim \sum_{\lambda'\in\Lambda'} \hat f(\lambda') e^{2\pi i \lambda' x}</math> where <math>\Lambda'</math> is the dual lattice to <math>\Lambda</math>. (Note that the Fourier series on the right-hand side need not converge in <math>L^1</math> or otherwise.)
Periodization of a function
Let <math display="inline">s\left( x \right)</math> be a smooth, complex valued function on <math>\mathbb R</math> which decays at infinity with all derivatives (Schwartz function), and its Fourier transform <math>S\left( f \right)</math>, defined as
<math display="block">S(f) = \int_{-\infty}^\infty s(x) e^{-2\pi i xf}dx.</math>
Then <math>S(f)</math> is also a Schwartz function, and we have the reciprocal relationship that
<math display="block">s(x) = \int_{-\infty}^\infty S(f) e^{2\pi i x f}df.</math>
The periodization of <math>s(x)</math> with period <math>P>0</math> is given by
<math display="block">s_{_P}(x) \triangleq \sum_{n=-\infty}^{\infty} s(x + nP).</math>
Likewise, the periodization of <math>S(f)</math> with period <math>1/T</math>, where <math>T>0</math>, is
<math display="block">S_{1/T}(f) \triangleq \sum_{k=-\infty}^{\infty} S(f + k/T).</math>
Then , <math>\sum_{n=-\infty}^\infty s(n)=\sum_{k=-\infty}^\infty S(k),</math> is a special case (P=1, x=0) of this generalization: (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 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>
One important such use of Poisson summation concerns theta functions: 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