# Poisson summation formula

Let $f:\mathbb{R}\to \mathbb{R}$ be an integrable function and let

$$\widehat{f}(\xi )={\int}_{\mathbb{R}}{e}^{-2\pi i\xi x}f(x)\mathit{d}x,\xi \in \mathbb{R}.$$ |

be its Fourier transform^{}. The Poisson summation formula is the assertion that

$$\sum _{n\in \mathbb{Z}}f(n)=\sum _{n\in \mathbb{Z}}\widehat{f}(n).$$ | (1) |

whenever $f$ is such that both of the above infinite sums are absolutely convergent.

Equation (1) is useful because it establishes a
correspondence between Fourier series^{} and Fourier integrals. To see
the connection, let

$$g(x)=\sum _{n\in \mathbb{Z}}f(x+n),x\in \mathbb{R},$$ |

be the periodic
function obtained by pseudo-averaging^{1}^{1}This terminology is at best a metaphor. The operation in question is not a genuine mean, in the technical sense of that word. $f$ relative to $\mathbb{Z}$ acting
as the discrete group of translations on $\mathbb{R}$. Since $f$ was
assumed to be integrable, $g$ is defined almost everywhere, and is
integrable over $[0,1]$ with

$${\parallel g\parallel}_{{L}^{1}[0,1]}\le {\parallel f\parallel}_{{L}^{1}(\mathbb{R})}.$$ |

Since $f$ is integrable, we may interchange integration and summation to obtain

$$\widehat{f}(k)=\sum _{n\in \mathbb{Z}}{\int}_{0}^{1}f(x+n){e}^{-2\pi ikx}\mathit{d}x={\int}_{0}^{1}{e}^{-2\pi ikx}g(x)\mathit{d}x$$ |

for every $k\in \mathbb{Z}$. In other words, the restriction of the Fourier transform of $f$ to the integers gives the Fourier coefficients of the averaged, periodic function $g$. Since we have assumed that the $\widehat{f}(k)$ form an absolutely convergent series, we have that

$$g(x)=\sum _{k\in \mathbb{Z}}\widehat{f}(k){e}^{2\pi ikx}$$ |

in the sense of
uniform convergence^{}. Evaluating the above equation at $x=0$, we
obtain the Poisson summation formula (1).

Title | Poisson summation formula |
---|---|

Canonical name | PoissonSummationFormula |

Date of creation | 2013-03-22 13:27:25 |

Last modified on | 2013-03-22 13:27:25 |

Owner | rmilson (146) |

Last modified by | rmilson (146) |

Numerical id | 16 |

Author | rmilson (146) |

Entry type | Theorem |

Classification | msc 42A16 |

Classification | msc 42A38 |

Synonym | Poisson summation |