Editing Poisson summation formula (section)

==Distributional formulation==

These equations can be interpreted in the language of [[distribution (mathematics)|distribution]]s<ref name="Córdoba"/><ref name="Hörmander"/>{{rp|at=§7.2}} for a function <math>s</math> whose derivatives are all rapidly decreasing (see [[Schwartz function]]). The Poisson summation formula arises as a particular case of the 
[[Convolution_theorem#Convolution theorem for tempered distributions| Convolution Theorem on tempered distributions]], using the [[Dirac comb]] distribution and its [[Fourier series]]:

<math display="block">\sum_{n=-\infty}^\infty \delta(x - nT) \equiv \sum_{k=-\infty}^\infty  \frac{1}{T}\cdot e^{- i 2\pi \frac{k}{T} x} \quad\stackrel{\mathcal{F}}{\Longleftrightarrow}\quad \frac{1}{T}\cdot \sum_{k=-\infty}^{\infty} \delta (f - k/T).</math>

In other words, the periodization of a [[Dirac delta]] <math>\delta,</math> resulting in a [[Dirac comb]], corresponds to the discretization of its spectrum which is constantly one. Hence, this again is a Dirac comb but with reciprocal increments.

For the case <math>T = 1,</math> {{EquationNote|Eq.1}} readily follows:

<math display="block">\begin{align}
\sum_{k=-\infty}^\infty S(k)
&= \sum_{k=-\infty}^\infty \left(\int_{-\infty}^{\infty} s(x)\ e^{-i 2\pi k x} dx \right)
= \int_{-\infty}^{\infty} s(x) \underbrace{\left(\sum_{k=-\infty}^\infty e^{-i 2\pi k x}\right)}_{\sum_{n=-\infty}^\infty \delta(x-n)} dx \\
&= \sum_{n=-\infty}^\infty  \left(\int_{-\infty}^{\infty} s(x)\ \delta(x-n)\ dx \right) = \sum_{n=-\infty}^\infty s(n).
\end{align}</math>

Similarly:

<math display="block">\begin{align}
\sum_{k=-\infty}^{\infty} S(f - k/T)
&= \sum_{k=-\infty}^{\infty} \mathcal{F}\left \{ s(x)\cdot e^{i 2\pi\frac{k}{T}x}\right \}\\
&= \mathcal{F} \bigg \{s(x)\underbrace{\sum_{k=-\infty}^{\infty} e^{i 2\pi\frac{k}{T}x}}_{T \sum_{n=-\infty}^{\infty} \delta(x-nT)}\bigg \}
= \mathcal{F}\left \{\sum_{n=-\infty}^{\infty} T\cdot s(nT) \cdot \delta(x-nT)\right \}\\
&= \sum_{n=-\infty}^{\infty} T\cdot s(nT) \cdot \mathcal{F}\left \{\delta(x-nT)\right \}
= \sum_{n=-\infty}^{\infty} T\cdot s(nT) \cdot e^{-i 2\pi nT f}.
\end{align}</math>

Or:<ref name="Oppenheim"/>{{rp|p=143}}

<math display="block">\begin{align}
\sum_{k=-\infty}^{\infty} S(f - k/T) &= S(f) * \sum_{k=-\infty}^{\infty} \delta(f - k/T) \\
&= S(f) * \mathcal{F}\left \{T \sum_{n=-\infty}^{\infty} \delta(x-nT)\right \} \\
&= \mathcal{F}\left \{s(x)\cdot T \sum_{n=-\infty}^{\infty} \delta(x-nT)\right \}
= \mathcal{F}\left \{\sum_{n=-\infty}^{\infty} T\cdot s(nT) \cdot \delta(x-nT)\right \} \quad \text{as above}.
\end{align}</math>

The Poisson summation formula can also be proved quite conceptually using the compatibility of [[Pontryagin duality]] with [[short exact sequence]]s such as<ref name="Deitmar"/> <math display="inline">0 \to \Z \to \R \to \R / \Z \to 0.</math>