Editing Poisson summation formula (section)

==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 {{EquationNote|Eq.1}}, <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:<ref name="Pinsky" /><ref name="Zygmund" />

{{Equation box 1
|indent=: |cellpadding= 0 |border= 0 |background colour=white
|equation={{NumBlk||<math>s_{_P}(x) = \sum_{k=-\infty}^{\infty} \underbrace{\frac{1}{P}\cdot S\left(\frac{k}{P}\right)}_{S[k]}\ e^{i 2\pi \frac{k}{P} x },</math> &nbsp; &nbsp;|{{EquationRef|Eq.2}}}}
}}

which is a [[Fourier series#Definition|Fourier series]] expansion with coefficients that are samples of the function <math>S(f).</math> Conversely, {{EquationNote|Eq.2}} follows from {{EquationNote|Eq.1}} by applying the known behavior of the Fourier transform under translations (see the [[Fourier transform#Basic properties|Fourier transform properties]] time scaling and shifting).

Similarly:

{{Equation box 1
|indent=: |cellpadding= 0 |border= 0 |background colour=white
|equation={{NumBlk||<math>S_{1/T}(f) = \sum_{n=-\infty}^{\infty} \underbrace{T\cdot s(nT)}_{s[n]}\ e^{-i 2\pi n Tf},</math> &nbsp; &nbsp;|{{EquationRef|Eq.3}}}}
}}

also known as the important '''[[Discrete-time Fourier transform]]'''.