Editing Fourier transform (section)

=== Fourier transform for periodic functions ===
The Fourier transform of a periodic function cannot be defined using the integral formula directly.  In order for integral in {{EquationNote|Eq.1}} to be defined the function must be [[Absolutely integrable function|absolutely integrable]].  Instead it is common to use [[Fourier series]].  It is possible to extend the definition to include periodic functions by viewing them as [[Distribution (mathematics)#Tempered distributions|tempered distributions]].

This makes it possible to see a connection between the [[Fourier series]] and the Fourier transform for periodic functions that have a [[Convergence of Fourier series|convergent Fourier series]].  If <math>f(x)</math> is a [[periodic function]], with period <math>P</math>, that has a convergent Fourier series, then:
<math display="block">
\widehat{f}(\xi) = \sum_{n=-\infty}^\infty c_n \cdot \delta \left(\xi - \tfrac{n}{P}\right),
</math>
where <math>c_n</math> are the Fourier series coefficients of <math>f</math>, and <math>\delta</math> is the [[Dirac delta function]].  In other words, the Fourier transform is a [[Dirac comb]] function whose ''teeth'' are multiplied by the Fourier series coefficients.