Editing Fourier analysis (section)

===Fourier series===
{{Main|Fourier series}}

The Fourier transform of a periodic function, <math>s_{_P}(t),</math> with period <math>P,</math> becomes a [[Dirac comb]] function, modulated by a sequence of complex [[coefficients]]''':'''

:<math>S[k] = \frac{1}{P}\int_{P} s_{_P}(t)\cdot e^{-i2\pi \frac{k}{P} t}\, dt, \quad k\in\Z,</math> &nbsp; &nbsp; (where <math>\int_{P}</math> is the integral over any interval of length <math>P</math>).

The inverse transform, known as '''Fourier series''', is a representation of <math>s_{_P}(t)</math> in terms of a summation of a potentially infinite number of harmonically related sinusoids or [[complex exponentials|complex exponential]] functions, each with an amplitude and phase specified by one of the coefficients''':'''

:<math>s_{_P}(t)\ \ =\ \ \mathcal{F}^{-1}\left\{\sum_{k=-\infty}^{+\infty} S[k]\, \delta \left(f-\frac{k}{P}\right)\right\}\ \ =\ \ \sum_{k=-\infty}^\infty S[k]\cdot e^{i2\pi \frac{k}{P} t}.</math>

Any <math>s_{_P}(t)</math> can be expressed as a [[periodic summation]] of another function, <math>s(t)</math>''':'''

:<math>s_{_P}(t) \,\triangleq\, \sum_{m=-\infty}^\infty s(t-mP),</math>

and the coefficients are proportional to samples of <math>S(f)</math> at discrete intervals of <math>\frac{1}{P}</math>''':'''

:<math>S[k] =\frac{1}{P}\cdot S\left(\frac{k}{P}\right).</math>{{efn-ua
|<math>\int_{P} \left(\sum_{m=-\infty}^{\infty} s(t-mP)\right) \cdot e^{-i2\pi \frac{k}{P} t} \,dt = \underbrace{\int_{-\infty}^{\infty} s(t) \cdot e^{-i2\pi \frac{k}{P} t} \,dt}_{\triangleq\, S\left(\frac{k}{P}\right)}</math>
}}

Note that any <math>s(t)</math> whose transform has the same discrete sample values can be used in the periodic summation.  A sufficient condition for recovering <math>s(t)</math> (and therefore <math>S(f)</math>) from just these samples (i.e. from the Fourier series) is that the non-zero portion of <math>s(t)</math> be confined to a known interval of duration <math>P,</math> which is the frequency domain dual of the [[Nyquist–Shannon sampling theorem]].

See [[Fourier series]] for more information, including the historical development.