A function f(x) can be expanded in terms of a Fourier expansion over a finite
interval 
as
where the expansion coefficients are obtained as
As an example of the use of Mathematica in Fourier expansion over a finite interval
In[12]:=
?FourierTrigSeries
FourierTrigSeries[expr, {x, x0, x1}, n] gives the
trigonometric series expansion of expr to order n.
In[14]:=
FourierTrigSeries[x^2, {x, 0, 1}, 3]
Out[14]=
1 Cos[2 Pi x] Cos[4 Pi x] Cos[6 Pi x] Sin[2 Pi x]
- + ----------- + ----------- + ----------- - ----------- -
3 2 2 2 Pi
Pi 4 Pi 9 Pi
Sin[4 Pi x] Sin[6 Pi x]
----------- - -----------
2 Pi 3 Pi
If we want to expand a function f(x) over an infinite domain
we turn to the Fourier transform. The Fourier Transform 
of a
function f(x) and the corresponding inverse Fourier Transform can be written
in a number of forms including the symmetric form via
Mathematica allows a variety of Fourier Transforms (and their inverses) to be employed for the analyis of data
Fourier FourierOverallConstant
FourierCosSeriesCoefficient FourierSample
FourierCosTransform FourierSinSeriesCoefficient
FourierExpSeries FourierSinTransform
FourierExpSeriesCoefficient FourierTransform
FourierFrequencyConstant FourierTrigSeries
FourierTransform[expr, t, w] gives a function of w, which is
the Fourier transform of expr, a function of t. It is
defined by FourierTransform[expr, t, w] = FourierOverallConstant *
Integrate[Exp[FourierFrequencyConstant I w t] expr, {t,-Infinity, Infinity}].
Applications of Fourier Series and Transforms arise in diffraction (e.g. scattering of radiation by particles) but time does not permit a treatment here.