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The Fourier transform of a function is defined as

(1) 

(2) 
A discrete Fourier transform takes a function known only at
discrete points
and gives back a function
known only at the discrete points

(3) 

(4) 
Note that the relationship between the discrete Fourier transform and
the continuous one is

(5) 
where is the spacing between points and the spacing
between frequencies is given by

(6) 
(Note that I am following the conventions of Numerical Recipes using
to denote frequency rather than angular frequency. The conversion
is simply
.) Formally the discrete Fourier transform
is periodic with period so can take any set of
consecutive values. Practically speaking, though, if the points
represent a typical region of the function then the results
give the frequency components in the range
(where
by periodicity). The next section
describes how these complex points are arranged in the output of the
routines fftc1 and fftcn. The following section describes the
additional information needed to interpret the results of the real
Fourier transform routines fftr1 and fftrn.
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