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In Fourier series, we had to assume that the function is periodic outside of the desired interval with a period of \(L\). Therefore, assuming that \(L\rightarrow\infty\) will allow us to work with any function. However, with this approximation, the fundamental frequency (\(\omega_0\)) or the frequency resolution that we discussed in Fourier series will tend to zero: \(\omega_0\rightarrow0\). In the equation to find \(c_m\), every \(m\) represented a frequency (multiple of \(\omega_0\)) and the integration on \(l\) removes the dependence of the right side of the equation on \(l\), making it only a function of \(m\) or frequency. Let’s define the following two variables:

$$\omega{\equiv}m\omega_0={2{\pi}m\over L}$$

$$F(\omega){\equiv}Lc_m$$

The equation to find the coefficients of each frequency in Fourier series thus becomes:

$$F(\omega)=\int_{-\infty}^{\infty}f(l)e^{-i{\omega}l}dl. $$

The function \(F(\omega)\) is thus the *Fourier transform*
of \(f(l)\) in the frequency domain. So through this
transformation, we can find (analyze) the magnitudes of the
constituting frequencies or the value in the frequency
space^{114} of our spatial input function. The great thing
is that we can also do the reverse and later synthesize the input
function from its Fourier transform. Let’s do it: with the
approximations above, multiply the right side of the definition of the
Fourier Series (Fourier series) with
\(1=L/L=({\omega_0}L)/(2\pi)\):

$$f(l)={1\over 2\pi}\displaystyle\sum_{n=-\infty}^{\infty}Lc_ne^{{2{\pi}in\over L}l}\omega_0={1\over 2\pi}\displaystyle\sum_{n=-\infty}^{\infty}F(\omega)e^{i{\omega}l}\Delta\omega $$

To find the right most side of this equation, we renamed \(\omega_0\) as \(\Delta\omega\) because it was our resolution, \(2{\pi}n/L\) was written as \(\omega\) and finally, \(Lc_n\) was written as \(F(\omega)\) as we defined above. Now, as \(L\rightarrow\infty\), \(\Delta\omega\rightarrow0\) so we can write:

$$f(l)={1\over 2\pi}\int_{-\infty}^{\infty}F(\omega)e^{i{\omega}l}d\omega $$

Together, these two equations provide us with a very powerful set of
tools that we can use to process (analyze) and recreate (synthesize)
the input signal. Through the first equation, we can break up our
input function into its constituent frequencies and analyze it, hence
it is also known as *analysis*. Using the second equation, we can
synthesize or make the input function from the known frequencies and
their magnitudes. Thus it is known as *synthesis*. Here, we
symbolize the Fourier transform (analysis) and its inverse (synthesis)
of a function \(f(l)\) and its Fourier Transform
\(F(\omega)\) as \({\cal F}[f]\) and \({\cal F}^{-1}[F]\).

As we discussed before, this ‘magnitude’ can be interpreted as the radius of the circle rotating at this frequency in the epicyclic interpretation of the Fourier series, see Figure 6.1 and Figure 6.2.

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GNU Astronomy Utilities 0.9 manual, April 2019.