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$\label{10.8.1}\text{O}=\text{T}\ast \text{I} .$
Here $$\text{O}, \text{ T and I}$$ are respectively the observed, true and instrumental profiles, and the asterisk denotes the convolution. The mathematical problem is to deconvolve this equation so that, given the instrumental profile and the observed profile it is possible to recover the true profile. This is done by making use of a mathematical theorem known as Borel’s theorem, which is that the Fourier transform of the convolution of two functions is equal to the product of the Fourier transforms of each. That is
$\label{10.8.2}\overline{\text{O}}=\overline{\text{T}} \times \overline{\text{I}},$
where the bar denotes the Fourier transform. Numerical fast Fourier transform computer programs are now readily available, so the procedure is to calculate the Fourier transforms of the observed and instrumental profile, divide the former by the latter to obtain $$\overline{\text{T}}$$ , and then calculate the inverse Fourier transform to obtain the true profile. This procedure is well known in radio astronomy, in which the observed map of a sky region is the convolution of the true map with the beam of the radio telescope, though, unlike the one-dimensional spectroscopic problem the corresponding radio astronomy problem is two-dimensional.