A basic property of optical instruments, such as telescopes and microscopes, is that they have a finite aperture. This implies that the images produced by these instruments are band-limited, i.e. their Fourier transform has a bounded support which is strictly related to the aperture of the instrument and is called the band of the instrument. As a consequence these images are characterized by a resolution distance which is essentially the sampling distance associated with the band of the instrument (Rayleigh distance). A problem which has been investigated for a long time is that of super-resolution, namely the possibility of improving the resolution by extrapolating the images out of the band of the instrument. This problem is ill-posed. The present paper summarizes the main results which have been obtained in the particular case where the object, whose image we wish to extrapolate, has a size of the order of the Rayleigh resolution distance. Such a situation occurs, for example, in the case of the image of a double star when the angular distance between the two stars is of the order of the angular resolution of the telescope.
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