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The aim of this paper is to exhibit a real Paley–Wiener space sitting inside the Schwartz space, and to give a quick and simple proof of a Paley–Wiener-type theorem. A simple and elementary proof of a theorem postulated by H. H. Bang is also given.
Hardy's uncertainty principle states that it is impossible for a function and its Fourier transform to be simultaneously very rapidly decreasing. In this paper we prove versions of this principle for the Jacobi transform and for the Fourier transform on real hyperbolic spaces.
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