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1 IPHT - Institut de Physique Théorique - UMR CNRS 3681

Abstract : Motivated by various problems in physics and applied mathematics, we look for constraints and properties of real Fourier-positive functions, i.e. with positive Fourier transforms. Properties of the -Dirac comb- distribution and of its tensor products in higher dimensions lead to Poisson resummation, allowing for a useful approximation formula of a Fourier transform in terms of a limited number of terms. A connection with the Bochner theorem on positive definiteness of Fourier-positive functions is discussed. As a practical application, we find simple and rapid analytic algorithms for checking Fourier-positivity in 1- and radial 2-dimensions among a large variety of real positive functions. This may provide a step towards a classification of positive positive-definite functions.





Author: Bertrand Giraud - Robi Peschanski -

Source: https://hal.archives-ouvertes.fr/



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