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From "Dirac combs" to Fourier-positivity

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.
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Submitted on : Friday, April 8, 2016 - 3:07:19 PM
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  • HAL Id : cea-01299916, version 1
  • ARXIV : 1509.02373

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Bertrand Giraud, Robi Peschanski. From "Dirac combs" to Fourier-positivity. Acta Physica Polonica B, Jagellonian University, Cracow, 2016, 47, pp.1075-1100. ⟨cea-01299916⟩

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