From "Dirac combs" to Fourier-positivity

Bertrand G. Giraud (1) , Robi Peschanski (1)

1 IPHT - Institut de Physique Théorique - UMR CNRS 3681

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.

Domains: Mathematical Physics [math-ph]; High Energy Physics - Phenomenology [hep-ph]; High Energy Physics - Theory [hep-th]

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