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arXiv · 1509.02373

From "Dirac combs" to Fourier-positivity

Bertrand G. Giraud·Robi Peschanski
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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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BibTeXRIS

Bertrand G. Giraud, Robi Peschanski. 2015-09-08. From "Dirac combs" to Fourier-positivity. https://doi.org/10.5506/aphyspolb.47.1075

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