From "Dirac combs" to Fourier-positivity

TL;DR

This paper investigates properties of Fourier-positive functions, introduces approximation methods based on Dirac combs and Poisson resummation, and develops algorithms for efficiently verifying Fourier-positivity in one and two dimensions.

Contribution

It presents new analytic algorithms for quickly checking Fourier-positivity, connecting Dirac comb properties with positive definiteness, aiding classification efforts.

Findings

Developed rapid algorithms for Fourier-positivity verification.

Linked Dirac comb properties with Bochner's theorem.

Provided approximation formulas for Fourier transforms.

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.