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Characterizing in a constructive way the set of real functions whose Fourier transforms are positive appears to be yet an open problem. Some sufficient conditions are known but they are far from being exhaustive. We propose two constructive sets of necessary conditions for positivity of the Fourier transforms and test their ability of constraining the positivity domain. One uses analytic continuation and Jensen inequalities and the other deals with Toeplitz determinants and the Bochner theorem. Applications are discussed, including the extension to the two-dimensional Fourier-Bessel transform and the problem of positive reciprocity, i.e. positive functions with positive transforms.
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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.
The Fractional Fourier Transform (FrFT) has widespread applications in areas like signal analysis, Fourier optics, diffraction theory, etc. The Holomorphic Fractional Fourier Transform (HFrFT) proposed in the present paper may be used in the same wide range of applications with improved properties. The HFrFT of signals spans a one-parameter family of (essentially) holomorphic functions, where the parameter takes values in the bounded interval $tin (0,pi/2)$. At the boundary values of the parameter, one obtains the original signal at $t=0$ and its Fourier transform at the other end of the interval $t=pi/2$. If the initial signal is $L^2 $, then, for an appropriate choice of inner product that will be detailed below, the transform is unitary for all values of the parameter in the interval. This transform provides a heat kernel smoothening of the signals while preserving unitarity for $L^2$-signals and continuously interpolating between the original signal and its Fourier transform.
In this work we verify the sufficiency of a Jensens necessary and sufficient condition for a class of genus 0 or 1 entire functions to have only real zeros. They are Fourier transforms of even, positive, indefinitely differentiable, and very fast decreasing functions. We also apply our result to several important special functions in mathematics, such as modified Bessel function $K_{iz}(a), a>0$ as a function of variable $z$, Riemann Xi function $Xi(z)$, and character Xi function $Xi(z;chi)$ when $chi$ is a real primitive non-principal character satisfying $varphi(u;chi)ge0$ on the real line, we prove these entire functions have only real zeros.
Fourier-positivity, i.e. the mathematical property that a function has a positive Fourier transform, can be used as a constraint on the parametrization of QCD dipole-target cross-sections or Wilson line correlators in transverse position (r) space. They are Bessel transforms of positive transverse momentum dependent gluon distributions. Using mathematical Fourier-positivity constraints on the limit r -> 0 behavior of the dipole amplitudes, we identify the common origin of the violation of Fourier-positivity for various, however phenomenologically convenient, dipole models. It is due to the behavior r^{2+epsilon}, epsilon>0, softer, even slightly, than color transparency. Fourier-positivity seems thus to conflict with the present dipole formalism when it includes a QCD running coupling constant alpha(r).
We present a new method, based on Gaussian process regression, for reconstructing the continuous $x$-dependence of parton distribution functions (PDFs) from quasi-PDFs computed using lattice QCD. We examine the origin of the unphysical oscillations seen in current lattice calculations of quasi-PDFs and develop a nonparametric fitting approach to take the required Fourier transform. The method is tested on one ensemble of maximally twisted mass fermions with two light quarks. We find that with our approach oscillations of the quasi-PDF are drastically reduced. However, the final effect on the light-cone PDFs is small. This finding suggests that the deviation seen between current lattice QCD results and phenomenological determinations cannot be attributed solely on the Fourier transform.
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