On positive functions with positive Fourier transforms
Abstract
Description
Using the basis of Hermite-Fourier functions (i.e. the quantum oscillator eigenstates) and the Sturm theorem, we derive the practical constraints for a function and its Fourier transform to be both positive. We propose a constructive method based on the algebra of Hermite polynomials. Applications are extended to the 2-dimensional case (i.e. Fourier-Bessel transforms and the algebra of Laguerre polynomials) and to adding constraints on derivatives, such as monotonicity or convexity.
12 pages, 23 figures. High definition figures can be obtained upon request at giraud@dsm-mail.saclay.cea.fr or pesch@dsm-mail.saclay.cea.fr
12 pages, 23 figures. High definition figures can be obtained upon request at giraud@dsm-mail.saclay.cea.fr or pesch@dsm-mail.saclay.cea.fr