  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.
