Using the basis of Hermite--Fourier functions (i.e. the quantum oscillator eigenstates) and the Sturm theorem, we derive constraints for a function and its Fourier transform to be both real and 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.

PACS numbers: 02.30.Gp, 02.30.Mv, 02.30.Nw, 12.38.--t