Problems on Schwartz Functions

Question

(1) What are all positive Schwartz Functions on $\mathbb R$ whose Fourier Transform is positive ?

(2) What are all Schwartz Functions on $\mathbb R$ whose Fourier Transform is positive ?

(3) What are all positive Schwartz Functions on $\mathbb R$ whose Fourier Transform is non-negative ?

(4) What are all Schwartz Functions on $\mathbb R$ whose Fourier Transform is non-negative ?

Answer 1

Reference: On positive functions with positive Fourier transforms develops a basis of functions which are positive along with their Fourier Transforms. In $\mathbb{R}$ the results are built from Hermite Polynomials and in $\mathbb{R}^2$ from radial Laguerre Polynomials.