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(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 ?

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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.

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