We are concerned with a function in 2-D Euclidean space. $\theta: \mathbb{R}^2 \rightarrow \mathbb{R} \ $, $\theta \in L^1(\mathbb{R}^2) \cap L^p(\mathbb{R}^2) $, $p\in (2,\infty)$.
As per the title, I am wondering if there are any properties $\theta$ attains, if we set $\hat{\theta}(\xi) \geq 0$, $\forall \xi \in \mathbb{R}^2$.
I am wondering if there are any other strange properties that might be forced on $\theta$ by messing with its FT in this way. I am not overly familiar with the intricacies of the FT, and so am very curious as to how general $\theta$ can still be with this condition applied to it.
To give some background, I am using the $\theta$ I am talking about here as the initial data to the Quasi-Geostrophic Equations; and so I am wary of forcing the FT to have this property, for fear of being left with very strong conditions on my initial data.
EDIT: It is sufficient for my purposes to simply set $\hat{\theta}$ positive, not bounded from below by a positive constant as was originally written!