I want to understand the property of a class of real valued function $\mathcal F(\eta,\gamma)$, where $\eta,\gamma$ is positive, and $\forall f \in \mathcal F$ $$ \int_{\mathbb R^d} f(x)e^{\eta\|x\|}\mathrm dx< +\infty,\int_{\mathbb R^d}|\hat f(w)|^2 e^{2\gamma ||w||}\mathrm dw <+\infty $$
My questions:
(1) Is this kind of function exists? If not, how can I relax the assumption to let such kind of function exists (and maybe has the property in (2))
(2) Is it dense in some typical space, like Schwartz Space or L1 Space? (Here maybe you can ignore the influence of $\eta,\gamma$, like I am just want the space to be dense in the $\{f\in L^1:\|f\|_1\leq 1\}$.)