I have a subgaussian r.v. with pdf $p(x) \in SG(a,C)$, where $SG(a,C)$ is the set of pdf of all random variables $X \in \mathbb R^d$ s.t. $\mathbb E e^{aX^2} < C,$ i.e. a set of subgaussian pdf.
I wanna characterize some property of $p$. What I can do is that my result has already held for function in $\mathcal F(k,L)$, where $$ \mathcal F(k,L) = \{f\in L^1(\mathbb R^d)\bigcap L^2(\mathbb R^d):f \text{ is continuous}, \|f|x|^k\|_1\leq L,\|\hat f |x|^k \|_1 \leq L\} . $$ Here $\hat f$ is the fourier transform of $f$, and $k>0,L>0$. This function class has the property that both itself and fourier transform decay polynomial. Of course, $p$ might not be in $\mathcal F(k,L)$ since it's fourier transform might be smooth.
My questions:
Given $a>0,C>0$, is it possible that there exists $k,L$ (relevant to $a,C$), so that $\mathcal F(k,L)$ is dense in $SG(a,C)$ with norm $\| \cdot\|_1$? In other words, there exists $\{f_n\} \in \mathcal F(k,L)$, so that $\|f_n - p\|_1 \to 0$?