It is well-known that real valued Schwartz functions on $\mathbb{R}_+$ $\mathcal{S}(\mathbb{R}_+)$ are dense in the set of square integrable functions on $\mathbb{R}_+$ $L^2(\mathbb{R}_+)$. We can further restrict this result to positive functions: any positive $L^2$ function on $\mathbb{R}_+$ can be approximated by a positive Schwartz function on $\mathbb{R}_+$.

Now $L^2(\mathbb{R}_+)$ functions can be expanded on the orthonormal basis of Laguerre functions $\{\mathscr{L}_n\}_{n \in \mathbb{N}}$ where $\mathscr{L}_n : x \in \mathbb{R}_+ \mapsto L_n(x) e^{\frac{-x}2}$ with $L_n$ is the n$^{\text{th}}$ Laguerre polynomial. So for $f \in L^2(\mathbb{R}_+) $ there exists a square integrable sequence $\{ f_n \} \in \ell^2$ such that: $$ f = \sum_n f_n \mathscr{L}_n. $$ And likewise we can expand a Schwartz function with a rapidly decreasing sequence.

My question is the following: can I approximate a positive $L^2$ function on $\mathbb{R}_+$ with non negative coefficients $\{ f_n \}$ by a positive Schwartz function on $\mathbb{R}_+$ with again non negative coefficients?

My fear is that if one of the coefficient $f_k$ is zero then the corresponding sequence of Schwartz functions converging to $f$ may have the k$^{\text{th}}$ converging to $f_k$ from below and thus being negative. So the additional constraint asking for non negative coefficients may be too much (though it would be true if I were asking for positive coefficients).

It is well-known that real valued Schwartz functions on $\mathbb{R}_+$ $\mathcal{S}(\mathbb{R}_+)$ are dense in the set of square integrable functions on $\mathbb{R}_+$ $L^2(\mathbb{R}_+)$. We can further restrict this result to nonnegative functions: any nonnegative $L^2$ function on $\mathbb{R}_+$ can be approximated by a nonnegative Schwartz function on $\mathbb{R}_+$.

Now $L^2(\mathbb{R}_+)$ functions can be expanded on the orthonormal basis of Laguerre functions $\{\mathscr{L}_n\}_{n \in \mathbb{N}}$ where $\mathscr{L}_n : x \in \mathbb{R}_+ \mapsto L_n(x) e^{\frac{-x}2}$ with $L_n$ is the n$^{\text{th}}$ Laguerre polynomial. So for $f \in L^2(\mathbb{R}_+) $ there exists a square integrable sequence $\{ f_n \} \in \ell^2$ such that: $$ f = \sum_n f_n \mathscr{L}_n. $$ And likewise we can expand a Schwartz function with a rapidly decreasing sequence.

My question is the following: can I approximate a nonnegative $L^2$ function on $\mathbb{R}_+$ with nonnegative coefficients $\{ f_n \}$ by a nonnegative Schwartz function on $\mathbb{R}_+$ with again nonnegative coefficients?

My fear is that if one of the coefficient $f_k$ is zero then the corresponding sequence of Schwartz functions converging to $f$ may have the k$^{\text{th}}$ converging to $f_k$ from below and thus being negative. So the additional constraint asking for non negative coefficients may be too much (though it would be true if I were asking for positive coefficients).