Context: let $f \in \mathcal{S}(\mathbb{R}^+) $ be a function of the Schwartz space (all functions whose derivatives are rapidly decreasing) on $\mathbb{R}^+$.
We already know that a generic such $f$ can be pointwise approximated by a linear combination of Laguerre functions because they form a basis of the Schwartz space. Note that $\text{n}^{\text{th}}$ Laguerre function $l_n$ is defined as:
$$ l_n(x) = e^{\frac{-x}2} L_n(x)$$
where $L_n$ is the $\text{n}^{\text{th}}$ Laguerre polynomials.
Question: now suppose that $f \in \mathcal{S}(\mathbb{R}^+)$ is positive on $\mathbb{R}^+$. Is it true that I can always pointwise converge to $f$ with a sequence of positive polynomials multiplied by a decreasing exponential?
Any reference will be appreciated.
If $f\ge 0$ then $f^{1/2}= \lim_{n\to \infty} P_n(x)e^{-x/2}$ and $f =\lim_{n\to \infty} P_n(x)^2 e^{-x}$
If you want it to be strictly positive then add $e^{-x}/n$.
If you meant that $f\ge 0 \implies \sum_{n=1}^N L_n\frac{\langle f,L_n\rangle}{\langle L_n,L_n\rangle} \ge 0$ then no, $L_2-L_1$ is bounded so $(c+L_2-L_1)e^{-x/2}\ge 0$ but $(c-L_1)e^{-x/2}$ is not $\ge 0$.