Consider the Hilbert space $L^2_w$ with scalar product $\langle f,g\rangle_w =\int_0^\infty f(x)g(x)w(x)dx$ where the weight $w$ is the density function of a log-normal distribution $$ w(x)=\frac{1}{\sqrt{2\pi}\sigma x}e^{-\frac{(\ln(x)-\mu)^2}{2\sigma^2}},$$ for some $\mu\in\mathbb{R}$ and $\sigma>0$. The log-normal distribution has finite moments of any order but it is well known that it is not determined by its moments and as a consequence we have that the polynomials do not lie dense in $L^2_w$. Are there any known results that describe subspaces of functions that lie in the colsure of the polynomials in this space? In particular I am interested to know whether or not the square root function lies in the polynomial closure?