Closure of polynomials in $L^2_w$ with log-normal weight function

Question

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?

Answer 1

By squaring this is the same question as is $X$ in the span of $1, X^2, X^4,...$. I will produce a function that has the property that $E(FX^{2j}) = 0, E(FX) \ne 0$. Set $Z = log(X)$ which is a normal r.v. Regarding $F$ as a function of Z, $E(FX^{2j}) = 0$ is equivalent to $E(F(Z+2j) ) = 0$ since $X^{2j} e^{ - 2j^2} $ is the likelihood ratio for $N(2j, 1)$ with respect to $N(0,1)$. Similiarly, $E(FX^{j}) \ne 0$ is equivalent to $E(F(Z+1) ) \ne 0$. Take $F(x) = 1, 0< x < 2, -1 2 < x < 4$ and extend to be periodic with period 4. Then I claim $F(Z) = F(log(X)) $ is as claimed. The function is antisymmetric so $E(F(Z)) = 0$ but also, $F(x-2j)$ is antisymmetric, so $E(F(Z-2j)) = 0$ and therefore $E(FX^{2j}) = 0$ . But $F(x - 1)$ is not antisymmetric and will not have 0 expectation.