Given constants $m_0,\ldots m_n$ and a measure $\mu$ on $\mathbb{R}$, how can I "recover" the integrals $\int f x^n d\mu$ of the maximum entropy distribution $f\in L^2(\mathbb{R})$ which satisfies $\int f^2 x^n d\mu=m_n$ ?
Let us consider the special case when $\mu$ is the Lebesgue measure on $\mathbb{R}$. The derivation then should be extendable to the general case without much effort.
We need to find the distribution $f$ that solves the following problem: $$\max_{f\in \mathcal{B}}\int_\mathbb{R}f(x)\ln f(x) dx\\\Leftrightarrow\min_{f\in \mathcal{B}}\int_\mathbb{R}f(x)\ln f(x) dx\\\mathrm{s.t.}\quad \int_\mathbb{R}f(x) dx=1,\ \int_\mathbb{R}f^2(x) x^kdx=m_k,\ k=0,1,\cdots,\ n$$ where $\mathcal{B}$ is the set of Borel measurable functions on $\mathbb{R}$. The corresponding Lagrangian is $$\mathcal{L}\left(f(x),\{\lambda_k\}_{k=0}^n\right)=\int_\mathbb{R}f(x)\ln f(x) dx\lambda_0\int_\mathbb{R}f(x) dx\sum_{k=1}^n \lambda_k \int_\mathbb{R}f^2(x) x^kdx$$ The maximizing distribution is obtained by equating the derivative of the Lagrangian to $0$ $$\frac{\partial \mathcal{L}}{\partial f}=1+\ln f(x)\lambda_02\sum_{k=1}^n\lambda_k f(x) x^k=0\\\implies \ln f(x)=\lambda_01+2f(x)\sum_{k=1}^n \lambda_k x^k\\\implies f(x)=a\exp\left(bf(x)\right)\\\implies f(x)=\frac{W(ab)}{b}$$ where $W$ is the Lambert's W function, $a=e^{\lambda_01},\ b=2\sum_{k=1}^n \lambda_k x^k$ Thus, we can express the integral $\int_\mathbb{R} f(x) x^k dx$ as below: $$\int_\mathbb{R} f(x) x^k dx=\int_\mathbb{R}\frac{W\left(2a\sum_{j=1}^n \lambda_j x^j\right)}{2\sum_{j=1}^n \lambda_j x^j}x^k dx$$ Beyond this point, the integral seems to be formidable, and probably a numerical approach is required, but in principle, this derivation produces some form of the maximizing distribution $f$, which one can then work with.

$\begingroup$ You also have to recover the $\lambda_i$'s somehow I suppose. $\endgroup$ Oct 22 '20 at 7:59

$\begingroup$ Yes, that can be done by solving the constraint equations, although I am not sure if nice closed form expressions are available for this. Numerically I think there will not be many issues. $\endgroup$ Oct 22 '20 at 10:06