I have a problem concerning reconstructing a function from the expectation of its powers. The scalar version of this problem is as follows:
Given a probability measure $\mu(x)$ over $\mathbb{R}^n$ and a sequence $(a_k)$, $k\in \{1,2,\dots\}$, find a function $f(x)$ such that $\int_x \left(f(x)\right)^k \mu(x) = a_k$.
The scalar version above is the warmup. The question I am actually interested in is for a vector function:
Given a probability measure $\mu(x)$ over $\mathbb{R}^n$ and a sequence $(A_k)$, $k\in \{1,2,\dots\}$, where $A_k$ is a symmetric tensor of order $k$, find a function $F: \mathbb{R}^n\rightarrow \mathbb{R}^n$ such that $\int_x \left(F(x)\right)^{\otimes k} \mu(x) = A_k$. (Where $(\cdots)^{\otimes k}$ denotes an iterated outer product.)
This problem originates from optimal transport, where it has to do with finding a transport map from moments of two distributions. It bears resemblance to the problem of moments, but I am unsure whether that connection is helpful.
How can we find the function $F$ from the sequence $A_k$? What is known about this problem?
I will accept progress on the scalar version of the problem if the vector version proves too hard.