Let $D$ be a probability distribution on the unit interval $[0,1]$ with moments $\mu_i=\mathbb{E}_D [x^i]$. Let $\delta(x)$ be a singleton probability distribution with all weight at $x\in [0,1]$. Let $C(n)=\lfloor (n+2)/2 \rfloor$.

Then we can match the first $n$ moments of any $D$ using a convex combination of $C(n)$ singleton distributions. In other words, for any $D$ and any $n$, there exists $\alpha_i, x_i$ such that the distribution $$ \sum_{i=1}^{C(n)} \alpha_i \delta(x_i) $$ has the same first $n$ moments as $D$ (where $\alpha_i,x_i \in [0,1]$ and $\sum_i \alpha_i=1$.) For details, see this Mathoverflow post. Note that we can force $x_1\leq x_2 \leq \cdots \leq x_n$ without changing anything.

Now, consider a map $f$ from the (non-decreasing) singletons to the corresponding moments, i.e. $$f(x_1,...,x_{C(n)},\alpha_1,...,\alpha_{C(n)-1})=(\mu_1,...,\mu_n)$$ (Note that we can compute $\alpha_{C(n)}$ as $1-\sum_{i=1}^{C(n)-1}\alpha_i$.) Let $\Delta_n$ be the domain of $f$.

The function $f$ maps $\Delta \subseteq \mathbb{R}^{2C(n)-1}$ to $\mathbb{R}^n$. When $n$ is odd, $2C(n)-1=n$. However, when $n$ is even, $2C(n)-1=n+1$, so it would appear that the domain has an "extra dimension". I am trying to understand if it is possible to reparameterize the problem when $n$ is even so that the domain is $n$-dimensional in order to construct an injection.

My questions are:

- If $n$ is even and we consider all convex combinations of $C(n)$ singleton distributions, but if we force $x_1=0$, can we still match the first $n$ moments of any distribution on the unit interval? (I.e. does the range of $f$ match the range of $f|_{x_1=0}$ when $n$ is even?)
- Suppose the answer to the previous question if "yes". Let $g=f$ when $n$ is odd, and let $g$ be the restriction of $f$ to $x_1=0$ when $n$ is even. Is $g$ invertible?

I realize that these specific question may be misguided-- perhaps there is a better way to think about a space of canonical representative distributions with target moments. (Note that there exist moments that can only be represented by non-continuous distributions, so I wasn't able to use entropy maximization "out of the box".)