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](https://mathoverflow.net/questions/213275/tighter-caratheodory-on-the-moment-curve). 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:


1. 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?)
2. 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".)



Matching moments in even dimensions
The answer to question #2 is no. Consider for example $n = 2$ with the measure $\delta(0)$. You need to choose $x\_1 = 0$, but the choice of $x\_2$ is arbitrary as $\alpha\_2 = 0$.


The answer to question #1 is yes. For a more detailed analysis, I refer to theorem 2.2.3 in Dette/ Studden's *The Theory of Canonical Moments with Applications in Statistics, Probability, and analysis*. Essentially the problem lies therein, that the measure (and its support points) is uniquely determined as soon as one of corresponding canonical moments is 0 or 1. If you try to add any more support points, the corresponding weights need to be 0 and the choice of support points is ambiguous.


