Consider a sequence of real numbers $s=(s\_0,s\_1,\ldots)$. When is there a Borel measure $\mu$ supported on $[-1,1]$ so that
$$
s\_k = \int\_{[-1,1]} x^k\,\mathrm{d}\mu,\quad \forall k\in\mathbb N\;?
$$


The well known [Hausdorff moment problem](https://en.wikipedia.org/wiki/Hausdorff_moment_problem) asks the same question on the interval $[0,1]$. In that case, $s$ is a moment sequence of a measure supported on $[0,1]$ if and only if $s$ is completely monotone. It follows that, for our problem, the even terms of $s$ must be completely monotone. What is known about the odd terms?



> 
> **Question:** Are the necessary and sufficient conditions for $s$ to be a moment sequence on $[-1,1]$ known? What are the simplest known necessary conditions?
> 
> 
> 


The following question maybe easier to solve.



> 
> Assume now that $s$ is a sequence of nonnegative reals. When is there measure supported on $[-1,1]$ with the moment sequence $s$?
> 
> 
> 


There is a well known way of obtaining necessary conditions: Let $P(x) = \sum a\_i x^i$ be a polynomial that is nonnegative on $[-1,1]$. Form the Hankel matrix $H\_P$
$$
H\_P(j,k):= \sum\_i a\_i s\_{i+j+k},\quad j,k\in \mathbb N.
$$
Then we observe


**Claim:** Suppose $P(x)\ge 0$ on $[-1,1]$. Then $H\_P$ is positive semi-definite.


**Pf.** Let $c\_0,c\_1,\ldots$ be a complex sequence with finitely many nonzero elements. We have
\begin{align\*}
cH\_Pc^\* &= \sum\_{j,k}c\_j\bar{c\_k}\int \sum\_i a\_i x^{i+j+k}\,\mathrm{d}\mu\\
&=\int\_{[-1,1]} P(x)\left|\sum\_j c\_j x^j\right|^2\,\mathrm{d}\mu \ge 0.
\end{align\*}


A result due to Riesz implies that if $H\_P$ is PSD for all polynomials $P$ that are nonnegative on $[-1,1]$ then $s$ is a moment sequence. However if the interval is $[0,1]$ this PSD criterion is equivalent to a much much simpler condition: that the sequence is completely monotone. Is there a simpler characterization for $[-1,1]$?



Moment problem on [-1,1]: necessary and sufficient conditions
If $\mu$ is a measure supported on $[-1,1]$, the change of variables $y = (1+x)/2$ gives you a measure $\rho$ supported on $[0,1]$, and the corresponding moments are related by


$$\int\_0^1 y^k\; d\rho(y) = \int\_{-1}^1 ((1+x)/2)^k \; d\mu(x) 
= 2^{-k} \sum\_{j=0}^k {k \choose j} \int\_{-1}^1 x^j \; d\mu(x)$$


Moreover, we can also go in the opposite direction to give $\int\_{-1}^1 x^k\; d\mu(x)$ in terms of $\int\_0^1 y^j\; d\rho(y)$ for $j = 0 \ldots k$.


So the necessary and sufficient condition is that 
$r\_k = 2^{-k} \sum\_{j=0}^k {k \choose j} s\_j$ are completely monotone.


