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 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 $[-1,1]$?

all$P$ with $P\ge 0$ on $[-1,1]$ implies existence of a measure as desired. You wouldn't expect a very easy characterization really since it's not straightforward to tell from the moments what the support of the measure is. $\endgroup$