= Moment problem
{title2=$m_n=\int x^n\,d\mu$}
A moment problem asks whether specified numbers $m_0,m_1,\ldots$ arise from a positive <Borel measure> $\mu$ on a prescribed subset of the real line:
$$
m_n=\int x^n\,d\mu(x).
$$
Existence, support restrictions and uniqueness are separate parts of the question. A probability measure must have $m_0=1$. A point mass at $a$ gives $m_n=a^n$, while the uniform probability measure on $[0,1]$ gives $m_n=1/(n+1)$.
Positivity imposes concrete constraints. For every finite coefficient vector $c$,
$$
\sum_{i,j=0}^N\overline{c_i}c_jm_{i+j}=\int\left|\sum_{j=0}^Nc_jx^j\right|^2d\mu(x)\ge0,
$$
so every associated <Hankel matrix> is <positive semidefinite>. On a fixed compact interval, the full moment sequence determines a finite measure uniquely: equality on polynomials extends to equality on continuous functions by the <Weierstrass approximation theorem>. A finite list of moments generally leaves many possible measures, which explains the distinction between moment reconstruction and finite-data fitting.
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