Moment problem ($m_n=\int x^{n}d\mu$)

A moment problem asks whether specified numbers $m_0,m_1,\dots$ arise from a positive Borel measure $\mu$ on a prescribed subset of the real line: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 $c$,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.