$$ s_n = \int_-\infty^\infty x^n , d\mu(x) \quad \textfor n = 0, 1, 2, \dots $$
And if such a measure exists, is it unique?
The first is the Hamburger Moment Problem . Here, the support of the measure is the entire real line $(-\infty, \infty)$. This is the most general case, allowing for measures that extend infinitely in both directions. It asks whether a sequence of numbers corresponds to the moments of a probability distribution over the entire continuum of real numbers. $$ s_n = \int_-\infty^\infty x^n , d\mu(x) \quad
These polynomials satisfy a three-term recurrence relation:
On $[0,1]$, the moment problem is always determinate—if a solution exists, it is unique. Hausdorff gave a beautiful characterization: A sequence $(m_n)$ comes from a measure on $[0,1]$ iff it is completely monotonic , i.e., all finite differences alternate in sign: This is the most general case, allowing for
| Problem | Domain | Conditions on moments | |---------|--------|------------------------| | | $\mathbbR$ | $m_2n \ge 0$, Hankel matrices positive semidefinite | | Stieltjes | $[0, \infty)$ | Same as Hamburger + condition on shifted Hankel | | Hausdorff | $[0,1]$ | Moments are completely monotonic or satisfy difference conditions |
The moment problem is not an isolated curiosity; it is deeply woven into other mathematical disciplines: The classical moment problem 1]$ iff it is completely monotonic
Two powerful criteria from analysis:
$$ m_n = \int_\mathbbR x^n , d\mu(x) $$
The classical moment problem has grown into a vibrant research area: