If such a measure exists, the sequence is called a moment sequence , and the measure is said to "solve" the moment problem. This seemingly singular inquiry branches into a rich network of theories involving linear algebra, complex analysis, orthogonal polynomials, and functional analysis. It is a cornerstone of 20th-century mathematics, bridging the gap between discrete algebraic data and continuous functional behavior.
$$ m_n = \int_\mathbbR x^n , d\mu(x) $$
This is intimately related to the Hausdorff moment theorem and the theory of Bernstein polynomials. If such a measure exists, the sequence is
Consider a finite sum of squares: $\int (\sum_k=0^N a_k x^k)^2 d\mu(x) \ge 0$. Expanding gives: $$ m_n = \int_\mathbbR x^n , d\mu(x) $$
; a solution exists if and only if the sequence is completely monotonic. Stieltjes (Half-line) Can be indeterminate (multiple measures for one sequence). Hamburger (Whole line) The most general case; can also be indeterminate . 3. Related Questions in Analysis can also be indeterminate . 3.
The answer depends critically on the domain of integration: