The Classical Moment Problem And Some Related Questions In Analysis ✮ «FAST»

For the Hausdorff problem (support in $[0,1]$), the condition becomes that the sequence is : the forward differences alternate in sign. Specifically, $\Delta^k m_n \ge 0$ for all $n,k\ge 0$, where $\Delta m_n = m_n+1 - m_n$. 3. Uniqueness: The Problem of Determinacy Even if a moment sequence exists, the measure might not be unique. This is the most subtle part of the theory.

$$ m_n = \int_\mathbbR x^n , d\mu(x) $$

$$ S(z) = \int_\mathbbR \fracd\mu(x)x - z, \quad z \in \mathbbC\setminus\mathbbR $$ For the Hausdorff problem (support in $[0,1]$), the

encodes all the moments. The measure is determinate iff the associated (a tridiagonal matrix) is essentially self-adjoint in $\ell^2$. Indeterminacy corresponds to a deficiency of self-adjoint extensions—a concept from quantum mechanics. Complex Analysis and the Stieltjes Transform Define the Stieltjes transform of $\mu$: Uniqueness: The Problem of Determinacy Even if a