Hausdorff moment problem

Template:Short description In mathematics, the Hausdorff moment problem, named after Felix Hausdorff, asks for necessary and sufficient conditions that a given sequence Template:Math be the sequence of moments

$m_n={\int }_0^1{x}^{n}d\mu (x)$

of some Borel measure Template:Mvar supported on the closed unit interval Template:Math. In the case Template:Math, this is equivalent to the existence of a random variable Template:Mvar supported on Template:Math, such that Template:Math.

The essential difference between this and other well-known moment problems is that this is on a bounded interval, whereas in the Stieltjes moment problem one considers a half-line Template:Math, and in the Hamburger moment problem one considers the whole line Template:Math. The Stieltjes moment problems and the Hamburger moment problems, if they are solvable, may have infinitely many solutions (indeterminate moment problem) whereas a Hausdorff moment problem always has a unique solution if it is solvable (determinate moment problem). In the indeterminate moment problem case, there are infinite measures corresponding to the same prescribed moments and they consist of a convex set. The set of polynomials may or may not be dense in the associated Hilbert spaces if the moment problem is indeterminate, and it depends on whether measure is extremal or not. But in the determinate moment problem case, the set of polynomials is dense in the associated Hilbert space.

Completely monotonic sequences

In 1921, Hausdorff showed that Template:Math is such a moment sequence if and only if the sequence is completely monotonic, that is, its difference sequences satisfy the equation

${\mathrm{\Delta }}^k{m}_n\ge 0$

for all Template:Math. Here, $\mathrm{\Delta }$ is the difference operator given by

$\mathrm{\Delta }{m}_n=m_n-m_{n+1},$

and in general for all $k\ge 1$,

${\mathrm{\Delta }}^k{m}_n=\sum _{i=0}^k(\genfrac{}{}{0ex}{}{k}{i})(-1{)}^i{m}_{n+i}.$

The necessity of this condition is easily seen by observing that ${\mathrm{\Delta }}^k{m}_n$ can be expressed as the integral of a positive function and therefore must be positive:

$0\le {\int }_0^1{x}^n(1-x{)}^{k}d\mu (x)={\mathrm{\Delta }}^k{m}_n.$

The equality in the identity above is justified by using the binomial theorem to expand the term $(1-x{)}^k=\sum _{i=0}^k(\genfrac{}{}{0ex}{}{k}{i})(-1{)}^i{x}^i$ within the integral and then Fubini's theorem to switch the resulting sum and the integral:

${\int }_0^1{x}^n(1-x{)}^{k}d\mu (x)=\sum _{i=0}^k(\genfrac{}{}{0ex}{}{k}{i})(-1{)}^i{\int }_0^1{x}^{n+i}d\mu (x)=\sum _{i=0}^k(\genfrac{}{}{0ex}{}{k}{i})(-1{)}^i{m}_{n+i}={\mathrm{\Delta }}^k{m}_n.$

For example, it is necessary to have

${\mathrm{\Delta }}^4{m}_6=m_6-4m_7+6m_8-4m_9+m_{10}=\int x^6(1-x{)}^{4}d\mu (x)\ge 0.$

See also

• Absolutely and completely monotonic functions and sequences
• Total monotonicity

References

• Hausdorff, F. "Summationsmethoden und Momentfolgen. I." Mathematische Zeitschrift 9, 74–109, 1921.
• Hausdorff, F. "Summationsmethoden und Momentfolgen. II." Mathematische Zeitschrift 9, 280–299, 1921.
• Feller, W. "An Introduction to Probability Theory and Its Applications", volume II, John Wiley & Sons, 1971.
• Shohat, J.A.; Tamarkin, J. D. The Problem of Moments, American mathematical society, New York, 1943.