Schwarz reflection principle

In mathematics, the Hausdorff moment problem, named after Felix Hausdorff, asks for necessary and sufficient conditions that a given sequence { m_n : n = 0, 1, 2, ... } be the sequence of moments

$m_n=\underset{0}{\overset{1}{\int }}{x}^{n}d\mu (x)$

of some Borel measure μ supported on the closed unit interval [0, 1]. In the case m₀ = 1, this is equivalent to the existence of a random variable X supported on [0, 1], such that E Xⁿ = m_n.

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 [0, ∞), and in the Hamburger moment problem one considers the whole line (−∞, ∞).

In 1921, Hausdorff showed that { m_n : n = 0, 1, 2, ... } is such a moment sequence if and only if the sequence is completely monotonic, i.e., its difference sequences satisfy the equation

$(-1{)}^k({\mathrm{\Delta }}^{k}m{)}_n\ge 0$

for all n,k ≥ 0. Here, Δ is the difference operator given by

$(\mathrm{\Delta }m{)}_n=m_{n+1}-m_n.$

The necessity of this condition is easily seen by the identity

$(-1{)}^k({\mathrm{\Delta }}^{k}m{)}_n=\underset{0}{\overset{1}{\int }}{x}^n(1-x{)}^{k}d\mu (x),$

which is ≥ 0, being the integral of an almost sure non-negative function. 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

• 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.

External links

• Moment Problem, on Mathworld