In mathematics, particularly, in

analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...

, Carleman's condition gives a sufficient condition for the determinacy of the

moment problem In mathematics, a moment problem arises as the result of trying to invert the mapping that takes a measure \mu to the sequence of moments :m_n = \int_^\infty x^n \,d\mu(x)\,. More generally, one may consider :m_n = \int_^\infty M_n(x) \,d\mu( ...

. That is, if a measure

\mu

satisfies Carleman's condition, there is no other measure

\nu

having the same moments as

\mu.

The condition was discovered by Torsten Carleman in 1922.

Hamburger moment problem

For the

Hamburger moment problem In mathematics, the Hamburger moment problem, named after Hans Ludwig Hamburger, is formulated as follows: given a sequence , does there exist a positive Borel measure (for instance, the measure determined by the cumulative distribution function o ...

(the moment problem on the whole real line), the theorem states the following: Let

\mu

be a measure on

\R

such that all the moments

m_n = \int_^ x^n \, d\mu(x)~, \quad n = 0,1,2,\cdots

are finite. If

\sum_^\infty m_^ = + \infty,

then the moment problem for

(m_n)

is ''determinate''; that is,

\mu

is the only measure on

\R

with

(m_n)

as its sequence of moments.

Stieltjes moment problem

For the

Stieltjes moment problem In mathematics, the Stieltjes moment problem, named after Thomas Joannes Stieltjes, seeks necessary and sufficient conditions for a sequence (''m''0, ''m''1, ''m''2, ...) to be of the form :m_n = \int_0^\infty x^n\,d\mu(x) for some measure ''&m ...

, the sufficient condition for determinacy is

\sum_^\infty m_^ = + \infty.

Generalized Carleman's condition

In, Nasiraee et al. showed that, despite previous assumptions,S. S. Shamai, “Capacity of a pulse amplitude modulated direct detection photon channel,” IEE Proceedings I (Communications, Speech and Vision), vol. 137, no. 6, pp. 424–430, Dec. 1990. when the integrand is an arbitrary function, Carleman's condition is not sufficient, as demonstrated by a counter-example. In fact, the example violates the bijection, i.e. determinacy, property in the probability sum theorem. When the integrand is an arbitrary function, they further establish a sufficient condition for the determinacy of the moment problem, referred to as the ''generalized Carleman's condition''.

Notes

References

* {{Cite book , first=N. I. , last=Akhiezer , title=The Classical Moment Problem and Some Related Questions in Analysis , publisher=Oliver & Boyd , year=1965 * Chapter 3.3, Durrett, Richard. ''Probability: Theory and Examples''. 5th ed. Cambridge Series in Statistical and Probabilistic Mathematics 49. Cambridge ; New York, NY: Cambridge University Press, 2019. Mathematical analysis Moments (mathematics) Probability theory Theorems in approximation theory