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 (3 ...

, 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 ''μ'' to the sequences of moments :m_n = \int_^\infty x^n \,d\mu(x)\,. More generally, one may consider :m_n = \int_^\infty M_n(x) ...

. 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 Torsten Carleman (8 July 1892, Visseltofta, Osby Municipality – 11 January 1949, Stockholm), born Tage Gillis Torsten Carleman, was a Swedish mathematician, known for his results in classical analysis and its applications. As the director of ...

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 (''m''0, ''m''1, ''m''2, ...), does there exist a positive Borel measure ''μ'' (for instance, the measure determined by ...

(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 In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and ...

, the sufficient condition for determinacy is

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

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 Mathematical analysis Moment (mathematics) Probability theory Theorems in approximation theory