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In [[mathematics]], a '''moment problem''' arises as the result of trying to invert the mapping that takes a [[measure (mathematics)|measure]] μ to the sequences of [[Moment (mathematics)|moment]]s

:<math>m_n = \int_{-\infty}^\infty x^n \,d\mu(x)\,.\,\!</math>

More generally, one may consider
:<math>m_n = \int_{-\infty}^\infty M_n(x) \,d\mu(x)\,\!</math>
for an arbitrary sequence of functions ''M''''n''.

== Introduction ==

In the classical setting, μ is a measure on the [[real line]], and ''M'' is in the sequence { ''x''''n'' : ''n'' = 0, 1, 2, ... } In this form the question appears in [[probability theory]], asking whether there is a [[probability measure]] having specified [[mean]], [[variance]] and so on, and whether it is unique.

There are three named classical moment problems: the [[Hamburger moment problem]] in which the [[support (mathematics)|support]] of μ is allowed to be the whole real line; the [[Stieltjes moment problem]], for <nowiki>[0, +∞)</nowiki>; and the [[Hausdorff moment problem]] for a bounded interval, which [[without loss of generality]] may be taken as [0, 1].

==Existence==

A sequence of numbers ''m''''n'' is the sequence of moments of a measure ''μ'' if and only if a certain positivity condition is fulfilled; namely, the [[Hankel matrices]] ''H''''n'',

:<math>(H_n)_{ij} = m_{i+j}\,,\,\!</math>

should be [[positive-definite matrix|positive semi-definite]]. A condition of similar form is necessary and sufficient for the existence of a measure <math>\mu</math> supported on a given interval [''a'', ''b''].

One way to prove these results is to consider the linear functional <math>\scriptstyle\varphi</math> that sends a polynomial

: <math>P(x) = \sum_k a_k x^k \,\!</math>

to

: <math>\sum_k a_k m_k.\,\!</math>

If ''m''''kn'' are the moments of some measure ''μ'' supported on [''a'', ''b''], then evidently

:{{NumBlk|:|''φ''(''P'') ≥ 0 for any polynomial ''P'' that is non-negative on [''a'', ''b''].|{{EquationRef|1}}}}

Vice versa, if ({{EquationNote|1}}) holds, one can apply the [[Riesz–Markov–Kakutani representation theorem|M. Riesz extension theorem]] and extend <math>\phi</math> to a functional on the space of continuous functions with compact support ''C''0([''a'', ''b'']), so that

:{{NumBlk|:|<math>\qquad \varphi(f) \ge 0\text{ for any } f \in C_0([a,b])</math>|{{EquationRef|2}}}}

such that ''&fnof;'' ≥ 0 on [''a'', ''b''].

By the [[Riesz_representation_theorem#The_representation_theorem_for_linear_functionals_on_Cc.28X.29| Riesz representation theorem]], ({{EquationNote|2}}) holds iff there exists a measure ''μ'' supported on [''a'', ''b''], such that

: <math> \phi(f) = \int f \, d\mu\,\!</math>

for every ''&fnof;'' ∈ ''C''0([''a'', ''b'']).

Thus the existence of the measure <math>\mu</math> is equivalent to ({{EquationNote|1}}). Using a representation theorem for positive polynomials on [''a'', ''b''],  one can reformulate ({{EquationNote|1}}) as a condition on [[Hankel matrices]].

See Refs. 1–3. for more details.

== Uniqueness (or determinacy) ==

The uniqueness of μ in the Hausdorff moment problem follows from the [[Weierstrass approximation theorem]], which states that [[polynomial]]s are [[dense set|dense]] under the [[uniform norm]] in the space of [[continuous functions]] on [0, 1]. For the problem on an infinite interval, uniqueness is a more delicate question; see [[Carleman's condition]], [[Krein's condition]] and Ref. 2.

== Variations ==

An important variation is the [[truncated moment problem]], which studies the properties of measures with fixed first ''k'' moments (for a finite ''k''). Results on the truncated moment problem have numerous applications to [[extremal problems]], optimisation and limit theorems in [[probability theory]]. See also: [[Chebyshev–Markov–Stieltjes inequalities]] and Ref. 3.

==References==
:'''1'''. Shohat, James Alexander; [[Jacob Tamarkin|Tamarkin, J. D.]]; ''The Problem of Moments'', American mathematical society, New York, 1943.
:'''2'''. [[Naum Akhiezer|Akhiezer, N. I.]], ''The classical moment problem and some related questions in analysis'', translated from the Russian by N. Kemmer, Hafner Publishing Co., New York 1965 x+253 pp.
:'''3'''. Krein, M. G.; Nudelman, A. A.; ''The Markov moment problem and extremal problems. Ideas and problems of P. L. Chebyshev and A. A. Markov and their further development.'' Translated from the Russian by D. Louvish. Translations of Mathematical Monographs, Vol. 50. American Mathematical Society, Providence, R.I., 1977. v+417 pp.

[[Category:Mathematical analysis]]
[[Category:Hilbert space]]
[[Category:Probability theory]]
[[Category:Theory of probability distributions]]
[[Category:Mathematical problems]]
[[Category:Real algebraic geometry]]
[[Category:Optimization in vector spaces]]

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