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| In [[mathematics]], the '''Hamburger [[moment problem]]''', named after [[Hans Hamburger|Hans Ludwig Hamburger]], is formulated as follows: given a sequence { ''m<sub>n</sub>'' : ''n'' = 1, 2, 3, ... }, does there exist a positive [[Borel measure]] ''μ'' on the real line such that
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| :<math>m_n = \int_{-\infty}^\infty x^n\,d \mu(x)\ ?</math>
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| In other words, an affirmative answer to the problem means that { ''m<sub>n</sub>'' : ''n'' = 0, 1, 2, ... } is the sequence of [[moment (mathematics)|moments]] of some positive Borel measure ''μ''.
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| The [[Stieltjes moment problem]], [[Vorobyev moment problem]], and the [[Hausdorff moment problem]] are similar but replace the real line by [0, +∞) (Stieltjes and Vorobyev; but Vorobyev formulates the problem in the terms of matrix theory), or a bounded interval (Hausdorff).
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| == Characterization ==
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| The Hamburger moment problem is solvable (that is, {''m<sub>n</sub>''} is a sequence of [[moment (mathematics)|moments]]) if and only if the corresponding Hankel kernel on the nonnegative integers
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| :<math>
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| A =
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| \left(\begin{matrix}
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| m_0 & m_1 & m_2 & \cdots \\
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| m_1 & m_2 & m_3 & \cdots \\
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| m_2 & m_3 & m_4 & \cdots \\
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| \vdots & \vdots & \vdots & \ddots \\
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| \end{matrix}\right)</math>
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| is [[positive definite kernel|positive definite]], i.e.,
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| :<math> \sum_{j,k\ge0}m_{j+k}c_j\bar c_k\ge0
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| </math>
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| for an arbitrary sequence {''c<sub>j</sub>''}<sub>''j'' ≥ 0</sub> of complex numbers with finite support (i.e.
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| ''c<sub>j</sub>'' = 0 except for finitely many values of ''j'').
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| The "only if" part of the claims can be verified by a direct calculation.
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| We sketch an argument for the converse. Let '''Z'''<sup>+</sup> be the nonnegative integers and ''F''<sub>0</sub>('''Z'''<sup>+</sup>) denote the family of complex valued sequences with finite support. The positive Hankel kernel ''A'' induces a (possibly degenerate) [[sesquilinear]] product on the family of complex valued sequences with finite support. This in turn gives a [[Hilbert space]]
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| :<math>(\mathcal{H}, \langle, \; \rangle)</math>
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| whose typical element is an equivalence class denoted by [''f''].
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| Let ''e<sub>n</sub>'' be the element in ''F''<sub>0</sub>('''Z'''<sup>+</sup>) defined by ''e<sub>n</sub>''(''m'') = [[Kronecker delta|''δ<sub>nm</sub>'']]. One notices that | |
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| :<math>\langle [e_{n+1}], [e_m] \rangle = A_{m,n+1} = m_{m+n+1} = \langle [e_n], [e_{m+1}]\rangle.</math>
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| Therefore the [[shift operator|"shift" operator]] ''T'' on <math>\mathcal{H}</math>, with ''T''[''e<sub>n</sub>''] = [''e''<sub>''n'' + 1</sub>], is [[symmetric operator|symmetric]].
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| On the other hand, the desired expression
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| :<math>m_n = \int_{-\infty}^\infty x^n\,d \mu(x).</math>
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| suggests that ''μ'' is the [[spectral measure]] of a [[self-adjoint operator]]. If we can find a "function model" such that the symmetric operator ''T'' is [[multiplication operator|multiplication by ''x'']], then the spectral resolution of a [[extensions of symmetric operators|self-adjoint extension]] of ''T'' proves the claim.
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| A function model is given by the natural isomorphism from ''F''<sub>0</sub>('''Z'''<sup>+</sup>) to the family of polynomials, in one single real variable and complex coefficients: for ''n'' ≥ 0, identify ''e<sub>n</sub>'' with ''x<sup>n</sup>''. In the model, the operator ''T'' is multiplication by ''x'' and a densely defined symmetric operator. It can be shown that ''T'' always has self-adjoint extensions. Let
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| :<math> \bar{T} \, </math>
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| be one of them and ''μ'' be its spectral measure. So
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| :<math>\langle \bar{T}^n [1], [1] \rangle = \int x^n d \mu(x).</math>
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| On the other hand,
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| :<math> \langle \bar{T}^n [1], [1] \rangle = \langle T^n [e_0], [e_0] \rangle = m_n. \, </math>
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| === Uniqueness of solutions ===
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| The solutions form a convex set, so the problem has either infinitely many solutions or a unique solution. | |
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| Consider the (''n'' + 1)×(''n'' + 1) [[Hankel matrix]]
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| :<math>\Delta_n=\left[\begin{matrix}
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| m_0 & m_1 & m_2 & \cdots & m_{n} \\
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| m_1 & m_2 & m_3 & \cdots & m_{n+1} \\
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| m_2 & m_3 & m_4 & \cdots & m_{n+2} \\
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| \vdots & \vdots & \vdots & \ddots & \vdots \\
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| m_{n} & m_{n+1} & m_{n+2} & \cdots & m_{2n}
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| \end{matrix}\right].</math>
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| Positivity of ''A'' means that for each ''n'', det(Δ<sub>''n''</sub>) ≥ 0. If det(Δ<sub>''n''</sub>) = 0, for some ''n'', then
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| :<math>(\mathcal{H}, \langle, \; \rangle)</math>
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| is finite dimensional and ''T'' is self-adjoint. So in this case the solution to the Hamburger moment problem is unique and ''μ'', being the spectral measure of ''T'', has finite support. | |
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| More generally, the solution is unique if there are constants ''C'' and ''D'' such that for all ''n'', |m<sub>''n''</sub>|≤ ''CD''<sup>''n''</sup>''n''! {{harv|Reed|Simon|1975|p=205}}. This follows from the more general [[Carleman's condition]].
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| There are examples where the solution is not unique.
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| == Further results ==
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| {{Expand section|date=June 2008}}
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| One can see that the Hamburger moment problem is intimately related to [[orthogonal polynomials]] on the real line. The [[Gram–Schmidt]] procedure gives a basis of orthogonal polynomials in which the operator
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| :<math> \bar{T} \, </math> | |
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| has a tridiagonal ''Jacobi matrix representation''. This in turn leads to a ''tridiagonal model'' of positive Hankel kernels.
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| An explicit calculation of the [[Cayley transform]] of ''T'' shows the connection with what is called the ''[[Nevanlinna class]]'' of analytic functions on the left half plane. Passing to the non-commutative setting, this motivates ''Krein's formula'' which parametrizes the extensions of partial isometries.
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| The cumulative distribution function and the probability density function can often be found by applying the inverse [[Laplace transform]] to the moment generating function
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| :<math>m(t)=\sum_{n=0}m_n\frac{t^n}{n!},</math>
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| provided that this function converges.
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| == References ==
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| *{{citation|first=Michael|last=Reed|first2=Barry|last2=Simon|title=Fourier Analysis, Self-Adjointness|year=1975|ISBN=0-12-585002-6|series=Methods of modern mathematical physics|volume=2|publisher=Academic Press|pages=145, 205}}
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| * {{citation|last=Shohat|first= J. A.|last2= Tamarkin|first2= J. D.|title=The Problem of Moments|publisher =American mathematical society|publication-place= New York|year= 1943|isbn=0-8218-1501-6}}.
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| {{DEFAULTSORT:Hamburger Moment Problem}}
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| [[Category:Probability theory]]
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| [[Category:Measure theory]]
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| [[Category:Functional analysis]]
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| [[Category:Theory of probability distributions]]
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| [[Category:Mathematical problems]]
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