Mean value theorem (divided differences) - Revision history

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WP:CHECKWIKI error fix for #64. Do general fixes if a problem exists. - using AWB (9991)

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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		Greetings. Let me start by telling you the writer's name - Phebe. For a whilst I've been in South Dakota and my mothers and fathers reside nearby. The thing she adores most is physique developing and now she is attempting to make cash with it. Managing individuals is what I do and the wage has been truly satisfying.<br><br>Feel free to surf to my site: [http://www.videoworld.com/blog/211112 at home std testing]

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

	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''.

	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).

	~~== Characterization ==~~

	~~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~~

	~~:<math>~~
	~~A =~~
	~~\left(\begin{matrix}~~
	~~m_0 & m_1 & m_2 & \cdots \\~~
	~~m_1 & m_2 & m_3 & \cdots \\~~
	~~m_2 & m_3 & m_4 & \cdots \\~~
	~~\vdots & \vdots & \vdots & \ddots \\~~
	~~\end{matrix}\right)</math>~~

	~~is [[positive definite kernel\|positive definite]], i.e.,~~

	~~:<math> \sum_{j,k\ge0}m_{j+k}c_j\bar c_k\ge0~~
	~~</math>~~

	~~for an arbitrary sequence {''c<sub>j</sub>''}<sub>''j'' ≥ 0</sub> of complex numbers with finite support (i.e.~~
	~~''c<sub>j</sub>'' = 0 except for finitely many values of ''j'').~~

	~~The "only if" part of the claims can be verified by a direct calculation.~~

	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]]

	~~:<math>(\mathcal{H}, \langle, \; \rangle)</math>~~

	~~whose typical element is an equivalence class denoted by [''f'']~~.

	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~~

	~~:<math>\langle [e_{n+1}], [e_m] \rangle = A_{m,n+1} = m_{m+n+1} = \langle [e_n], [e_{m+1}]\rangle.</math>~~

	~~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]].~~

	~~On the other hand, the desired expression~~

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

	~~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.

	~~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~~

	~~:<math> \bar{T} \, </math>~~

	~~be one of them and ''μ'' be its spectral measure. So~~

	~~:<math>\langle \bar{T}^n [1], [1] \rangle = \int x^n d \mu(x).</math>~~

	~~On the other hand,~~

	~~:<math> \langle \bar{T}^n [1], [1] \rangle = \langle T^n [e_0], [e_0] \rangle = m_n. \, </math>~~

	~~=== Uniqueness of solutions ===~~

	The ~~solutions form a convex set, so the problem has either infinitely many solutions or a unique solution.~~

	~~Consider the (''n'' + 1)×(''n'' + 1) [[Hankel matrix]]~~

	~~:<math>\Delta_n=\left[\begin{matrix}~~
	~~m_0 & m_1 & m_2 & \cdots & m_{n} \\~~
	~~m_1 & m_2 & m_3 & \cdots & m_{n+1} \\~~
	~~m_2 & m_3 & m_4 & \cdots & m_{n+2} \\~~
	~~\vdots & \vdots & \vdots & \ddots & \vdots \\~~
	~~m_{n} & m_{n+1} & m_{n+2} & \cdots & m_{2n}~~
	~~\end{matrix}\right].</math>~~

	~~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~~

	~~:<math>(\mathcal{H}, \langle, \; \rangle)</math>~~

	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~~.

	~~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]].~~

	~~There are examples where the solution is not unique.~~

	~~== Further results ==~~
	~~{{Expand section\|date=June 2008}}~~
	~~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~~

	:~~<math> \bar{T} \, <~~/~~math>~~

	~~has a tridiagonal ''Jacobi matrix representation''~~. ~~This in turn leads to a ''tridiagonal model'' of positive Hankel kernels~~.

	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.

	~~The cumulative distribution function and the probability density function can often be found by applying the inverse [[Laplace transform]] to the moment generating function~~
	~~:<math>m(t)=\sum_{n=0}m_n\frac{t^n}{n!},<~~/~~math>~~
	~~provided that this function converges.~~

	~~== References ==~~
	*{{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}}
	* {{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}}.

	~~{{DEFAULTSORT:Hamburger Moment Problem}}~~
	~~[[Category:Probability theory]]~~
	~~[[Category:Measure theory]]~~
	~~[[Category:Functional analysis]]~~
	~~[[Category:Theory of probability distributions]]~~
	~~[[Category:Mathematical problems]~~]

en>Snotbot: Fixing section headings (task 5)

2012-02-16T17:20:10Z

Fixing section headings (task 5)

New page

In [[mathematics]], the '''Hamburger [[moment problem]]''', named after [[Hans Hamburger|Hans Ludwig Hamburger]], is formulated as follows: given a sequence { ''mn'' : ''n'' = 1, 2, 3, ... }, does there exist a positive [[Borel measure]] ''μ'' on the real line such that

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

In other words, an affirmative answer to the problem means that { ''mn'' : ''n'' = 0, 1, 2, ... } is the sequence of [[moment (mathematics)|moments]] of some positive Borel measure ''μ''.

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).

== Characterization ==

The Hamburger moment problem is solvable (that is, {''mn''} is a sequence of [[moment (mathematics)|moments]]) if and only if the corresponding Hankel kernel on the nonnegative integers

:<math>
A =
\left(\begin{matrix}
m_0 & m_1 & m_2 & \cdots \\
m_1 & m_2 & m_3 & \cdots \\
m_2 & m_3 & m_4 & \cdots \\
\vdots & \vdots & \vdots & \ddots \\
\end{matrix}\right)</math>

is [[positive definite kernel|positive definite]], i.e.,

:<math> \sum_{j,k\ge0}m_{j+k}c_j\bar c_k\ge0
</math>

for an arbitrary sequence {''cj''}''j'' ≥ 0 of complex numbers with finite support (i.e.
''cj'' = 0 except for finitely many values of ''j'').

The "only if" part of the claims can be verified by a direct calculation.

We sketch an argument for the converse. Let '''Z'''+ be the nonnegative integers and ''F''0('''Z'''+) 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]]

:<math>(\mathcal{H}, \langle, \; \rangle)</math>

whose typical element is an equivalence class denoted by [''f''].

Let ''en'' be the element in ''F''0('''Z'''+) defined by ''en''(''m'') = [[Kronecker delta|''δnm'']]. One notices that

:<math>\langle [e_{n+1}], [e_m] \rangle = A_{m,n+1} = m_{m+n+1} = \langle [e_n], [e_{m+1}]\rangle.</math>

Therefore the [[shift operator|"shift" operator]] ''T'' on <math>\mathcal{H}</math>, with ''T''[''en''] = [''e''''n'' + 1], is [[symmetric operator|symmetric]].

On the other hand, the desired expression

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

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.

A function model is given by the natural isomorphism from ''F''0('''Z'''+) to the family of polynomials, in one single real variable and complex coefficients: for ''n'' ≥ 0, identify ''en'' with ''xn''. 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

:<math> \bar{T} \, </math>

be one of them and ''μ'' be its spectral measure. So

:<math>\langle \bar{T}^n [1], [1] \rangle = \int x^n d \mu(x).</math>

On the other hand,

:<math> \langle \bar{T}^n [1], [1] \rangle = \langle T^n [e_0], [e_0] \rangle = m_n. \, </math>

=== Uniqueness of solutions ===

The solutions form a convex set, so the problem has either infinitely many solutions or a unique solution.

Consider the (''n'' + 1)×(''n'' + 1) [[Hankel matrix]]

:<math>\Delta_n=\left[\begin{matrix}
m_0 & m_1 & m_2 & \cdots & m_{n} \\
m_1 & m_2 & m_3 & \cdots & m_{n+1} \\
m_2 & m_3 & m_4 & \cdots & m_{n+2} \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
m_{n} & m_{n+1} & m_{n+2} & \cdots & m_{2n}
\end{matrix}\right].</math>

Positivity of ''A'' means that for each ''n'', det(Δ''n'') ≥ 0. If det(Δ''n'') = 0, for some ''n'', then

:<math>(\mathcal{H}, \langle, \; \rangle)</math>

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.

More generally, the solution is unique if there are constants ''C'' and ''D'' such that for all ''n'', |m''n''|≤ ''CD''''n''''n''! {{harv|Reed|Simon|1975|p=205}}. This follows from the more general [[Carleman's condition]].

There are examples where the solution is not unique.

== Further results ==
{{Expand section|date=June 2008}}
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

:<math> \bar{T} \, </math>

has a tridiagonal ''Jacobi matrix representation''. This in turn leads to a ''tridiagonal model'' of positive Hankel kernels.

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.

The cumulative distribution function and the probability density function can often be found by applying the inverse [[Laplace transform]] to the moment generating function
:<math>m(t)=\sum_{n=0}m_n\frac{t^n}{n!},</math>
provided that this function converges.

== References ==
*{{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}}
* {{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}}.

{{DEFAULTSORT:Hamburger Moment Problem}}
[[Category:Probability theory]]
[[Category:Measure theory]]
[[Category:Functional analysis]]
[[Category:Theory of probability distributions]]
[[Category:Mathematical problems]]