Math 2.0

Integral transforms, operational calculus Moment problems Theorem

Nevanlinna parametrization of solutions

Builds on Hamburger moment problemIndeterminate moment problemBorel measureEntire functionHolomorphic functionUpper half-plane

Statement

Let (mn)n≥0(m_n)_{n \ge 0} be a sequence of real numbers for which the Hamburger moment problem is indeterminate. Let VV be the set of positive Borel measures μ\mu on ℝ\R with ∫ℝxn dμ(x)=mn\int_\R x^n \, d\mu(x) = m_n for all n≥0n \ge 0. A Pick function is a holomorphic function φ\varphi on the upper half-plane ℍ={z∈ℂ:Im⁡z>0}\mathbb{H} = \set{z \in \C : \operatorname{Im} z > 0} with Im⁡φ(z)≥0\operatorname{Im} \varphi(z) \ge 0 for all z∈ℍz \in \mathbb{H}. Let 𝒫\mathcal{P} be the set of Pick functions together with the constant ∞\infty. Then there are four entire functions A,B,C,DA, B, C, D, real on ℝ\R and determined by (mn)(m_n), with A(z)D(z)−B(z)C(z)=1A(z) D(z) - B(z) C(z) = 1 for all z∈ℂz \in \C, such that the following holds. For each φ∈𝒫\varphi \in \mathcal{P} there is exactly one μφ∈V\mu_\varphi \in V with ∫ℝdμφ(x)x−z=−A(z)φ(z)−C(z)B(z)φ(z)−D(z)for all z∈ℍ,\int_\R \frac{d\mu_\varphi(x)}{x - z} = -\frac{A(z) \varphi(z) - C(z)}{B(z) \varphi(z) - D(z)} \quad \text{for all } z \in \mathbb{H}, where for φ=∞\varphi = \infty the right side means −A(z)/B(z)-A(z)/B(z). The map φ↦μφ\varphi \mapsto \mu_\varphi is a bijection from 𝒫\mathcal{P} onto VV.

Remarks

The functions A,B,C,DA, B, C, D are built from the orthonormal polynomials pnp_n of the moment sequence and the polynomials qnq_n of the second kind. For example, B(z)=−1+z∑k≥0qk(0)pk(z)B(z) = -1 + z \sum_{k \ge 0} q_k(0) p_k(z) and D(z)=z∑k≥0pk(0)pk(z)D(z) = z \sum_{k \ge 0} p_k(0) p_k(z), and these series converge locally uniformly exactly because the problem is indeterminate. The left side is the Stieltjes transform of μφ\mu_\varphi, which determines μφ\mu_\varphi by the Stieltjes inversion formula. The parameters φ=t\varphi = t with t∈ℝ∪{∞}t \in \R \cup \set{\infty} give the N-extremal solutions, for which the polynomials are dense in L2(μt)L^2(\mu_t). Each μt\mu_t is a discrete measure supported on the zeros of Bt−DB t - D, or on the zeros of BB when t=∞t = \infty. The hypothesis of indeterminacy is needed: in the determinate case VV has only one element, so there is nothing to parametrize. Sign conventions for A,B,C,DA, B, C, D differ between books.

Going deeper

For a student

The parametrization reduces the study of an indeterminate moment problem to the study of four entire functions, the Nevanlinna matrix. These entire functions have order at most one and minimal exponential type. The parameters t∈ℝ∪{∞}t \in \R \cup \set{\infty} give solutions that are spectral measures of self-adjoint extensions of the Jacobi operator of the moment sequence. The Pick functions are the functions of the Herglotz representation theorem transplanted from the disc to the upper half-plane, so the parameter set has a concrete integral form.

Reading path

To understand this page from the ground up, read these 23 pages in order. The foundations come first.

Show the path as a graph
Borel measure, concept, no evidence levelBorel measureconceptComplex numbers, object, no evidence levelComplex numbersobjectEntire function, concept, no evidence levelEntire functionconceptGeneral linear group, object, no evidence levelGeneral linear groupobjectHolomorphic function, concept, no evidence levelHolomorphic functionconceptIdentity matrix, object, no evidence levelIdentity matrixobjectIndeterminate moment problem, concept, no evidence levelIndeterminate moment …conceptMatrix, definition, no evidence levelMatrixdefinitionMeasure, concept, no evidence levelMeasureconceptMoment, concept, no evidence levelMomentconceptMöbius transformation, concept, no evidence levelMöbius transformationconceptPositive semidefinite matrix, concept, no evidence levelPositive semidefinite…conceptSequence, concept, no evidence levelSequenceconceptVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionHamburger moment problem, problem, no evidence levelHamburger moment prob…problemLinear map, definition, no evidence levelLinear mapdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionSpecial linear group, object, no evidence levelSpecial linear groupobjectUpper half-plane, object, no evidence levelUpper half-planeobjectNevanlinna parametrization of solutions, statement, no evidence levelNevanlinna parametriz…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Borel measure concept
  2. Complex numbers object
  3. Entire function concept
  4. General linear group object
  5. Holomorphic function concept
  6. Identity matrix object
  7. Indeterminate moment problem concept
  8. Matrix definition
  9. Measure concept
  10. Moment concept
  11. Möbius transformation concept
  12. Positive semidefinite matrix concept
  13. Sequence concept
  14. Vector space definition
  15. Square matrix definition
  16. Hamburger moment problem problem
  17. Linear map definition
  18. Multilinear map definition
  19. Alternating multilinear map definition
  20. Determinant definition
  21. Special linear group object
  22. Upper half-plane object
  23. Nevanlinna parametrization of solutions statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Nevanlinna_function
About this page
TagYRBT3ZHSV3
URIm2://YRBT3ZHSV3
Kindstatement, theorem
WritingW0 stub
MSC44A60, 30E05
StatusThis copy of the site has no build data. Run m2 status YRBT3ZHSV3.
AboutHamburger moment problem, Indeterminate moment problem
UsesHamburger moment problem, Indeterminate moment problem, Borel measure, Entire function, Holomorphic function, Upper half-plane

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