Strong Hamburger Moment Problem Research Articles

Analytic functions associated with strong Hamburger moment problems

AbstractWe complete the investigation of growth properties of analytic functions connected with the Nevanlinna parametrization of the solutions of an indeterminate strong Hamburger moment problem.

Nevanlinna Matrices for the Strong Hamburger Moment Problem

Let {c n }∞n=-∞ be a doubly infinite sequence of real numbers. The strong Hamburger moment problem consists of finding positive measures σ on R such that c n = ∫∞ - ∞ t n dσ(t) for n = 0, ±1, ±2,.... The problem is indeterminate if there is more than one solution. For an indeterminate problem there is a one-to-one correspondence between all Pick functions φ and all solutions σ of the moment problem, expressed by ∫∞ - ∞t) -1 dσ(t) = [α(z)φ(z) - γ(z)][β(z)φ(z) - δ(z)] -1 . The functions α,β,γ,δ are holomorphic in the complex plane outside the origin. The purpose of this paper is to study growth properties of these functions a, β,γ,δ, analogous to properties of corresponding entire functions connected with the classical Hamburger moment problem.

Laurent–Jacobi Matrices and the Strong Hamburger Moment Problem

Let ℒ be the linear space of the Laurent polynomials and suppose that <⋅, ⋅<ℒ is a positive-definite Hermitian inner product in ℒ with the additional property that \( \left\langle {f\left( z \right),g\left( z \right)} \right\rangle {/Mathcal L} = \left\langle {f\left( z \right)\overline {g\left( z \right)} ,1} \right\rangle {/Mathcal L} \). Starting from the five-term recurrence relation for orthogonal Laurent polynomials with respect to <⋅, ⋅<ℒ, we derive Laurent–Jacobi matrices \({/Mathcal J} \) and \({/Mathcal K} \) for the multiplication operator and its inverse in ℒ. These matrices are real and symmetric, and \({/Mathcal J} \) generates a symmetric operator in the Hilbert space l2 with natural basis { en }n = 0∞. We show that this operator has deficiency indices (0, 0) or (1, 1) and that every self-adjoint extension A in l2 has simple spectrum with generating vector e0. Let E be the spectral measure of A. Then the measure μe0 given by μe0(Ω) =<E(Ω) e0, e0< for all Borel sets Ω in \({/Mathcal R} \), satisfies \(\int_{/Mathcal R} {f\overline g {d\mu }_{e_0 } } = \left\langle {f,g} \right\rangle {/Mathcal L} \) forf,gℒ. In this way, we obtain a solution μe0 of the Strong Hamburger Moment Problem (SHMP) for which ℒ is dense in L2(μe0). Some results concerning the relation between the deficiency indices andthe set of all solutions of the SHMP are established. Finally, we give an analogue of a theorem by M. H. Stone which tells us which self-adjoint operators are generatedby a Laurent–Jacobi matrix with deficiency indices (0, 0).

Nevanlinna matrices for the strong Stieltjes moment problem

Let ∗c n∗ n=−∞ ∞ be a doubly infinite sequence of real numbers. A solution of the strong Hamburger moment problem is a positive measure σ on (−∞, ∞) such that c n= ∫ −∞ ∞ u n dδ(u) for n = 0, ±1, ±2,…. A solution of the strong Stieltjes moment problem is a positive measure σ on [0, ∞) such that c n= ∫ 0 ∞ u n dδ(u) for n = 0, + 1 ̄ , + 2 ̄ ,… . A moment problem is indeterminate if there exists more than one solution. With an indeterminate strong Hamburger moment problem there is associated a Nevanlinna matrix of functions α, β, γ, δ holomorphic in ofC − ∗0∗. These functions have growth properties partly similar to properties of analogous entire functions associated with an indeterminate classical Hamburger moment problem. In this paper we obtain a stronger growth result in the case where the strong Stieltjes moment problem is solvable.

Solutions of the Strong Hamburger Moment Problem

The strong Hamburger moment problem for a bi-infinite sequence { c n : n=0, ±1, ±2,…} can be described as follows: (1) Find conditions for the existence of a (positive) measure μ on (−∞, ∞) such that c n =∫ −∞ ∞ t n dμ( t) for all n. (2) When there is a solution, find conditions for uniqueness of the solution. (3) When there is more than one solution, describe the family of all solutions. In this paper a theory concerning question (3) is developed. In particular, an analog to the Nevanlinna parametrization describing the solutions of the classical Hamburger moment problem is given.

Extremal solutions of the strong Stieltjes moment problem

A solution of the strong Stieltjes moment problem for the sequence { C n : n = o,±1, ±2,…} is a finite positive measure μ on [0, ∞) such that c n= ∫ 0 ∞ t ndμ(t) for all n, while a solution of the strong Hamburger moment problem for the same sequence is a finite positive measure μ on (−∞, ∞) such that c n= ∫ −∞ ∞ t ndμ(t) for all n. When the Hamburger problem is indeterminate, there exists a one-to-one correspondence between all solutions μ and all Nevanlinna functions ϕ, the constant ∞ included. The correspondence is given by Fμ(z)=− ∝(z)ϕ(z)−γ(z) β(z)ϕ(z)−δ(z) , where α, β, γ, δ are certain functions holomorphic in C − {0} . The extremal solutions are the solutions μ t corresponding to the constant functions ϕ( z) ≡ t, t ϵ R ∪ {∞} . The accumulation points of the (isolated) set Z t of zeros of β( z) t − δ( z) consists of 0 and ∞. The support of the extremal solution μ t is the set Z t ∪ {0}. There exists an interval [ t (0), t (∞)] such that the extremal solutions of the Stieltjes problem are exactly those μ t for which t ∈ [ t (0), t (∞)]. The measures μ t (0) and μ t (∞) are natural solutions, and the only ones. If ξ k ( n) denote the zeros of the orthogonal Laurent polynomials determined by { C n } ordered by size, then { ξ k ( n) } tends to 0 and { ξ n− k ( n) } tends to ∞ for arbitrary constant k when n tends to ∞.

Solution of the strong hamburger moment problem by Laurent continued fractions

The strong Hamburger moment problem (SHMP) is the following: Let { c n : n = 0, ±1, ±2,…} be a given double sequence of real numbers. Find a real-valued, bounded, nondecreasing function on the real line with an infinite number of points of increase, such that ∫ −∞ ∞ t n dψ(t) = (−1) nc n , n = 0, ±1, ±2,…. In this paper the questions of existence of solutions and uniqueness of solutions are treated and solved in terms of continued fractions. The continued fractions in question are contractive Laurent fractions. Their relationship with the series ∑ n=0 ∞ c nz −n and ∑ n=1 ∞ −c −nz n is utilized to obtain the existen uniqueness results.

The strong Hamburger moment problem and self-adjoint operators in Hilbert space

The strong Hamburger moment problem is solved using spectral theory of (unbounded) self-adjoint linear operators in Hilbert space. By means of orthogonal projections of finite dimensional range certain two-point Padé approximants to the moment generating function are obtained.

The strong Hamburger moment problem and self-adjoint operators in Hilbert space

The strong Hamburger moment problem is solved using spectral theory of (unbounded) self-adjoint linear operators in Hilbert space. By means of orthogonal projections of finite dimensional range certain two-point Padé approximants to the moment generating function are obtained.

Unique solvability of the strong Hamburger moment problem

AbstractMethods from the theory of orthogonal polynomials are extended to L-polynomials . By this means the authors and W. B. Jones (J. Math. Anal. Appl. 98 (1984), 528–554) solved the strong Hamburger moment problem, that is, given a double sequence , to find a distribution function ψ(t), non-decreasing, with an infinitenumber of points of increase and bounded on −∞ &lt; t &lt; ∞, such that for all integers . In this article further menthods such as analogues of the Lioville-Ostrogradski formula and of the Christoffel-Darboux formula are developed to investigated When the moment porblem has a unique solution. This will be the case if and only if a sequence of nested disks associated with the sequence has only a point as its intersection (the so called limit point case).

Orthogonal Laurent polynomials and the strong Hamburger moment problem

This paper is concerned with the strong Hamburger moment problem (SHMP): For a given double sequence of real numbers C = { c n } ∞ −∞, does there exist a real-valued, bounded, non-decreasing function ψ on (−∞, ∞) with infinitely many points of increase such that for every integer n, c n = ∝ ∞ −∞ (− t) n dψ( t)? Necessary and sufficient conditions for the existence of such a function ψ are given in terms of the positivity of certain Hankel determinants associated with C. Our approach is made through the study of orthogonal (and quasi-orthogonal) Laurent polynomials (referred to here as L-polynomials) and closely related Gaussian-type quadrature formulas. In the proof of sufficiency an inner product for L-polynomials is defined in terms of the given double sequence C. Since orthogonal L-polynomials are believed to be of interest in themselves, some examples of specific systems are considered.