Hamburger Moment Problem Research Articles

Positivstellensatz and flat functionals on path ∗-algebras

We consider the class of non-commutative ∗-algebras which are path algebras of doubles of quivers with the natural involutions. We study the problem of extending positive truncated functionals on such ∗-algebras. An analog of the solution of the truncated Hamburger moment problem (Curto and Fialkow, 1991 [Fia91]) for path ∗-algebras is presented and non-commutative positivstellensatz is proved. We also present an analog of the flat extension theorem of Curto and Fialkow for this class of algebras.

Stieltjes moment problem and fractional moments

Stieltjes moment problem is considered and a solution, consisting of the use of fractional moments, is proposed. More precisely, a determinate Stieltjes moment problem, whose corresponding Hamburger moment problem is determinate too, is investigated in the setup of Maximum Entropy. Condition number in entropy calculation is provided endowing both Stieltjes moment problem existence conditions and Hamburger moment problem determinacy conditions by a geometric meaning. Then the resorting to fractional moments is considered; numerical aspects are investigated and a stable algorithm for calculating fractional moments from integer moments is proposed.

On Hankel Nonnegative Definite Sequences, the Canonical Hankel Parametrization, and Orthogonal Matrix Polynomials

This paper continues recent investigations started in Dyukarev et al. (Complex anal oper theory 3(4):759–834, 2009) into the structure of the set \({\mathcal{H}_{q,2n}^{\ge}}\) of all Hankel nonnegative definite sequences, \({(s_{j})_{j=0}^{2n}}\), of complex q × q matrices and its important subclasses \({\mathcal{H}_{q,2n}^{\ge,{\rm e}}}\) and \({\mathcal{H}_{q,2n}^>}\) of all Hankel nonnegative definite extendable sequences and of all Hankel positive definite sequences, respectively. These classes of sequences arise quite naturally in the framework of matrix versions of the truncated Hamburger moment problem. In Dyukarev et al. (Complex anal oper theory 3(4):759–834, 2009) a canonical Hankel parametrization \({[(C_k)_{k=1}^n, (D_k)_{k=0}^n]}\) consisting of two sequences of complex q × q matrices was associated with an arbitrary sequence \({(s_{j})_{j=0}^{2n}}\) of complex q × q matrices. The sequences belonging to each of the classes \({\mathcal{H}_{q,2n}^{\ge}, \mathcal{H}_{q,2n}^{\ge,{\rm e}}}\), and \({\mathcal{H}_{q,2n}^>}\) were characterized in terms of their canonical Hankel parametrization (see, Dyukarev et al. in Complex anal oper theory 3(4):759–834, 2009; Proposition 2.30). In this paper, we will study further aspects of the canonical Hankel parametrization. Using the canonical Hankel parametrization \({[(C_k)_{k=1}^n, (D_k)_{k=0}^n]}\) of a sequence \({(s_{j})_{j=0}^{2n} \in \mathcal{H}_{q,2n}^{\ge}}\), we give a recursive construction of a monic right (resp. left) orthogonal system of matrix polynomials with respect to \({(s_{j})_{j=0}^{2n}}\) (see Theorem 5.5). The matrices \({[(C_k)_{k=1}^n, (D_k)_{k=0}^n]}\) will be expressed in terms of an arbitrary monic right (resp. left) orthogonal system with respect to \({(s_{j})_{j=0}^{2n}}\) (see Theorem 5.11). This result will be reformulated in terms of nonnegative Hermitian Borel measures on \({\mathbb{R}}\). In this way, integral representations for the matrices \({[(C_k)_{k=1}^n, (D_k)_{k=0}^n]}\) will be obtained (see Theorem 6.9). Starting from the monic orthogonal polynomials with respect to some classical probability distributions on \({\mathbb{R}}\), Theorem 6.9 is used to compute the canonical Hankel parametrization of their moment sequences. Moreover, we discuss important number sequences from enumerative combinatorics using the canonical Hankel parametrization.

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.

On Distinguished Solutions of Truncated Matricial Hamburger Moment Problems

We study two slightly different versions of the truncated matri- cial Hamburger moment problem. A central topic is the construction and investigation of distinguished solutions of both moment problems under con- sideration. These solutions turn out to be nonnegative Hermitian q × q Borel measures on the real axis which are concentrated on a finite number of points. These points and the corresponding masses will be explicitly described in terms of the given data. Furthermore, we investigate a particular class of se- quences (sj) ∞=0 of complex q × q matrices for which the corresponding infinite matricial Hamburger moment problem has a unique solution. Our approach is mainly algebraic. It is based on the use of particular matrix polynomials con- structed from a nonnegative Hermitian block Hankel matrix. These matrix polynomials are immediate generalizations of the monic orthogonal matrix polynomials associated with a positive Hermitian block Hankel matrix. We generalize a classical theorem due to Kronecker on infinite Hankel matrices of finite rank to block Hankel matrices and discuss its consequences for the nonnegative Hermitian case.

Generalized Friedland's theorem for [formula omitted]-semigroups

Friedland's characterization of bounded normal operators is shown to hold for infinitesimal generators of C 0 -semigroups. New criteria for normality of bounded operators are furnished in terms of Hamburger moment problem. All this is achieved with the help of the celebrated Ando's theorem on paranormal operators.

On L. Schwartz's Boundedness Condition for Kernels

In previous works we analysed conditions for linearization of Hermitian kernels. The conditions on the kernel turned out to be of a type considered previously by L. Schwartz in the related matter of characterizing the real linear space generated by positive definite kernels. The aim of the present note is to find more concrete expressions of the Schwartz type conditions: in the Hamburger moment problem for Hankel type kernels on the free semigroup, in dilation theory (Stinespring type dilations and Haagerup decomposability), as well as in multi-variable holomorphy.

The continuous symmetric Hahn polynomials found in Ramanujan's lost notebook

Askey and Wilson found F 2 3 Hahn polynomials which are orthogonal with respect to a positive absolutely continuous weight function. More than a half century earlier, Ramanujan recorded the Stieltjes transform of this weight function in terms of a continued fraction in his lost notebook. We provide two different proofs for this integration. One applies theories of the Hamburger moment problem. The other uses elementary integration techniques and a couple of transformation formulas for hypergeometric functions.

Reconstruction of thermally symmetrized quantum autocorrelation functions from imaginary-time data

In this paper, I propose a technique for recovering quantum dynamical information from imaginary-time data via the resolution of a one-dimensional Hamburger moment problem. It is shown that the quantum autocorrelation functions are uniquely determined by and can be reconstructed from their sequence of derivatives at origin. A general class of reconstruction algorithms is then identified, according to Theorem 3. The technique is advocated as especially effective for a certain class of quantum problems in continuum space, for which only a few moments are necessary. For such problems, it is argued that the derivatives at origin can be evaluated by Monte Carlo simulations via estimators of finite variances in the limit of an infinite number of path variables. Finally, a maximum entropy inversion algorithm for the Hamburger moment problem is utilized to compute the quantum rate of reaction for a one-dimensional symmetric Eckart barrier.

On the determinacy of complex Jacobi matrices

The aim of this paper is to explore to which extend the theory of the Hamburger moment problem for real Jacobi matrices generalizes to the case of complex Jacobi matrices. In particular, we characterize the indeterminacy in terms of uniqueness of closed extensions of Jacobi matrices, and discuss the link to the growth of the smallest singular values of the underlying Hankel matrices. As a byproduct, we give a positive answer to the open question whether determinacy is preserved under bounded perturbations.

Stieltjes Truncated Moment Problem with Point Constraints

AbstractThe truncated Hamburger and Stieltjes moment problems [1] were solved elsewhere (see [3] and references therein) using the methods of the extension theory of Hermitian operators and, in particular, the results on extensions of nonnegative operators and matrices. Moreover, applying methods of the extension theory we found a purely algebraic algorithm for the solution of both problems. Here we solve the corresponding moment problems with point constraints.

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.

A uniqueness criterion in the multivariate moment problem

A determinacy criterion for the multivariate Hamburger moment problem is derived from a recent existence by extension result, [10].

Maximum entropy solutions and moment problem in unbounded domains

The classical Stieltjes and Hamburger moment problems in the maximum entropy approach have been reconsidered. Incorrect conditions of existence, previously appeared in literature, have been reviewed and improved.

Discrete Sturm–Liouville problems, Jacobi matrices and Lagrange interpolation series

The close relationship between discrete Sturm–Liouville problems belonging to the so-called limit-circle case, the indeterminate Hamburger moment problem and the search of self-adjoint extensions of the associated semi-infinite Jacobi matrix is well known. In this paper, all these important topics are also related with associated sampling expansions involving analytic Lagrange-type interpolation series.

On the moment determinacy of the distributions of compound geometric sums

We deal with compound geometric sums of independent positive random variables and study the moment problem for the distributions of such sums (the Stieltjes moment problem). We find conditions under which the distributions are uniquely determined by their moments. We also treat related topics, including the Hamburger moment problem involving random variables on the whole real line. Some conjectures are outlined.

On the moment determinacy of the distributions of compound geometric sums

We deal with compound geometric sums of independent positive random variables and study the moment problem for the distributions of such sums (the Stieltjes moment problem). We find conditions under which the distributions are uniquely determined by their moments. We also treat related topics, including the Hamburger moment problem involving random variables on the whole real line. Some conjectures are outlined.

Codimension of polynomial subspace in $L_2(\mathbb {R},d\mu )$ for discrete indeterminate measure $\mu $

A calculation formula is established for the codimension of the polynomial subspace in L 2 (R,dμ) with discrete indeterminate measure μ. We clarify how much the masspoint of the n-canonical solution of an indeterminate Hamburger moment problem differs from the masspoint of the corresponding N-extremal solution at a given point of the real axis.

Matrix valued interpolation and truncated Hamburger moment problems

A solvability condition for matrix valued directional single-node interpolation problems of Loewner type is established, in terms of properties of Pick kernel. As a consequence, a solvability condition for matrix valued directional truncated Hamburger moment problems is obtained.

A Generalization of Hankel Operators

We introduce a class of operators, called λ-Hankel operators, as those that satisfy the operator equation S*X−XS=λX, where S is the unilateral forward shift and λ is a complex number. We investigate some of the properties of λ-Hankel operators and show that much of their behaviour is similar to that of the classical Hankel operators (0-Hankel operators). In particular, we show that positivity of λ-Hankel operators is equivalent to a generalized Hamburger moment problem. We show that certain linear spaces of noninvertible operators have the property that every compact subset of the complex plane containing zero is the spectrum of an operator in the space. This theorem generalizes a known result for Hankel operators and applies to λ-Hankel operators for certain λ. We also study some other operator equations involving S.