Stieltjes Moment Problem Research Articles

A truncated indefinite Stieltjes moment problem

A truncated indefinite Stieltjes moment problem in the class $$ {\mathrm{N}}_{\kappa}^k $$ of generalized Stieltjes functions is studied. The set of solutions of the Stieltjes moment problem is described by the Schur stepby-step algorithm, which is based on the expansion of the solutions in a generalized Stieltjes continued fraction. The resolvent matrix is represented in terms of generalized Stieltjes polynomials. A factorization formula for the resolvent matrix is found.

Poly-element equations reducible to moment problem for entire functions of class A

We consider poly-element linear functional equations in the class of analytic functions in the complex plane with cuts along certain segments of imaginary axis. The obtained results are applied for study of the Stieltjes moment problem for entire functions of exponential type in the class A.

On Resolvent Matrix, Dyukarev–Stieltjes Parameters and Orthogonal Matrix Polynomials via $$[0, \infty )$$ [ 0 , ∞ ) -Stieltjes Transformed Sequences

By using Schur transformed sequences and Dyukarev–Stieltjes parameters we obtain a new representation of the resolvent matrix corresponding to the truncated matricial Stieltjes moment problem. Explicit relations between orthogonal matrix polynomials and matrix polynomials of the second kind constructed from consecutive Schur transformed sequences are obtained. Additionally, a non-negative Hermitian measure for which the matrix polynomials of the second kind are the orthogonal matrix polynomials is found.

Symmetric moment problems and a conjecture of Valent

In 1998 Valent made conjectures about the order and type of certain indeterminate Stieltjes moment problems associated with birth and death processes which have polynomial birth and death rates of degree . Romanov recently proved that the order is as conjectured. We prove that the type with respect to the order is related to certain multi-zeta values and that this type belongs to the interval which also contains the conjectured value. This proves that the conjecture about type is asymptotically correct as . The main idea is to obtain estimates for order and type of symmetric indeterminate Hamburger moment problems when the orthonormal polynomials and those of the second kind satisfy and , where may be different, and and are positive constants. In this case the order of the moment problem is majorized by the harmonic mean of and . Here means that . This also leads to a new proof of Romanov's Theorem that the order is . Bibliography: 19 titles.

On a simultaneous approach to the even and odd truncated matricial Stieltjes moment problem II: An α-Schur–Stieltjes-type algorithm for sequences of holomorphic matrix-valued functions

The main goal of this paper is to achieve a simultaneous treatment of the even and odd truncated matricial Stieltjes moment problems in the most general case. These results are generalizations of results of Chen and Hu [5,19] which considered the particular case α=0. Our approach is based on Schur analysis methods. More precisely, we use two interrelated versions of Schur-type algorithms, namely an algebraic one and a function-theoretic one. The algebraic version was worked out in a former paper of the authors. It is an algorithm which is applied to finite or infinite sequences of complex matrices. The construction and investigation of the function-theoretic version of our Schur-type algorithm is a central theme of this paper. This algorithm will be applied to relevant subclasses of holomorphic matrix-valued functions of the Stieltjes class. Using recent results on the holomorphicity of the Moore–Penrose inverse of matrix-valued Stieltjes functions, we obtain a complete description of the solution set of the moment problem under consideration in the most general situation.

On a simultaneous approach to the even and odd truncated matricial Stieltjes moment problem I: An α-Schur–Stieltjes-type algorithm for sequences of complex matrices

The characterization of the solvability of matrix versions of truncated Stieltjes-type moment problems led to the class of α-Stieltjes non-negative definite sequences of complex q×q matrices. In [22], a parametrization of this class was introduced, the so-called α-Stieltjes parametrization. The main topic of this first part of the paper is the construction of a Schur-type algorithm which produces exactly the α-Stieltjes parametrization.

Entropy convergence of finite moment approximations in Hamburger and Stieltjes problems

The Hamburger and Stieltjes moment problem and the sequence of maximum entropy (MaxEnt) approximates with given moments are considered. MaxEnt approximate converges in entropy to the unknown distribution when the Hamburger (or Stieltjes) moment problem is determinate and the underlying distribution has finite entropy.

Shell Polynomials and Dual Birth-Death Processes

This paper aims to clarify certain aspects of the relations between birth-death processes, measures solving a Stieltjes moment problem, and sets of parameters defining polynomial sequences that are orthogonal with respect to such a measure. Besides giving an overview of the basic features of these relations, revealed to a large extent by Karlin and McGregor, we investigate a duality concept for birth-death processes introduced by Karlin and McGregor and its interpretation in the context of shell polynomials and the corresponding orthogonal polynomials. This interpretation leads to increased insight in duality, while it suggests a modification of the concept of similarity for birth-death processes.

Moment Problem and Inverse Cauchy Problems for Heat Equation

The solution of Hamburger and Stieltjes moment problem can be thought of as the solution of a certain inverse Cauchy problem. The solution of the inverse Cauchy problem for heat equation is founded in the form of Hermite polynomial series. The author reveals, the formulas obtained by him for the solution of inverse Cauchy problem have a symmetry with respect to the formulas for corresponding direct Cauchy problem. Obtained formulas for solution of the inverse problems can serve as a basis for the solution of the moment problem.

On the roots of generalized Wills $\mu$-polynomials

We investigate the roots of a family of geometric polynomials of convex bodies associated to a given measure \mu on the non-negative real line \mathbb R_{\geq 0} , which arise from the so called Wills functional. We study its structure, showing that the set of roots in the upper half-plane is a closed convex cone, containing the non-positive real axis \mathbb R_{\leq0} , and strictly increasing in the dimension, for any measure \mu . Moreover, it is proved that the 'smallest' cone of roots of these \mu -polynomials is the one given by the Steiner polynomial, which provides, for example, additional information about the roots of \mu -polynomials when the dimension is large enough. It will also give necessary geometric conditions for a sequence \{m_i\colon i=0,1,\dots\} to be the moments of a certain measure on \mathbb R_{\geq0} , a question regarding the so called (Stieltjes) moment problem.

Criterion for complete indeterminacy of limiting interpolation problem of Stieltjes type in terms of orthonormal matrix-functions

The main goal of the paper is to study the ordered sequences of generalized interpolation problems and the limiting interpolation problem in the Stieltjes class by means of the orthonormal matrix-functions. We obtain explicit formulas for the orthonormal matrix-functions. The main result of the paper is a criterion for the complete indeterminacy of limiting interpolation problem in the Stieltjes class. General constructions are illustrated by examples of the Stieltjes moment problem and the Nevanlinna-Pick problem in the Stieltjes class. DOI: 10.3103/S1066369X15040015

On Dyukarev's resolvent matrix for a truncated Stieltjes matrix moment problem under the view of orthogonal matrix polynomials

The main result of this paper is a new representation of Yu.M. Dyukarev's resolvent matrix for the non-degenerate truncated matricial version of the classical Stieltjes moment (TMCSM) problem. This new representation is based on the use of a quadruple of q×q matrix polynomials which are characterized by certain orthogonality properties. In this way, useful new interrelations between the moment problems of Stieltjes and Hamburger which are associated with a Stieltjes positive definite sequence (sj)j=02n of complex q×q matrices are found. This includes an explicit formula connecting the resolvent matrices used by Yu.M. Dyukarev and I.V. Kovalishina, respectively. Furthermore, the coefficients in the recurrence formulas for the orthogonal polynomials with respect to a Stieltjes positive definite sequence are expressed in terms of the Dyukarev–Stieltjes parameters. Additionally, we obtain two different representations for each extremal solution of the TMCSM problem via matrix continued fraction expansions.

Hamburger and Stieltjes Moment Problems for Operators

Solutions to some operator-valued, unidimensional, Hamburger and Stieltjes moment problems in this paper are given. Necessary and sufficient conditions on some sequences of bounded operators being Hamburger, respectively, Stieltjes operator-valued moment sequences are obtained. The determinateness of the operator-valued Hamburger and Stieltjes moment sequence is studied.

Composite convergence bounds based on Chebyshev polynomials and finite precision conjugate gradient computations

The conjugate gradient method (CG) for solving linear systems of algebraic equations represents a highly nonlinear finite process. Since the original paper of Hestenes and Stiefel published in 1952, it has been linked with the Gauss-Christoffel quadrature approximation of Riemann-Stieltjes distribution functions determined by the data, i.e., with a simplified form of the Stieltjes moment problem. This link, developed further by Vorobyev, Brezinski, Golub, Meurant and others, indicates that a general description of the CG rate of convergence using an asymptotic convergence factor has principal limitations. Moreover, CG is computationally based on short recurrences. In finite precision arithmetic its behaviour is therefore affected by a possible loss of orthogonality among the computed direction vectors. Consequently, any consideration concerning the CG rate of convergence relevant to practical computations must include analysis of effects of rounding errors. Through the example of composite convergence bounds based on Chebyshev polynomials, this paper argues that the facts mentioned above should become a part of common considerations on the CG rate of convergence. It also explains that the spectrum composed of small number of well separated tight clusters of eigenvalues does not necessarily imply a fast convergence of CG or other Krylov subspace methods.

Completeness for coherent states in a magnetic–solenoid field

This paper completes our study of coherent states in the so-called magnetic–solenoid field (a collinear combination of a constant uniform magnetic field and Aharonov–Bohm solenoid field) presented in Bagrov et al (2010 J. Phys. A: Math. Theor. 43 354016, 2011 J. Phys. A: Math. Theor. 44 055301). Here, we succeeded in proving nontrivial completeness relations for non-relativistic and relativistic coherent states in such a field. In addition, we solve here the relevant Stieltjes moment problem and present a comparative analysis of our coherent states and the well-known, in the case of pure uniform magnetic field, Malkin–Man’ko coherent states.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

The truncated Stieltjes moment problem solved by using kernel density functions

In this work the problem of the approximate numerical determination of a semi-infinite supported, continuous probability density function (pdf) from a finite number of its moments is addressed. The target space is carefully defined and an approximation theorem is proved, establishing that the set of all convex superpositions of appropriate Kernel Density Functions (KDFs) is dense in this space. A solution algorithm is provided, based on the established approximate representation of the target pdf and the exploitation of some theoretical results concerning moment sequence asymptotics. The solution algorithm also permits us to recover the tail behavior of the target pdf and incorporate this information in our solution. A parsimonious formulation of the proposed solution procedure, based on a novel sequentially adaptive scheme is developed, enabling a very efficient moment data inversion. The whole methodology is fully illustrated by numerical examples.

Oscillators in the Framework of Unified (q, α, β, γ, ν)-Deformation and Their Oscillator Algebras

The aim of this paper is to review our results on the description of multiparameter deformed oscillators and their oscillator algebras. We define generalized (q; α, β, γ, ν)-deformed oscillator algebras and study their irreducible representations. The Arik–Coon oscillator with the main relation aa+ – qa+a = 1, where q >1, is embedded in this framework. We have found the connection of this oscillator with the Askey q–1-Hermite polynomials. We construct a family of generalized coherent states associated with these polynomials and give their explicit expression in terms of standard special functions. By means of the solution of the appropriate classical Stieltjes moment problem, we prove the property of (over)completeness of these states.

The matrix version of Hamburger’s theorem

Necessary and sufficient conditions for the Stieltjes moment problem to have a unique solution and for the Hamburger moment problem with the same moments to have infinitely many solutions were obtained in Hamburger’s papers on the classical moment problem. In this paper, we obtain an analog of Hamburger’s criterion for the matrix moment problem.

Supersymmetric Quantum Mechanics with Lévy Disorder in One Dimension

We consider the Schrödinger equation with a random potential of the form where w is a Lévy noise. We focus on the problem of computing the so-called complex Lyapunov exponent where N is the integrated density of states of the system, and γ is the Lyapunov exponent. In the case where the Lévy process is non-decreasing, we show that the calculation of Ω reduces to a Stieltjes moment problem, we ascertain the low-energy behaviour of the density of states in some generality, and relate it to the distributional properties of the Lévy process. We review the known solvable cases—where Ω can be expressed in terms of special functions—and discover a new one.

Natural solutions of rational Stieltjes moment problems

In the strong or two-point Stieltjes moment problem, one has to find a positive measure on [0,∞) for which infinitely many moments are prescribed at the origin and at infinity. Here we consider a multipoint version in which the origin and the point at infinity are replaced by sequences of points that may or may not coincide. In the indeterminate case, two natural solutions μ0 and μ∞ exist that can be constructed by a limiting process of approximating quadrature formulas. The supports of these natural solutions are disjoint (with possible exception of the origin). The support points are accumulation points of sequences of zeros of even and odd indexed orthogonal rational functions. These functions are recursively computed and appear as denominators in approximants of continued fractions. They replace the orthogonal Laurent polynomials that appear in the two-point case. In this paper we consider the properties of these natural solutions and analyze the precise behavior of which zero sequences converge to which support points.