Indeterminate Moment Problems Research Articles

Some Cubic Birth and Death Processes and their Related Orthogonal Polynomials

The orthogonal polynomials with recurrence relation (λ,n +μn-z)Fn(z) = μn+1Fn+1(z)+λn-1Fn-1(z) with two kinds of cubic transition rates λn and μn, corresponding to indeterminate Stieltjes moment problems, are analyzed. We derive generating functions for these two classes of polynomials, which enable us to compute their Nevanlinna matrices. We discuss the asymptotics of the Nevanlinna matrices in the complex plane.

Preface

In the field of orthogonal polynomials theory, the classical Markov theorem shows that for determinate moment problems the spectral measure is under control of the polynomials asymptotics. The situation is completely different for indeterminate moment problems, in which case the interesting spectral measures are to be constructed using Nevanlinna theory. Nevertheless it is interesting to observe that some spectral measures can still be obtained from weaker forms of Markov theorem. The exposition will be illustrated by orthogonal polynomials related to elliptic functions: in the determinate case by examples due to Stieltjes and some of their generalizations and in the indeterminate case by more recent examples.

From Asymptotics to Spectral Measures: Determinate Versus Indeterminate Moment Problems

In the field of orthogonal polynomials theory, the classical Markov theorem shows that for determinate moment problems the spectral measure is under control of the polynomials asymptotics. The situation is completely different for indeterminate moment problems, in which case the interesting spectral measures are to be constructed using Nevanlinna parametrization. Nevertheless it is interesting to observe that some spectral measures can still be obtained from weaker forms of the Markov theorem. The exposition will be illustrated by orthogonal polynomials related to elliptic functions: in the determinate case by examples due to Stieltjes and some of their generalizations and in the indeterminate case by more recent examples.

Orthogonal polynomials and analytic functions associated to positive definite matrices

For a positive definite infinite matrix A, we study the relationship between its associated sequence of orthonormal polynomials and the asymptotic behaviour of the smallest eigenvalue of its truncation A n of size n × n . For the particular case of A being a Hankel or a Hankel block matrix, our results lead to a characterization of positive measures with finite index of determinacy and of completely indeterminate matrix moment problems, respectively.

Indeterminate moment problems related to birth and death processes with quartic rates

We consider indeterminate symmetric moment problems related to birth and death processes with quartic rates. The Nevanlinna matrices are presented and the entire functions B and D are expressed in terms of the trigonometric functions of order 4. Several examples of symmetric and nonsymmetric solutions are given.

Indeterminacy Criteria for the Stieltjes Matrix Moment Problem

In this paper, we obtain criteria for the indeterminacy of the Stieltjes matrix moment problem. We obtain explicit formulas for Stieltjes parameters and study the multiplicative structure of the resolvent matrix. In the indeterminate case, we study the analytic properties of the resolvent matrix of the moment problem. We describe the set of all matrix functions associated with the indeterminate Stieltjes moment problem in terms of linear fractional transformations over Stieltjes pairs.

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.

The Moment Problem Associated with the q -Laguerre Polynomials

Abstract. We consider the indeterminate Stieltjes moment problem associated with the q -Laguerre polynomials. A transformation of the set of solutions, which has all the classical solutions as fixed points, is established and we present a method to construct, for instance, continuous singular solutions. The connection with the moment problem associated with the Stieltjes—Wigert polynomials is studied; we show how to come from q -Laguerre solutions to Stieltjes—Wigert solutions by letting the parameter α —> ∞ , and we explain how to lift a Stieltjes—Wigert solution to a q -Laguerre solution at the level of Pick functions. Based on two generating functions, expressions for the four entire functions from the Nevanlinna parametrization are obtained.

The moment problem associated with the Stieltjes–Wigert polynomials

We consider the indeterminate Stieltjes moment problem associated with the Stieltjes–Wigert polynomials. After a presentation of the well-known solutions, we study a transformation T of the set of solutions. All the classical solutions turn out to be fixed under this transformation but this is not the case for the so-called canonical solutions. Based on generating functions for the Stieltjes–Wigert polynomials, expressions for the entire functions A, B, C, and D from the Nevanlinna parametrization are obtained. We describe T(n)(μ) for n∈N when μ=μ0 is a particular N-extremal solution and explain in detail what happens when n→∞.

Small eigenvalues of large Hankel matrices: The indeterminate case

In this paper we characterize the indeterminate case by the eigenvalues of the Hankel matrices being bounded below by a strictly positive constant. An explicit lower bound is given in terms of the orthonormal polynomials and we find expressions for this lower bound in a number of indeterminate moment problems.

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.

The Nevanlinna parametrization for a matrix moment problem

We obtain the Nevanlinna parametrization for an indeterminate matrix moment problem, giving a homeomorphism between the set $V$ of solutions to the matrix moment problem and the set $\mathcal V$ of analytic matrix functions in the upper half plane such that $V(\lambda )^*V(\lambda )\le I$. We characterize the N-extremal matrices of measures (those for which the space of matrix polynomials is dense in their $L^2$-space) as those whose corresponding matrix function $V(\lambda )$ is a constant unitary matrix.

The discrete Kramer sampling theorem and indeterminate moment problems

In this paper we propose candidates to be the kernel appearing in the discrete Kramer sampling theorem. These kernels arise either from orthonormal polynomials associated with indeterminate Hamburger or Stieltjes moment problems, or from the second kind orthogonal polynomials associated with the former ones. The sampling points are given by the zeros of the denominator in the Nevanlinna parametrization of the N-extremal measures. Explicit formulae are given associated with some cases where the Nevanlinna parametrization is known explicitly.

Shell polynomials and indeterminate moment problems

Shell polynomials and indeterminate moment problems

On the analytic form of the discrete Kramer sampling theorem

The classical Kramer sampling theorem is, in the subject of self‐adjoint boundary value problems, one of the richest sources to obtain sampling expansions. It has become very fruitful in connection with discrete Sturm‐Liouville problems. In this paper a discrete version of the analytic Kramer sampling theorem is proved. Orthogonal polynomials arising from indeterminate Hamburger moment problems as well as polynomials of the second kind associated with them provide examples of Kramer analytic kernels.

N-Extremal Matrices of Measures for an Indeterminate Matrix Moment Problem

In this paper we study the N-extremal matrices of measures associated to a completely indeterminate matrix moment problem, i.e., those matrices of measure W, solutions of a completely indeterminate matrix moment problem for which the linear space of matrix polynomials is dense in the corresponding L2(W).

SzegÖ polynomials on the real axis

Szegö polynomials orthogonal on the unit circle satisfy the recurrence relation with reflection parameters . We construct special classes of Szegö polynomials . These polynomials appear to be orthogonal on the positive (or negative) real axis. We present explicit examples of such polynomials that are connected with the Askey–Wilson polynomials. In contrast to the case of the Szegö polynomials on the unit circle, there exist Szegö polynomials on the real axis with an indeterminate moment problem. A simple criterion of determinacy of the moment problem is found.

Some Basic Lommel Polynomials

The basic Lommel polynomials associated to the1ϕ1q-Bessel function and the Jacksonq-Bessel functions are considered as orthogonal polynomials inqν, whereνis the order of the corresponding basic Bessel functions. The corresponding moment problems are both indeterminate and determinate depending on a parameter. Using techniques of Chihara and Maki we derive an explicit orthogonality measure, which is discrete and unbounded. For the indeterminate moment problem this measure is N-extremal. Some results on the zeros of the basic Bessel functions, both as functions of the order and of the argument are obtained. Precise asymptotic behaviour of the zeros of the1ϕ1q-Bessel function is obtained.

On some indeterminate moment problems for measures on a geometric progression

We consider some indeterminate moment problems which all have a discrete solution concentrated on geometric progressions of the form cq k , k ∈ Z or ± cq k , k ∈ Z . It turns out to be possible that such a moment problem has infinitely many solutions concentrated on the geometric progression.

Theq-Laguerre Polynomials and Related Moment Problems

We study two indeterminate Hamburger moment problems associated withq-Laguerre polynomials. The coefficients in their recurrence relations are of exponential growth. This completes earlier work started by D. Moak. Some of the measures that the polynomials are orthogonal with respect to are found and lead to the evaluation of certain integrals. A symmetric set of orthogonal polynomials is also considered.