Indeterminate Moment Problems Research Articles

Hermitean Matrix Ensembles and Orthogonal Polynomials

In this article, we investigate orthogonal polynomials associated with complex Hermitean matrix ensembles using the combination of the methods of Coulomb fluid (or potential theory), chain sequences, and Birkhoff–Trjitzinsky theory. We give a general formula for the largest eigenvalue of the N×N Jacobi matrices (which is equivalent to estimating the largest zero of a sequence of orthogonal polynomials) and the two‐level correlation function for the α ensembles (α>0) introduced previously for α>1. In the case of 0<α<1, we give a natural representation for the weight function that is a special case of the general Nevanlinna parametrization. We also discuss Hermitean matrix ensembles associated with general indeterminate moment problems.

Some Indeterminate Moment Problems and Freud-Like Weights

We study two indeterminate Hamburger moment problems and the corresponding orthogonal polynomials. The coefficients in their recurrence relations are of exponential growth or are polynomials of degree 2. The entire functions in the Nevanlinna parametrization are found. The orthogonal polynomials with polynomial recurrence coefficients resemble the Freud polynomials with a = 1/2 . Inequalities are given for the largest zero and the asymptotic behavior of the largest zero is established.

Co-Recursivity and karlin-McGregor duality for indeterminate moment problems

We analyze co-recursivity for indeterminate Hamburger moment problems and the duality transformation of Karlin and McGregor for indeterminate Stieltjes moment problems. In both cases the transformed Nevanlinna matrix is given and the Nevanlinna extremal measures are discussed. An example involving associated polynomials, relevant for a quartic birth and death process, is worked out.

Co-Recursivity and Karlin-McGregor Duality for Indeterminate Moment Problems

Co-Recursivity and Karlin-McGregor Duality for Indeterminate Moment Problems

Indeterminate moment problems and the theory of entire functions

The set of solutions to an indeterminate Hamburger moment problem is given by the Nevanlinna parametrization involving four entire functions A, B, C, D. We review the properties of these functions, and discuss recent progress in the still open problem of characterizing the class of entire functions which occur in this way. They belong to the Krein class and are of minimal exponential type. We deduce the analogous parametrization of the set of solutions to an indeterminate Stieltjes moment problem from the corresponding symmetric Hamburger problem. The survey ends with a discussion of three special cases, where the functions A, B, C, D are known explicitly.

GAP FORMATION PROBABILITY OF THE α-ENSEMBLE

In this paper we employ the continuum approximation of Dyson to determine the asymptotic gap formation probability in the spectrum of N×N Hermitian matrices associated with orthogonal polynomials that are solutions of indeterminate moment problems. We have shown that the probability, Eβ[J], that an interval J is free of eigenvalues is [Formula: see text] where [Formula: see text]

Nevanlinna extremal measures for some orthogonal polynomials related to birth and death processes

We consider two classes of birth and death processes which lead to indeterminate Stieltjes moment problems, and we calculate the entire functions C and D from the corresponding Nevanlinna matrix. This permits a direct computation of a Nevanlinna extremal orthogonality measure for each of the processes.

On the order and type of the entire functions associated with an indeterminate Hamburger moment problem

On the order and type of the entire functions associated with an indeterminate Hamburger moment problem

𝑞-Hermite polynomials, biorthogonal rational functions, and 𝑞-beta integrals

We characterize the solutions of the indeterminate moment problem associated with the continuousqq-Hermite polynomials whenq>1q > 1in terms of their Stieltjes transforms. The extremal measures are found explicitly. An analog of the Askey-Wilson integral is evaluated. It involves integrating a kernel, similar to the Askey-Wilson kernel, against any solution of theqq-Hermite moment problem, provided that certain integrability conditions hold. This led to direct evaluation of severalqq-beta integrals and their discrete analogs as well as a generalization of Bailey’s6ψ6{}_6{\psi _6}, sum containing infinitely many parameters. A system of biorthogonal rational functions is also introduced.

Extremal measures for a system of orthogonal polynomials

We characterize the extremal measures of an indeterminate moment problem associated with a system of orthogonal polynomials defined by a three-term recurrence relation.

Existence of smooth solutions to the classical moment problems

Let s ( 0 ) , s ( 1 ) , … s(0),s(1), \ldots be a given sequence, and define s ( n ) = s ( − n ) ¯ s(n) = \overline {s( - n)} for n > 0 n > 0 . If Σ Σ ξ ¯ n ξ m s ( m − n ) ≥ 0 \Sigma \Sigma {\overline \xi _n}{\xi _m}s(m - n) \geq 0 holds for all finite sequences ( ξ n ) n ∈ Z {({\xi _n})_{n \in \mathbb {Z}}} , then it is known that there is a positive Borel measure μ \mu on the circle T \mathbb {T} such that s ( n ) = ∫ − π π e i n t d μ ( t ) s(n) = \smallint _{ - \pi }^\pi {{e^{int}}d\mu (t)} , and conversely. Our main theorem provides a necessary and sufficient condition on the sequence ( s ( n ) ) (s(n)) that the measure μ \mu may be chosen to be smooth. A measure μ \mu is said to be smooth if it has the same spectral type as the operator i d / d t id/dt acting on L 2 ( T ) {L^2}(\mathbb {T}) with respect to Haar measure d t dt on T \mathbb {T} : Equivalently, μ \mu is a superposition (possibly infinite) of measures of the form | w ( t ) | 2 d t |w(t){|^2}dt with w ∈ L 2 ( T ) w \in {L^2}(\mathbb {T}) such that d w / d t ∈ L 2 ( T ) dw/dt \in {L^2}(\mathbb {T}) . The condition is stated purely in terms of the initially given sequence ( s ( n ) ) (s(n)) : We show that a smooth representation exists if and only if, for some ε ∈ R + \varepsilon \in {\mathbb {R}_ + } , the a priori estimate \[ ∑ ∑ s ( m − n ) ξ ¯ n ξ m ≥ ε | ∑ n s ( n ) ξ n | 2 \sum {\sum {s(m - n){{\overline \xi }_n}{\xi _m} \geq \varepsilon {{\left | {\sum {ns(n){\xi _n}} } \right |}^2}} } \] is valid for all finite double sequences ( ξ n ) ({\xi _n}) . An analogous result is proved for the determinate (Hamburger) moment problem on the line. But the corresponding result does not hold for the indeterminate moment problem.

Indeterminate symmetric moment problems

Let {n,(x)} be a sequence of orthogonal polynomials with respect to a real distribution &. According to a classical theorem of Reisz 191, (x,(x)] is complete in L: if and only if w is either the solution of a determined Hamburger moment problem or an extremal solution of an indeterminate moment problem. In the special case where the zeros of the orthogonal polynomials have a finite infimum a and the Hamburger moment problem is indeterminate, there is an extremal solution which is essentially the only solution whose spectrum is a subset of [a, co) [4]. This solution is, in a sense, the “best” solution from the viewpoint of orthogonal polynomials since its spectrum consists of the limit points of the zeros of the polynomials. Askey remarked to the author that in view of Riesz’ theorem, it is surprising that so little attention has been paid to the question of extremal solutions. (Moak [8] has recently studied the extremal solutions associated with the “q-Laguerre,” nee generalized Stieltjes-Wigert, polynomials.) Askey suggested it would be desirable to investigate the existence of “best” extremal solutions in the general case or, as a first step, even in the “symmetric case” (moments of odd order are 0). In this paper, we will consider this special case and the related systems of orthogonal polynomials. In particular, we will show that an indeterminate symmetric moment problem has precisely two “symmetric” extremal solutions. This result, which we found surprizing at first, appears after reflection quite natural (and perhaps obvious) since the two solutions are the symmetrizations of the “best” extremal solutions of the related moment problems on [0, co).

The Nevanlinna parametrization for some indeterminate Stieltjes moment problems associated with birth and death processes

The Nevanlinna parametrization for some indeterminate Stieltjes moment problems associated with birth and death processes