Matrix Moment Problem Research Articles

On rank variation of block matrices generated by Nevanlinna matrix functions

AbstractWithin the framework of the multiple Nevanlinna–Pick matrix interpolation and its related matrix moment problem, we study the rank of block moment matrices of various kinds, generalized block Pick matrices and generalized block Loewner matrices, as well as their Potapov modifications, generated by Nevanlinna matrix functions, and derive statements either on rank (or inertia) invariance in different senses or on rank variation of such types of block matrices (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Boundary Nevanlinna–Pick interpolation for Nevanlinna matrix functions and the related Hamburger matrix moment problem

This paper is concerned with a boundary Nevanlinna–Pick interpolation for Nevanlinna matrix functions and the Hamburger matrix moment problem but with specified constraints that the nonnegative matrix-valued measure has no mass distributions at a finite number of real points. Intrinsic connections between these two problems are established by the so-called Hankel vector approach. The connections, together with the theory of matrix moments and the theory of matrix polynomials with respect to a positive Hermitian block Hankel matrix due to H. Dym, enable us to get a solvability criterion and a parameterized description of all the solutions for each of these two problems.

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.

Vector coherent states with matrix moment problems

Canonical coherent states can be written as infinite series in powers of a single complex number $z$ and a positive integer $\rho(m)$. The requirement that these states realize a resolution of the identity typically results in a moment problem, where the moments form the positive sequence of real numbers $\{\rho(m)\}_{m=0}^\infty$. In this paper we obtain new classes of vector coherent states by simultaneously replacing the complex number $z$ and the moments $\rho(m)$ of the canonical coherent states by $n \times n$ matrices. Associated oscillator algebras are discussed with the aid of a generalized matrix factorial. Two physical examples are discussed. In the first example coherent states are obtained for the Jaynes-Cummings model in the weak coupling limit and some physical properties are discussed in terms of the constructed coherent states. In the second example coherent states are obtained for a conditionally exactly solvable supersymmetric radial harmonic oscillator.

Tangential Nevanlinna-Pick Interpolation and Its Connection with Hamburger Matrix Moment Problem

This paper is concerned with the solution of a certain tangential Nevanlinna-Pick interpolation for Nevanlinna functions. We use the so-called block Hankel vector method to establish two intrinsic connections between the tangential Nevanlinna-Pick interpolation in the Nevanlinna class and the truncated Hamburger matrix moment problem associated with the block Hankel vector under consideration: one is a congruent relationship between their information matrices, and the other is a divisor-remainder connection between their solutions. These investigations generalize our previous work on the Nevanlinna-Pick interpolation and power matrix moment problem.

Matrix Christoffel Functions

We introduce and study matrix Christoffel functions for a matrix weight W. We find an explicit expression of the matrix Christoffel functions in terms of any sequence of orthonormal matrix polynomials with respect to W. An extremal property related to the matrix moment problem defined by W is established for the matrix Christoffel functions. We finally find the relative asymptotic behavior of the matrix Christoffel functions associated to matrix weights in the matrix Nevai class.

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.

On the Relation between the Scalar Moment Problem and the Matrix Moment Problem on *-Semigroups

There is a countable cancellative commutative *-semigroup S withzero (in fact, a *-subsemigroup of G × N0 for some abelian group G carrying the inverse involution) such that the answer to the question “if f is a function on S , with values in Md(C) (the square matrices of order d) and such that $\sum^{n}_{j,k=1} \lbrak f(s^*_k s_j)\xi_j, \xi_k \rbrak \ge 0$ for all n in N, s1, . . . , sn in S , and $\xi_1$, . . . , $\xi_n$ in Cd, does it follow that $f(s) = \int_{S^*}\sigma (s) d\mu(\sigma) (s \memb S)$ for some measure $\mu$ (with values in Md(C)+ , the positive semidenite matrices) on the space S of hermitian multiplicative functions on S?” is “yes” if d = 1 but “no” if d = 2 (hence also for d > 2).

Matrix moment problems with constraints

Matrix moment problems with constraints

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.

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).

The Nevanlinna–Pick interpolation problems and power moment problems for matrix-valued functions III: The infinitely many data case

The objective of this paper is to establish an intrinsic and simple correspondence between a multiple Nevanlinna–Pick matrix interpolation (NP) problem in the Nevanlinna class N p with a denumerable set of nodes and a certain full Hamburger matrix moment problem. With the help of the indicated correspondence, the solution of an infinite NP problem in N p can be reduced to what amounts to the study of the associated full Hamburger matrix moment problem with the so-called block Hankel vector of the former, and vice versa. These investigations can be regarded as a natural extension of our previous work on both the NP problems with a finite set of interpolation data and the truncated power matrix moment problems.

Riesz's Theorem for Orthogonal Matrix Polynomials

We describe the image through the Stieltjes transform of the set of solutions V of a matrix moment problem. We extend Riesz's theorem to the matrix setting, proving that those matrices of measures of V for which the matrix polynomials are dense in the corresponding $ {\cal L} $ 2 space are precisely those whose Stieltjes transform is an extremal point (in the sense of convexity) of the image set.

On the multiple Nevanlinna-Pick matrix interpolation in the class [formula omitted]p and the Carathéodory matrix coefficient problem

In the present paper an intrinsic one-to-one correspondence between the (finite) multiple Nevanlinna-Pick matrix interpolation (NP) problem in the Carathéodory class and the Carathéodory matrix coefficient (CC) problem (or, equivalently, the truncated trigonometric matrix moment (TM) problem) is established. Thanks to this correspondence, the NP problem is reduced to what amounts to solving the CC problem (or the TM problem) associated with the so-called Toeplitz block-vector of the former in both the nondegenerate and degenerate cases simultaneously.

The truncated Hamburger matrix moment problems in the nondegenerate and degenerate cases, and matrix continued fractions

The present paper deals simultaneously with the nondegenerate and degenerate truncated Hamburger matrix moment problems in a unified way based on the use of the Schur algorithm involving matrix continued fractions. A full analysis of them together with a relative matrix moment problem on the real axis is given. With the help of the correspondence between the moment problem on the real axis and the Nevanlinna-Pick (NP) interpolation, the solutions of the nontangential NP interpolation in the Nevanlinna class are derived as an application.

Density Questions for the Truncated Matrix Moment Problem

AbstractFor a truncated matrix moment problem, we describe in detail the set of positive definite matrices of measures μ in V2n (this is the set of solutions of the problem of degree 2n) for which the polynomials up to degree n are dense in the corresponding space L2(μ). These matrices of measures are exactly the extremal measures of the set Vn.

Degenerate Stieltjes Moment Problem and Associated $J$-Inner Polynomials

In this paper we consider the truncated Stieltjes matrix moment problem when the so-called information matrices are degenerate and describe the set of all solutions in terms of the linear fractional transformation using the fundamental matrix inequality method.

Description of the solutions of a degenerate moment problem on the axis and the semiaxis

Schur's stepwise algorithm is constructed for degenerate matrix moment problems on the axis and the semiaxis, yielding the complete description of their solutions.

On the avoidance of cancellations in the matrix moment problem

Completes a line of argument begun in an earlier paper (see ibid., vol.4, p.835, 1978) which shows that avoidance of the cancellations that destroy the stability of the standard solutions of the matrix moment problem is possible and that it leads naturally to the Lanczos method.

Matrix moment methods in perturbation theory, boson quantum field models, and anharmonic oscillators

A matrix moment problem is considered in connection with anyx 2m(m=2, 3, 4, ...) anharmonic oscillator as well as the (:φ2m(x):g(x))2 (m=2, 3) field theory models, whose Rayleigh-Schrödinger perturbation expansions for the ground state eigenvalue are known to diverge. The approximants related to such a problem are proven to converge to the eigenvalue, when applied to an expansion of the Brillouin-Wigner type. These approximants, whose construction involves only matrix elements occurring in the Rayleigh-Schrödinger expansion, are the approximants of aJ-type matrix continued fraction, i.e. the [N−1,N] matrix Padé approximants. The explicit analytical expression of matrix continued fraction is found in the anharmonic oscillators case.