共查询到20条相似文献,搜索用时 31 毫秒
1.
Yuriy M. Dyukarev Bernd Fritzsche Bernd Kirstein Conrad Mädler Helge C. Thiele 《Complex Analysis and Operator Theory》2009,3(4):759-834
We study two slightly different versions of the truncated matricial Hamburger moment problem. A central topic is the construction
and investigation of distinguished solutions of both moment problems under consideration. 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 sequences (sj)∞j = 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 constructed 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. 相似文献
2.
G. Valent 《Constructive Approximation》1996,12(4):531-553
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. 相似文献
3.
A. G. García M. A. Hernndez-Medina 《Journal of Mathematical Analysis and Applications》2003,280(2):221-231
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. 相似文献
4.
We characterize the orthogonal polynomials in a class of polynomials defined through their generating functions. This led to three new systems of orthogonal polynomials whose generating functions and orthogonality relations involve elliptic functions. The Hamburger moment problems associated with these polynomials are indeterminate. We give infinite families of weight functions in each case. The different polynomials treated in this work are also polynomials in a parameter and as functions of this parameter they are orthogonal with respect to unique measures, which we find explicitly. Through a quadratic transformation we find a new exactly solvable birth and death process with quartic birth and death rates. 相似文献
5.
The unified approach to the matrix inversion problem initiated in this work is based on the concept of the generalized Bezoutian for several matrix polynomials introduced earlier by the authors. The inverse X–1 of a given block matrix X is shown to generate a set of matrix polynomials satisfying certain conditions and such that X–1 coincides with the Bezoutian associated with that set. Thus the inversion of X is reduced to determining the underlying set of polynomials. This approach provides a fruitful tool for obtaining new results as well as an adequate interpretation of the known ones. 相似文献
6.
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. 相似文献
7.
In this paper, matrix orthogonal polynomials in the real line are described in terms of a Riemann–Hilbert problem. This approach provides an easy derivation of discrete equations for the corresponding matrix recursion coefficients. The discrete equation is explicitly derived in the matrix Freud case, associated with matrix quartic potentials. It is shown that, when the initial condition and the measure are simultaneously triangularizable, this matrix discrete equation possesses the singularity confinement property, independently if the solution under consideration is given by the recursion coefficients to quartic Freud matrix orthogonal polynomials or not. 相似文献
8.
We show that many well-known counting coefficients in combinatorics are Hamburger moment sequences in certain unified approaches and that Hamburger moment sequences are infinitely convex. We introduce the concept of the q-Hamburger moment sequence of polynomials and present some examples of such sequences of polynomials. We also suggest some problems and conjectures. 相似文献
9.
I. V. Kapalin V. V. Fomichev 《Moscow University Computational Mathematics and Cybernetics》2011,35(3):116-123
We consider problems close to that of the minimal stabilization of a linear vector (i.e., MISO or SIMO) dynamic system; more
specifically, the problem of determining the number of common roots of a family of polynomials, and investigating the properties
of the so-called generalized Sylvester matrix. The classical definition of the Sylvester matrix is valid for two polynomials,
and there are different methods for defining the generalized (extended) Sylvester matrix for a family of polynomials. In this
work, we consider a definition of the generalized Sylvester matrix and its properties in the context of their potential future
application for solving the minimal stabilization problem. 相似文献
10.
Matrix Szeg? biorthogonal polynomials for quasi‐definite matrices of Hölder continuous weights are studied. A Riemann‐Hilbert problem is uniquely solved in terms of the matrix Szeg? polynomials and its Cauchy transforms. The Riemann‐Hilbert problem is given as an appropriate framework for the discussion of the Szeg? matrix and the associated Szeg? recursion relations for the matrix orthogonal polynomials and its Cauchy transforms. Pearson‐type differential systems characterizing the matrix of weights are studied. These are linear systems of ordinary differential equations that are required to have trivial monodromy. Linear ordinary differential equations for the matrix Szeg? polynomials and its Cauchy transforms are derived. It is shown how these Pearson systems lead to nonlinear difference equations for the Verblunsky matrices and two examples, of Fuchsian and non‐Fuchsian type, are considered. For both cases, a new matrix version of the discrete Painlevé II equation for the Verblunsky matrices is found. Reductions of these matrix discrete Painlevé II systems presenting locality are discussed. 相似文献
11.
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.
April 24, 1996. Date revised: March 3, 1997. 相似文献
12.
该文描述带有矩量序列{v_m}_0~∞■C~(q×q)的完全不确定Hamburger矩阵矩量问题:v_m=integral from n=-∞to∞x~m dρ(x),m=0,1,…的有限阶解,即该问题的那些解ρ,使得C~(q×q)-值多项式的线性空间P在对应的空间L~2(R,dρ/E(x))内稠密,这里E(x)为在实轴R上取正值的某个数值多项式.作为预备知识,作者考虑所谓广义Akhiezer插值的矩阵变种与它的相关矩阵矩量问题之间的一种关系. 相似文献
13.
H. Gzyl 《Applied mathematics and computation》2010,216(11):3307-3318
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. 相似文献
14.
Todd Kapitula Elizabeth Hibma Hwa-Pyeong Kim Jonathan Timkovich 《Linear algebra and its applications》2013
There is a well-established instability index theory for linear and quadratic matrix polynomials for which the coefficient matrices are Hermitian and skew-Hermitian. This theory relates the number of negative directions for the matrix coefficients which are Hermitian to the total number of unstable eigenvalues for the polynomial. Herein we extend the theory to ?-even matrix polynomials of any finite degree. In particular, unlike previously known cases we show that the instability index depends upon the size of the matrices when the degree of the polynomial is greater than two. We also consider Hermitian matrix polynomials, and derive an index which counts the number of eigenvalues with nonpositive imaginary part. The results are refined if we consider the Hermitian matrix polynomial to be a perturbation of a ?-even polynomials; however, this refinement requires additional assumptions on the matrix coefficients. 相似文献
15.
《Journal of Computational and Applied Mathematics》2001,127(1-2):1-16
The computation of zeros of polynomials is a classical computational problem. This paper presents two new zerofinders that are based on the observation that, after a suitable change of variable, any polynomial can be considered a member of a family of Szegő polynomials. Numerical experiments indicate that these methods generally give higher accuracy than computing the eigenvalues of the companion matrix associated with the polynomial. 相似文献
16.
S. M. Zagorodnyuk 《Ukrainian Mathematical Journal》2010,62(4):537-551
We obtain necessary and sufficient conditions for the solvability of the strong matrix Hamburger moment problem. We describe
all solutions of the moment problem by using the fundamental results of A. V. Shtraus on generalized resolvents of symmetric
operators. 相似文献
17.
《Applied Mathematics Letters》2001,14(7):859-866
An error analysis of the so-called signal zeros of polynomials linked to exponentially damped signals is performed and error bounds are derived. The analysis uses the link between polynomials and companion matrices and allows us to show that the related companion matrix eigenvalue problem is governed by the condition number of a rectangular Vandermonde matrix which has the zeros of interest as nodes. Conditions under which the zeros are well conditioned are discussed. 相似文献
18.
The notion of infinite companion matrix is extended to the case of matrix polynomials (including polynomials with singular leading coefficient). For row reduced polynomials a finite companion is introduced as the compression of the shift matrix. The methods are based on ideas of dilation theory. Connections with systems theory are indicated. Applications to the problem of linearization of matrix polynomials, solution of systems of difference and differential equations and new factorization formulae for infinite block Hankel matrices having finite rank are shown. As a consequence, any system of linear difference or differential equations with constant coefficients can be transformed into a first order system of dimension n = deg det D. 相似文献
19.
V. M. Prokip 《Journal of Mathematical Sciences》1996,81(6):3020-3023
We study the problem of the decomposition of a matrix polynomial over an arbitrary field into a product of factors of lower
degrees with preassigned characteristic polynomials. We find necessary conditions for the existence of the required factorization,
which are also sufficient for certain classes of matrix polynomials. The proposed method makes it possible to solve the problem
completely for matrix polynomials with one nonconstant invariant factor.
Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995. 相似文献
20.
V. N. Kublanovskaya 《Journal of Mathematical Sciences》2004,121(4):2511-2518
The paper considers the problem of computing the invariant polynomials of a general (regular or singular) one-parameter polynomial matrix. Two new direct methods for computing invariant polynomials, based on the W and V rank-factorization methods, are suggested. Each of the methods may be regarded as a method for successively exhausting roots of invariant polynomials from the matrix spectrum. Application of the methods to computing adjoint matrices for regular polynomial matrices, to finding the canonical decomposition into a product of regular matrices such that the characteristic polynomial of each of them coincides with the corresponding invariant polynomial, and to computing matrix eigenvectors associated with roots of its invariant polynomials are considered. Bibliography: 5 titles. 相似文献