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1.
Bernd Fritzsche 《Journal of Computational and Applied Mathematics》2010,235(4):1008-1041
The main theme of this paper is the discussion of a family of extremal solutions of a finite moment problem for rational matrix functions in the nondegenerate case. We will point out that each member of this family is extremal in several directions. Thereby, the investigations below continue the studies in Fritzsche et al. (in press) [1]. In doing so, an application of the theory of orthogonal rational matrix functions with respect to a nonnegative Hermitian matrix Borel measure on the unit circle is used to get some insights into the structure of the extremal solutions in question. In particular, we explain characterizations of these solutions in the whole solution set in terms of orthogonal rational matrix functions. We will also show that the associated Riesz-Herglotz transform of such a particular solution admits specific representations, where orthogonal rational matrix functions are involved. 相似文献
2.
The main theme of this paper is the discussion of a parametrized family of solutions of a finite moment problem for rational matrix‐valued functions in the nondegenerate case. We will show that each member of this family is extremal in several directions concerning some point of the open unit disk. These investigations are inspired by the authors' paper [23], where a similar topic is studied in the context of the matricial Carathéodory problem. We will see that larger parts of the results presented there can be extended to the rational case studied here.The main theme of this paper is the discussion of a parametrized family of solutions of a finite moment problem for rational matrix‐valued functions in the nondegenerate case. We will show that each member of this family is extremal in several directions concerning some point of the open unit disk. These investigations are inspired by the authors' paper [23], where a similar topic is studied in the context of the matricial Carathéodory problem. We will see that larger parts of the results presented there can be extended to the rational case studied here (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
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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. 相似文献
6.
Skew‐selfadjoint Dirac systems with rational rectangular Weyl functions: explicit solutions of direct and inverse problems and integrable wave equations 下载免费PDF全文
B. Fritzsche M. A. Kaashoek B. Kirstein A. L. Sakhnovich 《Mathematische Nachrichten》2016,289(14-15):1792-1819
In this paper we study direct and inverse problems for discrete and continuous skew‐selfadjoint Dirac systems with rectangular (possibly non‐square) pseudo‐exponential potentials. For such a system the Weyl function is a strictly proper rational matrix function and any strictly proper rational matrix function appears in this way. In fact, extending earlier results, given a strictly proper rational matrix function we present an explicit procedure to recover the corresponding potential using techniques from mathematical system and control theory. We also introduce and study a nonlinear generalized discrete Heisenberg magnet model, extending earlier results for the isotropic case. A large part of the paper is devoted to the related discrete systems of which the pseudo‐exponential potential depends on an additional continuous time parameter. Our technique allows us to obtain explicit solutions for the generalized discrete Heisenberg magnet model and evolution of the Weyl functions. 相似文献
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In this paper we discuss Weyl matrix balls in the context of the matricial versions of the classical interpolation problems named after Carathéodory and Schur. Our particular focus will be on studying the monotonicity of suitably normalized semi-radii of the corresponding Weyl matrix balls. We, furthermore, devote a fair bit of attention to characterizing the case in which equality holds for particular matricial inequalities. Solving these problems will provide us with a new perspective on the role of the central functions for the classes of Carathéodory and Schur. 相似文献
9.
I. Gohberg M. A. Kaashoek A. L. Sakhnovich 《Integral Equations and Operator Theory》1998,30(3):338-377
This paper solves explicitly the direct and inverse spectral problems for Sturm-Liouville systems with rational Weyl functions. As in the previous papers of the authors on canonical and pseudocanonical systems, the method employed in the present paper is based on state space techniques and uses the idea of realization from mathematical system theory. For a subclass of rational potentials a property of modified bispectrality is proven. A wide class of solutions of the matrix Korteweg-de Vries equation is derived.To the memory of M.G. Kreîn, one of the founding fathers of modern spectral theory of differential operators, with admiration and gratitude. 相似文献
10.
B. Fritzsche B. Kirstein A. L. Sakhnovich 《Integral Equations and Operator Theory》2010,66(2):231-251
Inverse problem to recover the skew-self-adjoint Dirac-type system from the generalized Weyl matrix function is treated in
the paper. Sufficient conditions under which the unique solution of the inverse problem exists, are formulated in terms of
the Weyl function and a procedure to solve the inverse problem is given. The case of the generalized Weyl functions of the
form f(l) exp{-2ilD}{\phi(\lambda)\,{\rm exp}\{-2i{\lambda}D\}}, where f{\phi} is a strictly proper rational matrix function and D = D* ≥ 0 is a diagonal matrix, is treated in greater detail. Explicit formulas for the inversion of the corresponding semiseparable
integral operators and recovery of the Dirac-type system are obtained for this case. 相似文献
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Andreas Lasarow 《Mathematische Nachrichten》2008,281(1):75-108
We consider an interpolation problem of Nevanlinna–Pick type for matrix‐valued Carathéodory functions, where the values of the functions and its derivatives up to certain orders are given at finitely many points of the open unit disk. For the non‐degenerate case, i.e., in the particular situation that a specific block matrix (which is formed by the given data in the problem) is positive Hermitian, the solution set of this problem is described in terms of orthogonal rational matrix‐valued functions. These rational matrix functions play here a similar role as Szegő's orthogonal polynomials on the unit circle in the classical case of the trigonometric moment problem. In particular, we present and use a connection between Szegő and Schur parameters for orthogonal rational matrix‐valued functions which in the primary situation of orthogonal polynomials was found by Geronimus. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
13.
In this paper, we study a class of Jacobi matrices with very rapidly decreasing weights. It is shown that the Weyl function (the matrix element of the resolvent of the operator) for the class under study can be expressed as the ratio of two entire transcendental functions of order zero. It is shown that the coefficients in the expansion of these functions in Taylor series are proportional to the generating functions of the number of integral solutions defined by certain Diophantine equations. An asymptotic estimate for the eigenvalues is obtained. 相似文献
14.
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
15.
A. Bultheel P. González-Vera E. Hendriksen O. Njåstad 《Journal of Approximation Theory》2010,162(12):2184-2201
We consider rational moment problems on the real line with their associated orthogonal rational functions. There exists a Nevanlinna-type parameterization relating to the problem, with associated Nevanlinna matrices of functions having singularities in the closure of the set of poles of the rational functions belonging to the problem. We prove results related to the growth at the singularities of the functions in a Nevanlinna matrix, and in particular provide bounds on the growth analogous to the corresponding result in the classical polynomial case, when the number of singularities is finite. 相似文献
16.
A mixed problem for the nonlinear Bogoyavlenskii system on the half-line is studied by the inverse problem method. The solution of the mixed problem is reduced to the solution of the inverse spectral problem of recovering a forth-order differential operator on the half-line from the Weyl matrix. We derive evolution equations for the elements of the Weyl matrix and give an algorithm for the solution of the mixed problem. Evolution equations of the elements of the Weyl matrix are nonlinear. It is shown that they can be reduced to a nested system of three successively solvable matrix Riccati equations. 相似文献
17.
V. I. Nalimov 《Journal of Applied and Industrial Mathematics》2011,5(2):259-270
In order to find approximate solutions to some continuum mechanics problems that admit variational statements, we use an approach
that is based on restricting the class of functions in which we seek an extremal for the action functional. We demonstrate
the method by some examples for the problem of forced oscillations of a nonlinear elastic membrane (in particular, a string),
the problem of a fluid flow through a porous obstacle, and the problem of stationary waves on the surface of a heavy fluid. 相似文献
18.
P. López-Rodríguez 《Constructive Approximation》1999,15(1):135-151
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
2
space are precisely those whose Stieltjes transform is an extremal point (in the sense of convexity) of the image set.
May 20, 1997. Date revised: January 8, 1998. 相似文献
19.
The problem of cancelling a specified part of the zeros of a completely general rational matrix function by multiplication with an appropriate invertible rational matrix function is investigated from different standpoints. Firstly, the class of all factors that dislocate the zeros and feature minimal McMillan degree are derived. Further, necessary and sufficient existence conditions together with the construction of solutions are given when the factor fulfills additional assumptions like being J-unitary, or J-inner, either with respect to the imaginary axis or to the unit circle. The main technical tool are centered realizations that deliver a sufficiently general conceptual support to cope with rational matrix functions which may be polynomial, proper or improper, rank deficient, with arbitrary poles and zeros including at infinity. A particular attention is paid to the numerically-sound construction of solutions by employing at each stage unitary transformations, reliable numerical algorithms for eigenvalue assignment and efficient Lyapunov equation solvers. 相似文献