Showing papers by "Israel Gohberg published in 1998"
Book•
18 Feb 1998
TL;DR: In this paper, a general completion theorem is applied to interpolation parameterization of all solutions of the three-chains completion problem, and a nonstationary interpolation and time-varying system is presented.
Abstract: Part 1 Interpolation and time-invariant system: interpolation problems for time-valued functions proofs using the commutant lifting theorem time invariant systems central commutant lifting central state space solutions parametization of intertwinning and its applications applications to control systems. Part 2 Nonstationary interpolation and time-varying systems nonstationary interpolation theorems nonstationary systems and point evaluation reduction techniques - from nonstationary to stationary and vice versa proofs of the nonstationary interpolation theorems by reduction to the stationary case a general completion theorem applications of the three chains completion theorem to interpolation parameterization of all solutions of the three chains completion problem.
147 citations
TL;DR: In this paper, the authors extended earlier results of the authors for canonical systems with spectral functions of which the absolutely continuous part has a rational derivative to a class of differential systems with skew selfadjoint potentials.
69 citations
TL;DR: In this article, the direct spectral problem of canonical differential systems for a special class of potentials is solved explicitly, where the corresponding spectral function may have jumps and its absolutely continuous part has a rational derivative possibly with zeros on the real line.
Abstract: This paper solves explicitly the direct spectral problem of canonical differential systems for a special class of potentials. For a potential from this class the corresponding spectral function may have jumps and its absolutely continuous part has a rational derivative possibly with zeros on the real line. A direct and self-contained proof of the diagonalization of the associated differential operator is given, including explicit formulas for the diagonalizing operator and the spectral function. This proof also yields an explicit formula for the solution of the inverse problem. As an application new representations are derived for a large class of solutions of nonlinear integrable partial differential equations. The method employed is based on state space techniques and uses the idea of realization from mathematical system theory.
56 citations
TL;DR: In this article, the direct and inverse spectral problems for Sturm-Liouville systems with rational Weyl functions were solved explicitly using state space techniques and using the idea of realization from mathematical system theory.
Abstract: 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.
37 citations
TL;DR: In this article, exact formulas for the potential associated to a Sturm-Liouville equation when the reflection coefficient function is rational are given in terms of a minimal realization of the reflection coefficients function.
Abstract: In this paper we give exact formulas for the potential associated to a Sturm-Liouville equation when the reflection coefficient function is rational. The solution is given in terms of a minimal realization of the reflection coefficient function.
17 citations
01 Jan 1998
6 citations
TL;DR: Singular values for pencils were introduced and studied for selfadjoint matrices in this paper, where the singular values describe the widths of certain unbounded sets of a set.
Abstract: Singular values are introduced and studied for pencils A — λG of selfadjoint matrices which for some values of λ are positive definite These singular values describe the widths of certain unbounded sets
2 citations
01 Jan 1998
TL;DR: This chapter introduces discrete time-invariant input-output systems, and various notions connected with them, and shows that point and operator evaluation naturally occur in linear systems.
Abstract: Throughout this book the interconnection between interpolation and input-output systems in state space form plays a fundamental role This chapter introduces discrete time-invariant input-output systems, and various notions connected with them The chapter also serves as a further motivation for the interpolation problems considered earlier An introduction to the state space theory is given Moreover, it is shown that point and operator evaluation naturally occur in linear systems State space techniques are used to compute the norm of certain Hankel operators, which is precisely the error in the corresponding Nehari interpolation problem State space techniques are also used to give explicit state space formulas to connect the Nevanlinna-Pick problem to the Sarason problem, and the Nudelman problem to the two-sided Sarason problem The last section presents some aspects of unitary systems that will be used later
1 citations
TL;DR: A time-variant version of the maximum principle for the central solution in the commutant lifting theorem is given in this paper, where the main result is illustrated on the Parrott completion problem.
Abstract: A time-variant version of the maximum principle for the central solution in the commutant lifting theorem is given The main result is illustrated on the Parrott completion problem
1 citations
TL;DR: Inversion formulas for infinite block Toeplitz plus Hankel matrices were obtained in this paper for matrix-valued singular integral operators (SIVO) and finite block ToEplitz matrices.
Abstract: “Inversion formulas are obtained for a certain class of infinite matrices that possess displacement structure similar to that of finite block Toeplitz matrices. Consequences are symmetric inversion formulas for matrix-valued singular integral operators and infinite Toeplitz plus Hankel matrices”.
1 citations
TL;DR: Mark G. Krein this article was born into a Jewish family of modest means in Kiev on April 3, 1907 and was accepted for doctoral studies by N.G. Chebotarev at Odessa University.
Abstract: Mark G. Krein was born into a Jewish family of modest means in Kiev on April 3, 1907. His father was a lumber merchant. As a youngster he exhibited a talent for mathematics. At the age of 14 he was already attending research seminars. He never obtained an undergraduate degree. In 1924 he ran away from home to Odessa and in 1926 he was accepted for doctoral studies by N.G. Chebotarev at Odessa University. He completed the degree requirements in 1929.
TL;DR: In this paper, a direct connection between finite dimensional operator blocks and a certain class of quadruples of subspaces was made and it was shown that the problem of classifying operator blocks under block similarity is equivalent to classifying the corresponding quadruples under isomorphism of subspace quadruples.