On the complex moment problem

Berezansky, Yu. M.; Dudkin, M. E.

doi:10.1002/mana.200410464

Cited by 8 publications

(6 citation statements)

References 26 publications

(39 reference statements)

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“…We note that some results similar to those described above in this section were obtained earlier in [39], where consideration was given to time-dependent pentadiagonal unitary matrices J(t) having the structure (5), (39), (40), (42) in the usual space 2 . In this case, the Lax equation (17) with a matrix A(t) distinct from (32) was integrated.…”

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confidence: 67%

“…However, these coefficients are the (complex) moments of the measure dρ(ζ) and admit the estimate |s m,n | J m+n , m, n ∈ N 0 . It follows from the theory of the complex moment problem (e.g., see [8] and [42]) that in this case the measure dρ(ζ) is uniquely reconstructed from (s m,n ) ∞ m,n=0 and therefore from M (z) as well. Thus, for every t ∈ [0, T ] the spectral measure dρ(ζ; t), ζ ∈ S , is reconstructed uniquely from the function (43).…”

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confidence: 99%

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Integration of some differential-difference nonlinear equations using the spectral theory of normal block Jacobi matrices

Berezansky

Mokhon’ko

2008

Funct Anal Its Appl

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The following method for integrating the Cauchy problem for a Toda lattice on the half-line is well known: to a solution u(t), t ∈ [0, ∞), of the problem, one assigns a self-adjoint semiinfinite Jacobi matrix J(t) whose spectral measure dρ(λ; t) undergoes simple evolution in time t. The solution of the Cauchy problem goes as follows. One writes out the spectral measure dρ(λ; 0) for the initial value u(0) of the solution and the corresponding Jacobi matrix J(0) and then computes the time evolution dρ(λ; t) of this measure. Using the solution of the inverse spectral problem, one reconstructs the Jacobi matrix J(t) from dρ(λ; t) and hence finds the desired solution u(t).In the present paper, this approach is generalized to the case in which the role of J(t) is played by a block Jacobi matrix generating a normal operator in the orthogonal sum of finite-dimensional spaces with spectral measure dρ(ζ; t) defined on the complex plane. Some recent results on the spectral theory of these normal operators permit one to use the integration method described above for a rather wide class of differential-difference nonlinear equations replacing the Toda lattice.To integrate the Cauchy problem with respect to time t for the nonlinear Korteweg-de Vries equation, there exists a well-known inverse spectral problem method for the self-adjoint SturmLiouville equation on the half-line. This method was put forward and developed in the classical papers by I. M. Gelfand, B. M. Levitan, V. A. Marchenko, and M. G. Krein (for its presentation, see [1]). Using some results for a finite Toda lattice (see [2], [3]), one of the authors [4]-[6]suggested a similar integration method for the Cauchy problem with respect to t in the case of a nonlinear difference equation, a half-infinite Toda lattice, on the basis of the simpler spectral theory for the difference analog of the Sturm-Liouville equation, a semi-infinite self-adjoint Jacobi matrix. We point out that in both cases use was made of self-adjoint operators.The main objective of the present paper is to show that a similar integration theory for the corresponding classes of nonlinear difference equations can be developed on the basis of the recently constructed spectral theory of normal semi-infinite Jacobi matrices (see [7], [8]). In this approach, there necessarily arise matrix (non-Abelian) nonlinear equations; this is related to the stringent property that normal tridiagonal matrices are necessarily matrices of block Jacobi structure with (as a rule) increasing finite dimensions of the blocks rather than to the desire for generalizations.We now give some remarks on the spectral theory of block self-adjoint Jacobi matrices. This theory was created under the influence of the note [9] by M. G. Krein, published in 1949. In particular, it was developed in [10]-[12] and found application in the integration of the Cauchy problem for some classes of matrix difference equations [13]- [16]. (Spectral results of some type that are related to the general block Jacobi matrices and developed in [17]...

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confidence: 67%

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Integration of some differential-difference nonlinear equations using the spectral theory of normal block Jacobi matrices

Berezansky

Mokhon’ko

2008

Funct Anal Its Appl

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“…The situation is much more complicated with the complex moment problem. By solving the complex moment problem we obtain a block three-diagonal Jacobi-type matrix corresponding to a (pre)normal operator (prenormal is a densely defined operator that has a normal extension in the same space) [7,8]. In this case, the blocks of the matrix are growing.…”

Section: Introductionmentioning

confidence: 99%

“…We remark that, actually, the trigonometric and the strong Hamburger moment problems are two particular cases. In general, we do not have any description of elements of matrices that would correspond to the usual, the strong, the half-strong two-dimensional moment problems [9,10,11], the complex moment problem in the exponential form [13,12]. These problems are still open.…”

Section: Introductionmentioning

confidence: 99%

The inner structure of the block Jacobi type matrix related to the complex moment problem with the measure supported on the second order curve

Dudkin,

Dyuzhenkova,

Kozak

2022

Methods of Functional Analysis and Topology

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We present an exact inner structure of the block Jacobi type matrix related to the complex moment problem with the corresponding measure supported on an arbitrary second order curve in the complex plane. For completeness of the study we also present a solution of the direct and inverse spectral problems for such matrices. In this the way, we give a necessary and sufficient condition under which a matrix in the CMV-form generates a (pre)normal operator, namely, not obligatory a unitary one.Надано точну внутрiшню структуру блочної якоюiєвої матрицi, яка пов'язана з комплексною проблемою моментiв i мiрою, що мiє носiй на довiльнiй кривiй другого порядку в комплекснiй площинi. Для повноти дослiдження подаємо також розв'язок прямої та оберненої спектральних задачi для таких матриць. Ми також даємо необхiдну i достатню умову зв якої CMV-матриця породжує (пре)нормальний оператор, а саме не обов'язково унiтарний. 2020 Mathematics Subject Classification. 44A60, 47A57, 47A70.

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“…Но эти коэффициенты являются (комплексными) моментами меры dρ(ζ), допускающими оценку |s m,n | J m+n , m, n ∈ N 0 . Из теории комплексной проблемы моментов (см., например, [8], [42]…”

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Интегрирование Некоторых Дифференциально-Разностных Нелинейных Уравнений С Помощью Спектральной Теории Блочных Якобиевых Нормальных Матриц

Березанский¹,

Berezanskii²,

Mokhon’ko³

et al. 2008

Функц. анализ и его прил.

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On the complex moment problem

Abstract: The article is devoted to the solution of the infinite‐dimensional variant of the complex moment problem, and to the uniqueness of the solution. The main approach is illustrated for the best explanation on the one‐dimensional case. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Cited by 8 publications

References 26 publications

Integration of some differential-difference nonlinear equations using the spectral theory of normal block Jacobi matrices

Integration of some differential-difference nonlinear equations using the spectral theory of normal block Jacobi matrices

The inner structure of the block Jacobi type matrix related to the complex moment problem with the measure supported on the second order curve

Интегрирование Некоторых Дифференциально-Разностных Нелинейных Уравнений С Помощью Спектральной Теории Блочных Якобиевых Нормальных Матриц

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