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Kovalyov I. M. A truncated indefinite Stieltjes moment problem [Електронний ресурс] / I. M. Kovalyov // Український математичний вісник. - 2016. - Т. 13, № 4. - С. 473-498. - Режим доступу: http://nbuv.gov.ua/UJRN/UMvis_2016_13_4_3
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Derkach V. A. An operator approach to indefinite Stieltjes moment problem [Електронний ресурс] / V. A. Derkach, I. M. Kovalyov // Український математичний вісник. - 2017. - Т. 14, № 1. - С. 42-85. - Режим доступу: http://nbuv.gov.ua/UJRN/UMvis_2017_14_1_5 In the present paper we solve the indefinite Stieltjes moment problem <$EMP sub kappa sup k (s)> within the M. G. Krein theory of u-resolvent matrices applied to a Pontryagin space symmetric operator <$EA sub [0,N]> generated by <$EJ sub [0,N]>. The u-resolvent matrices of the operator <$EA sub [0,N]> are calculated in terms of generalized Stieltjes polynomials using the boundary triple's technique. Criterions for the problem <$EMP sub kappa sup k (s)> to be solvable and indeterminate are found. Explicit formulae for Pade approximants for generalized Stieltjes fraction in terms of generalized Stieltjes polynomials are also presented.
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Kovalyov I. M. Shifted Darboux transformations of the generalized Jacobi matrices, I [Електронний ресурс] / I. M. Kovalyov // Український математичний вісник. - 2018. - Т. 15, № 4. - С. 490-515. - Режим доступу: http://nbuv.gov.ua/UJRN/UMvis_2018_15_4_6 Let J be a monic generalized Jacobi matrix, i.e. a threediagonal block matrix of a special form. We find conditions for a monic generalized Jacobi matrix J to admit a factorization <$EJ~=~LU~+~alpha I> with L and U being lower and upper triangular two-diagonal block matrices of the special form. In this case the shifted Darboux transformation without parameter of J defined by <$EJ sup (p) ~=~UL~+~alpha I> is shown to be also a monic generalized Jacobi matrix. Analogues of Christoffel formulas for polynomials of the first and second kind, corresponding to the Darboux transformation J<^>(p) are found.
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